Anja Niels

Stephanie Sanne

Jaouad, Etienne, Hélène, Manon Andrea Aymeric

Massil Romain Raphaël

Laure Francis

Etienne Lise

Sylvain, Omar Sébastien Baptiste

En plus d'être une aventure scientifique, le doctorat est une aventure humaine, et même une tranche de vie. Mon parcours de doctorant m'a permis de côtoyer de nombreuses belles personnes, que je souhaite saluer et remercier ici.

Je me limiterai aux personnes rencontrées dans le cadre de mon doctorat ou des activités qui y sont liées. Les proches que je connais depuis longtemps, ou que j'ai rencontrés par ailleurs, ne sont donc pas cités ici, je pense néanmoins bien à eux.

Ma gratitude va d'abord à mes encadrants, Guillaume Lecué et Gilles Stoltz. Vous avez su réaliser un encadrement complet : à la fois de grande qualité scientifique, adapté à ma situation, m'aidant à construire un projet professionnel, et enfin, très humain. Vous avez été des rayons de soleil dans des périodes parfois difficiles.

Guillaume, tu as toujours été ouvert à mes idées et encourageant pour les mettre en oeuvre. Ton optimisme et ton enthousiasme m'ont motivé tout au long de ce doctorat.

Gilles, ton pragmatisme rassurant et tes conseils précieux m'ont aiguillé efficacement et m'ont amené à découvrir d'autres manières de réfléchir. Merci également pour ta relecture très minutieuse de mes documents.

Pour avoir aidé nombre de doctorants ne bénéficiant pas d'un encadrement de cette qualité, je mesure ma chance d'avoir eu deux encadrants tels que vous. Merci encore. J'ai également de la gratitude envers mes "grands frères de thèse" Sébastien, Pierre, Paul et Emilien : vos conseils et vos encouragements m'ont bien aidé ! J'inclus également dans ce salut amical mes "petits frères et soeur de thèse", notamment Pierre, Malo, Margaux : bonne fin de doctorat ! Je remercie sincèrement les membres de mon jury. Vincent, Olivier, merci d'avoir pris le temps

La fin de la rédaction de ma thèse s'étant effectuée en parallèle d'un travail de data scientist, j'en profite pour saluer mes collègues de l'équipe INS/DAT : Jean, Clément, Eric, Khémon, Pierre, Charlotte, Laurent, Guillaume, Lucile, Chloé, Jaouad, Lorraine, Dimitri. . . ainsi bien sûr que le chef de l'équipe, Louis.

Je souhaiterais rendre hommage à quelques-uns de mes enseignants en mathématiques, qui ont marqué mon parcours chacun à leur manière : Mme Orgogozo (qui sait mettre en évidence le plaisir des mathématiques), Mme Lasserre (d'une rigueur exemplaire), Mme Gouteyron (dont certaines discussions m'ont amené définitivement à choisir les mathématiques), Mme Picard et M. Alquier (qui m'ont initié aux statistiques et à l'apprentissage).

Enfin, je conclurai par une pensée très tendre et affectueuse pour mes parents. Cette thèse vous est dédiée.

Have a nice reading! Veel leesplezier! Bonne lecture !

Table des Matières

Introduction générale

Présentation générale de la thèse

Cette thèse se situe dans le domaine des statistiques mathématiques, avec une focalisation particulière sur l'apprentissage. Son fil conducteur est la régression linéaire, dont elle cherche à mettre en valeurs la multiplicité des approches et des variations selon le problème statistique à traiter -mais aussi les logiques communes. Elle s'intéresse en particulier aux bénéfices de relier les cadres (notamment "batch stochastique" et "séquentiel déterministe") entre eux et d'en transférer des idées.

Deux types de méthodes jouent un rôle central dans les travaux présentés : les approches dites "par régularisation", et celles dites "par agrégation".

Cette introduction générale (chapitre 1) présente les différents cadres dans lesquels se situent les travaux de recherche et leurs enjeux. Elle indique également les apports de cette thèse, et met en évidence des pistes de recherche possibles pour le futur.

L'introduction mathématique (chapitre 2) présente les outils mathématiques utilisés dans les différents chapitres.

Le chapitre 3 présente une construction d'une régularisation optimale dans un cadre dit "batch stochastique", améliorant notamment, selon certains critères qui seront présentés, l'estimateur LASSO, très utilisé à l'heure actuelle. Le chapitre 4 s'intéresse à un cadre plus général, en un certain sens, que la régression linéaire : l'optimisation convexe séquentielle. Nous présentons tout d'abord des améliorations pour un algorithme récent et prometteur, MetaGrad, et ensuite proposons une approche pour convertir cet algorithme d'un des cadres de cette thèse ("séquentiel", utilisé aux chapitres 5 et 6) vers l'autre cadre ("batch", utilisé au chapitre 3).

Le chapitre 5 aborde un problème a priori classique, la création d'intervalles de prévision, mais dans un cadre original : les suites individuelles. Il propose une méthodologie nouvelle, permettant d'adapter les algorithmes existants. Cette méthodologie requiert une optimisation, dont on présente des méthodes pour l'effectuer sur trois algorithmes (régression Ridge, algorithme EWA et son extension Fixed-Share EWA).

Le chapitre 6 met en oeuvre des algorithmes d'agrégation en suites individuelles, et de prévision par intervalles (selon la méthodologie développée au chapitre 5), sur un jeu de données de production pétrolière.

Les différents chapitres ont été rédigés de façon à pouvoir être lus indépendamment.

Introduction générale

Protocole 1 Processus de prévision "batch" Phase d'apprentissage : Le statisticien dispose d'un échantillon d'apprentissage (x 1 , y 1 ), . . . , (x N , y N ) Phase de prévision :

1. L'environnement révèle le vecteur x des variables explicatives 2. Le statisticien propose une valeur y en utilisant l'échantillon d'apprentissage (x 1 , y 1 ), . . . , (x N , y N )

3. L'environnement révèle la vraie valeur y 4. Le statisticien subit la perte ( y, y) d'un échantillon d'apprentissage avant la première prévision -mais pas toujours. L'objectif est bien entendu d'essayer de prévoir correctement à chaque tour, mais la performance sera surtout évaluée sur l'erreur cumulée, plutôt que sur une prévision précise. On s'intéresse ainsi à la perte cumulée :

L T ( y 1 , . . . , y T ) := T t=1 t ( y t ) où les t sont des fonctions de perte, correspondant souvent à l'erreur : t ( y t ) = ( y t , y t ), parfois à un coût. De nombreuses situations réelles peuvent être vues dans ce cadre : prévisions heure par heure (consommation nationale d'électricité : [START_REF] Devaine | Forecasting the electricity consumption by aggregation of specialized experts; application to Slovakian and French country-wide (half-)hourly predictions[END_REF]), quotidiennes (prévision de propagation de maladie : [START_REF] Chan | Daily forecast of dengue fever incidents for urban villages in a city[END_REF]), mensuelles, trimestrielles (indicateurs économiques tels que la croissance : [START_REF] Bessec | Étalonnages du taux de croissance du PIB français sur la base des enquêtes de conjoncture[END_REF] s'intéresse à des données mensuelles et trimestrielles), annuelles (prévision des coûts liés à une infrastructure : [START_REF] Vernet | A model to forecast data centre infrastructure costs[END_REF]). . . Dans le cadre de cette thèse, on s'intéresse dans le chapitre 6 au problème de la prévision mensuelle de plusieurs grandeurs liées à la production de pétrole dans un champ pétrolifère, sur une période de dix ans.

Influence de la modélisation, ou non, des données Modélisation stochastique. La connaissance que l'on possède sur les phénomènes étudiés et la façon dont les données (notamment les variables à prévoir) sont générées permet parfois de proposer une modélisation stochastique, ce qui permet d'obtenir des résultats assez précis "avec grande probabilité" ou en espérance (en moyenne). Ainsi, au chapitre 3, on dispose d'observations Y i et de vecteurs explicatifs X i dont on sait qu'il existe un vecteur t * tel que :

Y i = t * , X i + ξ i
où ξ i ∼ σN (0, 1) avec σ connu, mais où t * est inconnu et cherché. On présente alors une méthodologie "sur-mesure" et optimale (en un sens qui sera défini) permettant de tirer parti de cette "bonne" connaissance des données.

1.2. Cadre mathématique de la thèse Protocole 2 Processus de prévision "séquentiel" (ou "online") Phase d'apprentissage facultative : Le statisticien dispose parfois d'un échantillon d'apprentissage. Phase de prévision : L'erreur cumulée initiale est nulle : L 0 = 0. for t = 1, 2, . . . 1. L'environnement révèle le vecteur des variables explicatives x t 2. Le statisticien propose une valeur y t en utilisant les valeurs observées précédemment y 1 , . . . , y t-1 et les variables explicatives présentes et passées x 1 , . . . , x t (ainsi que l'échantillon d'apprentissage s'il en dispose)

3. L'environnement révèle la vraie valeur y t 4. Le statisticien subit la perte ( y t , y t ), sa perte cumulée est incrémentée :

L t = L t-1 + ( y t , y t )
Suites individuelles. En revanche, dans certaines situations, on ne fait pas de modélisation stochastique sur les données, que l'on considère alors comme des suites déterministes. Cela peut venir d'une connaissance insuffisante des phénomènes étudiés, d'une volonté de "limiter les risques", ou permettre de gérer des situations antagonistes, où la valeur de la variable observée dépend (de manière défavorable) de la prévision qui en a été faite. Les hypothèses peuvent par exemple se limiter à des bornes sur les pertes, de type :

∀ t = 1..T, ∀ y ∈ R, t (y) ∈ [m, M ].
On parle pour ce type de modèles de suites individuelles (ou suites arbitraires), c'est le cadre des chapitres 5, 6 et d'une partie du chapitre 4 (ce dernier traitant d'un cadre un peu plus général, l'optimisation convexe séquentielle). Dans ce domaine, la monographie [START_REF] Cesa | Prediction, Learning, and Games[END_REF] constitue une référence importante.

Adaptation pratique de l'approche "suites individuelles". Il faut noter que cette approche, "pire cas" dans l'état d'esprit, est par définition très conservative. Dans sa mise en oeuvre pratique sur des jeux de données réels, quelques adaptations plus souples et "optimistes", permettant de mieux s'adapter au degré de stochasticité des données, aboutissent généralement à un gain de performance. En effet, hors situations antagonistes évoquées plus haut, dans le monde physique "le pire n'est jamais certain" et n'est pas le résultat le plus fréquent. . . Typiquement, on choisira parfois les paramètres des algorithmes en fonction des données observées plutôt qu'en fonction des bornes théoriques, par exemple en choisissant pour une prévision à un instant t, une calibration de paramètre qui aurait donné de bons résultats jusqu'à l'instant précédent t -1.

Introduction générale

Comparaison des cadres "batch stochastique" et "suites individuelles" Dans les deux cas on a tendance à chercher à se rapprocher des vecteurs "performants", c'est pourquoi on retrouve nombre d'algorithmes similaires. Le lien est même plus profond, avec des résultats théoriques transposables d'un cadre à l'autre (voir les sections de cette thèse évoquant la conversion "online-to-batch"). Mais le raisonnement théorique sous-jacent diffère quelque peu. Dans le cadre batch stochastique, la performance sur la période d'apprentissage donne des indications sur la loi des performances. Dans le cadre des suites individuelles, "les performances passées ne présagent pas des performances futures", mais faire des prévisions proches des experts performants jusqu'alors permet d'éviter des différences fortes avec eux et donc d'empêcher le regret (voir Section 1.2.2) d'augmenter fortement.

Batch Séquentiel

Chapitre 3 Modélisation stochastique

Section "Online to Section "Online to Batch" 4.3 Batch" 4.3 des données du chapitre 4 du chapitre 4

Chapitre 5 Chapitre 4

Modélisation déterministe

Section "Intervalles de prévision" Chapitre 5 des données 6.5 du chapitre 6 Chapitre 6

Figure 1.1: Différents cadres pour les processus de prévision et la modélisation des données

Influence des variables explicatives

Les connaissances sur les variables explicatives jouent un rôle déterminant : les a priori que l'on a sur leurs capacités prédictives, notamment la confiance que l'on porte (ou non) à leurs performances individuelles, peuvent amener à limiter la recherche des coefficients de la régression linéaire à un sous-domaine de l'espace R d (dans le cas de d variables explicatives réelles).

Le chapitre 3 s'intéresse à une recherche des coefficients sur l'espace R d tout entier. Cette démarche est naturelle lorsqu'on ignore tout des capacités prédictives des variables explicatives. Toutefois, on verra que la régularisation 1 étudiée dans ce chapitre vise précisément à restreindre implicitement la zone de recherche (à une "boule 1 ").

Le chapitre 4 s'intéresse à des ensembles convexes, bornés mais potentiellement assez grands.

Enfin, les chapitres 5 et 6 présentent le cas de l'agrégation d'experts, où l'on suppose que certaines variables explicatives ont individuellement de bonnes capacités prédictives (d'où le terme "experts"), et où par conséquent des performances a minima correctes sont attendues 1.2. Cadre mathématique de la thèse sur le simplexe des combinaisons convexes, ce qui n'empêchera pas d'utiliser également des coefficients plus généraux.

La figure 1.2 illustre ces différences dans le domaine de recherche des coefficients. 1.2.2. Un point commun : la mesure des performances est souvent relative "Faire au mieux avec les données disponibles. . . "

Les estimateurs de régression linéaire s'appuient par définition sur des prédicteurs (experts, variables explicatives, . . . ), leurs performances dépendent donc des qualités de prévision de l'ensemble de ces prédicteurs. C'est pourquoi les mesures de performance (et notamment les garanties théoriques) s'effectuent souvent comparativement à un point de référence, un étalon, lié aux prédicteurs. Ce dernier est souvent un oracle, c'est-à-dire qu'il est lié à des quantités inconnues à l'avance.

1. Introduction générale ". . . pour ne pas avoir (trop ?) de regret" La différence entre la perte du statisticien et celle du point de référence est appelée regret (ou, si on parle de son espérance,"excès de risque") : R = L algorithme -L référence ;

Il est souvent difficile en théorie de garantir un regret négatif, en particulier dans toutes les situations où le point de référence peut être optimal, et on cherche plutôt dans les résultats théoriques à borner le regret par une quantité positive, la plus faible possible. Le théorème suivant (prouvé dans l'introduction mathématique : voir Corollary 2.3) est un exemple classique pour l'algorithme Exponentially Weighted Average ("EWA"). Cet algorithme convexe repose sur un paramètre d'apprentissage η et ses coefficients p k,t vérifient, dans le cas où l'on dispose de K prédicteurs à agréger : p k,1 = 1/K et pour tout instant t 2 :

p k,t =
exp(-ηL k,t-1 ) K i=1 exp(-ηL i,t-1 ) •

Theorem 1.1. On note f k,t la prévision du k-ième prédicteur à l'instant t, et on suppose que les pertes des K prédicteurs pour chacun des T instants de prévision sont toutes comprises dans l'intervalle [m, M ]. Alors l'algorithme "EWA", effectué avec un paramètre d'apprentissage constant η = 8 log(K)/ ((Mm) 2 T ) garantit un regret : (f k,t , y t ) (Mm) T log(K) 2 .

En revanche, dans les applications pratiques, il arrivera régulièrement qu'on obtienne des regrets négatifs. Ainsi, dans le chapitre 6, lorsqu'on parvient à faire mieux que le meilleur expert, cela signifie qu'on a mieux prédit (en terme d'erreur cumulée) que toutes les simulations fournies par l'IFPEN, autrement dit on a bonifié (en terme de prévisions) les données fournies.

1.3. Chapitre 3 : Obtention d'une régularisation optimale dans un cadre batch stochastique 1.3.1. Cadre mathématique

Présentation générale

Le chapitre 3 de cette thèse s'intéresse à la régularisation, dans un cadre batch stochastique. Dans ce cadre, on cherche à relier les deux variables aléatoires X et Y d'un couple (X, Y ) par une fonction g appartenant à un ensemble F , telle que g(X) soit proche de Y (au sens du risque quadratique). On cherche donc à minimiser sur un ensemble F de fonctions le risque quadratique : g -→ E (Yg(X)) 2 (1.3.1)

1.3. Chapitre 3 : Obtention d'une régularisation optimale dans un cadre batch stochastique et on s'intéresse au minimiseur g * correspondant (dont l'existence sera garantie par les hypothèses utilisées). Dans le cas de la régression linéaire, sur lequel nous nous centrons, g(X) = t, X pour t ∈ T ⊂ R d . Dans le chapitre 3, on ne disposera pas d'information a priori sur T , le vecteur cherché pourra être n'importe quel élément de R d .

Pour atteindre l'objectif souhaité, on dispose d'un échantillon d'apprentissage de N couples i.i.d. (indépendants et identiquement distribués) : (X 1 , Y 1 ), . . . , (X N , Y N ), de même loi que (X, Y ). Dans ce chapitre, on recherchera des résultats valables avec grande probabilité par rapport à cet échantillon d'apprentissage.

Design aléatoire

Quand les X i sont aléatoires, on parle alors de "design aléatoire". Le risque quadratique (1.3.1) d'un algorithme est conditionné par l'aléa de l'échantillon d'apprentissage (à la fois les X i et les Y i ). Dans le chapitre 3, on s'intéressera à un design gaussien pour les X i . Ce design est classique et possède la propriété géométrique d'isotropie : si X est un vecteur gaussien standard de R d , pour tout t ∈ R d , E t, X 2 = t 2 2 .

Design fixe

On s'intéressera également dans le chapitre 3 (Section 3.4) au cas du design fixe. Dans ce cas les X i sont fixés (seuls les Y i sont aléatoires), et le risque quadratique est alors défini par rapport au design (i.e., conditionnellement aux X i ) et devient :

E 1 N N i=1 Y i -t, X i 2 X 1 , . . . , X N .
1. Introduction générale

Régularisation

Minimiseur du risque empirique

Le minimiseur du risque empirique (ici, l'"estimateur des moindres carrés"), ou "ERM" (Empirical Risk Minimizer), est un candidat naturel :

t ERM ∈ argmin t∈T N i=1 Y i -t, X i 2 •
Cet estimateur a été largement étudié (voir par exemple [START_REF] Stein | Inadmissibility of the usual estimator for the mean of a multivariate normal distribution[END_REF] pour le fameux paradoxe de Stein).

Sous certaines hypothèses (T convexe, cadre sous-gaussien), [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] montrent que l'ERM est optimal (minimax) en déviation sur T . Cela ne veut pas dire qu'il est optimal sur tous les sous-ensembles de T , en particulier ceux de taille beaucoup plus réduites. En effet, lorsque T est très grand, l'ERM a tendance à faire du sur-apprentissage : il "colle" trop à l'échantillon d'apprentissage, se laissant influencer par le bruit et les éventuels données atypiques, et s'éloignant ainsi de la cible théorique

t * ∈ argmin t∈T E Y -t, X 2 . (1.3.2)
Dans ce qui suit, on fera la supposition (valable presque sûrement, sous certaines hypothèses) que cet argmin est réduit à un singleton t * . Si l'on a une idée assez précise de la zone où se situe t * , une solution consiste bien sûr à restreindre l'ensemble de recherche T à cette zone. Mais dans le cadre du chapitre 3, on ne suppose pas avoir cette connaissance a priori et on prend T = R d tout entier.

Régularisation

Une idée très utilisée pour pallier ce problème correspond d'une certaine manière à "régulariser", c'est-à-dire à ajouter au risque empirique d'un vecteur t un "coût" supplémentaire, le terme de régularisation Ψ(t). L'estimateur devient alors un minimiseur du risque empirique régularisé :

t Ψ ∈ argmin t∈T N i=1 Y i -t, X i 2 + Ψ(t) . (1.3.3)
On introduit ainsi un biais favorisant les zones où Ψ est faible (souvent, les zones proches de l'origine). Une fois défini par (1.3.3) t Ψ vérifie la propriété suivante (qui ne peut être une définition vue sa récursivité) :

Lemma 1.2.

t Ψ ∈ argmin t∈T : Ψ(t) Ψ( t Ψ ) N i=1 Y i -t, X i 2 .
(1.3.4)

1.3. Chapitre 3 : Obtention d'une régularisation optimale dans un cadre batch stochastique En effet, s'il existait un élément ayant à la fois un risque empirique strictement inférieur à celui de t Ψ et une régularisation inférieure ou égale à Ψ( t Ψ ), cet élément aurait alors un risque empirique régularisé strictement inférieur à celui de t Ψ , en contradiction avec la définition (1.3.3).

On peut interpréter cette expression (1.3.4) comme le fait qu'un minimiseur du risque empirique régularisé est aussi, implicitement, un minimiseur du risque empirique sur un ensemble (inconnu à l'avance) plus restreint : {t : Ψ(t) Ψ( t RERM )}. On verra que tout l'enjeu d'un bon choix de régularisation est précisément que cet ensemble restreint ait une taille adéquate, proche de celle de l'ensemble lié à l'estimateur "oracle" {t : Ψ(t) Ψ(t * )}.

Apports de la thèse

Les travaux présentés au chapitre 3 définissent d'abord un critère d'optimalité pour les fonctions de régularisation (voir Section 1.3.4).

L'apport principal du chapitre est une "preuve de concept", appliquée à la régularisation 1 et dans le cadre défini ci-dessus, d'une approche permettant de construire une fonction de régularisation optimale (voir Section 1.3.5 pour les grandes lignes de cette construction).

Définition d'un critère d'optimalité pour la fonction de régularisation

Norme et fonction de régularisation

Le choix du terme de régularisation Ψ(t) est très important. Il est généralement fonction (croissante) d'une norme du vecteur t : Ψ(t) = g( t ). Le choix de cette norme peut venir d'un a priori sur la cible : si l'on sait que l'argmin (1.3.2) est réduit à un singleton t * tel que t * est faible pour une certaine norme • , on utilisera une régularisation fondée sur cette norme . Le choix peut aussi venir d'une volonté du statisticien d'utiliser les propriétés liées à une certaine norme. Ainsi, [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF] introduit la régularisation LASSO, qui utilise le terme de régularisation : Ψ(t) = λ t 1 (avec un paramètre λ fixé par le statisticien) et permet souvent, grâce aux propriétés géométriques de la norme . 1 , d'obtenir un vecteur parcimonieux ("sparse"). Cela peut se révéler utile, par exemple dans un contexte de haute dimensionnalité ou de données massives ("Big Data"). L'estimateur LASSO est donc :

t LASSO ∈ argmin t∈R d N i=1 Y i -t, X i 2 + λ t 1 .
(on peut aussi, par convention, comme dans le chapitre 3, diviser la quantité à minimiser par N et adapter λ, ce qui bien sûr est sans conséquence). Une autre régularisation classique, la régression Ridge (introduite dans [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF], utilisée également par [START_REF] Tikhonov | Methods for solving ill-posed problems[END_REF]), utilise, elle, le carré de la norme 2 : Ψ(t) = λ t 2 2 . Elle peut parfois surpasser le LASSO en prévision, et peut être utile quand on dispose de variables fortement corrélées et qu'on veut éviter qu'il n'y ait qu'une seule d'entre elles qui soit sélectionnée. Ces éléments poussent [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF] à proposer de sommer les régularisations 1 et 2 , aboutissant à un estimateur nommé "Elastic Net". Sa régularisation est λ 1 . 1 + λ 2 . 2 2 avec λ 1 > 0 et λ 2 > 0 éventuellement différents.

Introduction générale

Une fois la norme choisie, il reste à décider d'une fonction de régularisation, qui transformera cette norme en un terme de régularisation. On peut dès le départ se poser la question suivante : quelles sont les caractéristiques d'une "bonne" fonction de régularisation ? Le premier apport du chapitre 3 est donc de proposer un critère d'optimalité pour une fonction de régularisation, pour une norme donnée.

Un critère d'optimalité

Comme on l'a vu en (1.3.4), la régularisation aboutit en fait à la combinaison d'une restriction de la région de l'espace R d où l'estimateur est cherché, et d'une minimisation du risque empirique sur cette zone restreinte. Lorsque Ψ est fonction croissante d'une norme, les zones restreintes possibles sont les boules (pour la norme considérée) centrées en 0, et de rayons différents. Le modèle optimal, que l'on notera B * , est la boule centrée en 0 et de rayon t * -qui est inconnu. L'ERM restreint à cette boule B * (qui est donc un modèle oracle) est lui-même minimax sur cette boule B * (cf. [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF]). On définit alors dans le chapitre 3 un critère d'optimalité pour une fonction de régularisation. Il s'agit pour l'estimateur régularisé t Ψ de parvenir à obtenir un risque inférieur, à une constante multiplicative absolue près, au risque de cet ERM oracle sur la boule B * (noté t B * ERM ) :

∃ C : ∀ t * , E Y -t Ψ , X 2 C E Y -t B * ERM , X 2
(1.3.5) Tout l'enjeu, pour arriver à remplir ce critère, est donc de définir une fonction de régularisation capable de "sélectionner" une région de l'espace proche du modèle optimal.

Précisons que ce critère est un critère théorique, et qu'une fonction de régularisation le remplissant peut éventuellement être difficile à calculer (non-convexité, etc.)

Construction d'une fonction de régularisation optimale

Cette section décrit les grandes lignes du raisonnement qui fonde le chapitre 3.

Idée directrice

Le chapitre 3 se veut une "preuve de concept" du principe suivant : "pour obtenir une régularisation optimale pour un problème donné, il faut créer une régularisation "sur-mesure" pour le problème en question, en déterminant "les zones où elle est nécessaire, et sa valeur nécessaire sur ces zones". C'est cette idée de régulariser "suffisamment mais pas plus que nécessaire pour le problème étudié", qui guide notre approche.

Dans ce chapitre, nous appliquons ce principe à un cadre classique gaussien, et à la norme de régularisation 1 (qui est, comme indiqué plus haut, très utilisée pour ses propriétés de parcimonie). Plusieurs données du problème (par exemple la densité -ou même la log-densitégaussienne) ne sont en rien linéaire vis-à-vis des normes des vecteurs de l'espace, pourquoi donc le choix habituel "LASSO" d'une régularisation linéaire en la norme (de type λ . 1 ) serait-il optimal ? (Effectivement, il ne l'est pas.) 1.3. Chapitre 3 : Obtention d'une régularisation optimale dans un cadre batch stochastique On cherche à obtenir une régularisation permettant de sélectionner, avec grande probabilité, un modèle de taille proche de celle du modèle optimal B * . Cette régularisation doit donc être :

• suffisante pour exclure les modèles trop grands,

• mais suffisamment faible pour ne pas trop biaiser l'estimateur vers l'origine et continuer à inclure les modèles de taille proche de celle de B * .

Probabilités et géométrie, localisation

L'essentiel de notre approche consistera à contrôler "au plus juste" (à des constantes absolues multiplicatives près), sur les modèles les plus grands, les processus aléatoires, grâce à la régularisation. Plus précisément, on va garantir (avec grande probabilité) que tout vecteur t vérifiant t 1 ω t * 1 (pour une certaine constante absolue ω > 1) a un risque régularisé strictement plus grand que celui de t * , et ne peut donc être t.

Pour cela, on va introduire dans nos raisonnements, pour tout vecteur t, la différence entre le risque empirique régularisé de t et celui de t * . Cette quantité, qu'on notera P N L Ψ t , est un intermédiaire de calcul inconnu, car dépendant de t * . Elle vérifie par construction P N L Ψ t P N L Ψ t * = 0, et on veut montrer que, sur les grands modèles, tous les vecteurs de trop grande norme (que l'on veut donc exclure) vérifient P N L Ψ t > 0 et ainsi ne peuvent pas être t. On introduit la décomposition suivante : Le rayon r de la boule 2 joue un rôle crucial ; on verra qu'on le déterminera à partir de deux équations de points fixes, faisant intervenir la "fenêtre gaussienne" des ensembles localisés :

P N L Ψ t = P N L t + R t,
1. Introduction générale où G est un vecteur gaussien standard de R d .

Le théorème principal du chapitre 3 (théorème 3.1.4) introduit une fonction de régularisation judicieusement choisie (dépendant notamment des deux équations de points fixes évoqués cidessus), et montre qu'elle est optimale pour le critère (1.3.5). Les quantités en jeu dans (1.3.5) dépendent elles aussi des deux équations de points fixes.

t * (A) (A) (A) (A) (B) ρB d 1 + t * rB d 2 + t *

Une idée importante pour le design fixe

Dans le cas du design fixe, la matrice des variables explicatives X, de taille N × d et dont les lignes successives sont X 1 , . . . , X N , est déterministe.

L'idée-clé permettant l'adaptation de la preuve dans le cas de ce design fixe est le transfert de la localisation : on passe d'une localisation dans l'espace de départ R d (dans la partie "design aléatoire") à une localisation dans l'espace d'arrivée R N (dans la partie "design fixe"). Ce choix est finalement assez naturel, car en termes de prévisions ce ne sont pas les vecteurs t qui interviennent, mais uniquement leur image Xt par la matrice déterministe X. On intersecte ainsi, non pas le modèle ρB d 1 mais son image ρXB d 1 ⊂ R N , par une boule 2 de R N , en l'occurrence rB N 2 .

1.4. Chapitre 4 : Amélioration d'un algorithme adaptatif : MetaGrad

Pistes de recherche futures

Ce chapitre se veut une "preuve de concept" d'une approche pour construire des régularisations optimales, aussi une première piste serait-elle de l'appliquer sous d'autres hypothèses. En particulier, on pourrait changer la géométrie du problème :

• en remplaçant les hypothèses permettant une certaine isométrie 2 (design gaussien, hypothèse d'isométrie restreinte) ;

• en utilisant une autre norme que la norme 1 .

S'agissant de l'estimateur présenté dans ce chapitre, une étude de ses propriétés de parcimonie (par exemple comparativement au LASSO) serait judicieuse.

Enfin, sa mise en pratique (non triviale, car nécessitant une optimisation non convexe) sur des jeux de données réels pourrait être riche d'enseignements. le choix de la distance d dépendant de l'algorithme. Cette projection peut s'avérer coûteuse en terme de temps de calcul. L'utilisation de domaines "faiblement bornés" dans le chapitre est ainsi motivée par l'existence d'une formule explicite permettant une projection rapide. On peut aussi appliquer des techniques inspirées de l'optimisation : descente de gradient, méthode de Newton modifiée, etc., en les adaptant au problème.

Importance de la perte

Les méthodes et performances optimales dépendent des hypothèses sur les pertes t , notamment leur courbure. La convexité garantit une certaine sécurité dans le comportement des algorithmes, mais sous des hypothèses de courbure plus fortes, on peut obtenir des performances meilleures. L'hypothèse de stricte convexité est peu utilisée sur le plan théorique (elle permet de garantir l'unicité du minimum, mais apporte peu sur la convergence vers ce minimum, vu que son "écart" avec la convexité simple n'est pas quantifié). En revanche, deux hypothèses plus fortes permettent d'obtenir de meilleurs résultats. Tout d'abord, l'(α-)expconcavité : la perte t est dite α-exp-concave (pour α > 0) si x → exp (-α t (x)) est concave. Cette hypothèse garantit en particulier que t est convexe. Une hypothèse encore plus forte est la (λ-)convexité forte : pour λ > 0, la perte t est dite λ-fortement convexe si :

∀x, y, t (y) t (x) + ∇ t (x), y -x + λ 2 y -x 2 2 .
Enjeux autour de la perte et de la calibration des paramètres Des méthodes différentes ont été étudiées pour obtenir des performances optimales selon chacune des hypothèses ci-dessus, et choisir judicieusement l'une ou l'autre d'entre elles demande donc de déterminer la courbure de la perte. Step" (introduite dans [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]). Chaque algorithme auxiliaire utilise un paramètre d'apprentissage différent (d'où le nom MetaGrad, pour "Multiple Eta Gradient"), et le méta-algorithme maître parvient à obtenir des garanties proches de celle du meilleur algorithme auxiliaire. L'algorithme MetaGrad parvient à supprimer le problème de la calibration du paramètre d'apprentissage (il contient d'autres paramètres : hyper-paramètres de grille et paramètres d'initialisation, mais ceux-ci sont beaucoup plus faciles à calibrer). Il parvient également dans le cadre de l'optimisation convexe séquentielle "déterministe", à obtenir, en ce qui concerne la dépendance en la dimension d et le nombre T des prévisions, des bornes quasi-optimales sous hypothèses de convexité simple, et optimales (à constante multiplicative près) sous hypothèses d'exp-concavité.

Accélérations de MetaGrad

MetaGrad possède néanmoins un point faible : il est (relativement) lent, a fortiori en grandes dimensions. L'un des axes de recherche consiste à proposer des versions modifiées, plus rapides mais aux performances et garanties proches. Un premier exemple était présenté dans l'article originel (une version diagonale de l'algorithme), dans cette thèse nous proposons plusieurs autres versions modifiées, inspirées notamment du "sketching" de [START_REF] Luo | Efficient second order online learning by sketching[END_REF], que l'on présente ci-dessous, et nous étudions leurs garanties théoriques.

L'algorithme MetaGrad nécessite à chaque instant t le calcul d'une matrice de covariance de taille d×d : (εI d +G t G t ) -1 , où G t est une matrice de taille t×d. La formule de Woodbury permet d'écrire :

(εI d + G t G t ) -1 = 1 ε I d -G t (εI t + G t G t ) -1 G t .
Le sketching consiste à remplacer G t G t par une approximation de rang moindre, pour accélérer les calculs. Plus précisément, on va chercher à introduire une matrice S t de taille m × d vérifiant les conditions suivantes :

• m < t et, autant que possible, m d,

• S t S t est une "bonne" approximation de G t G t (pour des critères qui seront présentés dans le chapitre 4)

1. Introduction générale

(εI d + S t S t ) -1 = 1 ε I d -S t (εI t + S t S t ) -1 S t
ce qui accélère fortement les calculs correspondants. Une piste pour le futur consisterait à chercher d'autres "accélérations" possibles pour MetaGrad, et à les tester sur des jeux de données en s'intéressant à la fois à leurs améliorations en termes de temps de calculs, et à l'éventuelle dégradation (ou, qui sait, amélioration) de leurs performances pratiques par rapport au MetaGrad initial.

Conversion online-to-batch "améliorée" de MetaGrad Un autre axe de travail présenté consiste à adapter l'algorithme MetaGrad, conçu pour un cadre séquentiel déterministe, au cadre batch stochastique. La méthode de conversion "online-to-batch" classique, appliquée à MetaGrad, permet d'obtenir sous des hypothèses d'exp-concavité et de convexité forte une erreur dont l'espérance est de l'ordre de d log(T )/T , alors que les ordres de grandeur optimaux sont respectivement pour ces deux hypothèses de d/T et 1/T . En nous inspirant de Hazan and Kale [2014], nous proposons une conversion "online-to-batch" plus performante, garantissant une erreur de l'ordre de d log log(T )/T -mais pour l'instant sous l'hypothèse de convexité forte.

Cette conversion utilise une division de l'échantillon d'apprentissage en périodes ("epochs"), dont la taille suit une croissance géométrique : ainsi, la taille T j de la période j vérifie T j = 2 j T 0 . Sur chaque période, l'algorithme MetaGrad est appliqué, avec à chaque fois une mise à jour des paramètres d'initialisation (plus pertinente dans le cas de MetaGrad que la mise à jour du paramètre d'apprentissage utilisée par [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF]). Le point de départ x j+1 1 de la période j + 1 est la moyenne des vecteurs produits par l'algorithme sur la période précédente :

x j+1 1 = 1 T j T j t=1
x j t où x j t désigne le vecteur produit par l'algorithme au t-ième instant de la période j. L'analyse de l'algorithme MetaGrad permet d'obtenir une borne sur le regret dépendant uniquement de paramètres déterministes (paramètres du problème, paramètres de l'algorithme sur cette période) et de la distance x j 1x * 2 entre le point de départ et la cible :

E F (x j+1 1 ) -F x * g 1 x j 1 -x * 2
, paramètres déterministes pour une certaine fonction g 1 , avec F la fonction qu'on cherche à minimiser (espérance des pertes), et x * son minimum. Une hypothèse de convexité forte permet ensuite de relier x j 1x * 2 au regret correspondant E F (x j 1 ) -F x * , et ainsi obtenir une récurrence :

E F (x j+1 1 ) -F x * g 2 E F (x j 1 ) -F x * 1.
F : x ∈ K -→ E[f 1 (x)],
dont on suppose qu'elle admet un minimum :

x * = argmin x∈K F (x).
Alors le vecteur x T résultant de l'application de l'algorithme "Epoch MetaGrad" vérifie, en ce qui concerne la dépendance en d et T :

E [F ( x T ) -F (x * )] = O d log log(T ) T •
Si on connaît le paramètre de convexité forte λ, on peut utiliser l'algorithme "Epoch Online Newton Step" présenté au chapitre 4 qui garantira une vitesse en O(d/T ) : voir Théorème 4.9. On pourra également, comme signalé en fin de chapitre 4, utiliser une version modifiée d'"Epoch MetaGrad" qui utilise explicitement λ mais garantit une vitesse en O(d/T ).

Une piste de recherche pour le futur serait de chercher à adapter l'analyse dans le but de montrer que cette borne en d log log(T )/T , voire même en d/T , reste valable sous l'hypothèse plus faible d'exp-concavité. On pourra aussi chercher à supprimer le terme en d dans le cas de la convexité forte, mais cet objectif paraît plus difficile.

Pistes de recherches futures

Il serait judicieux de tester l'algorithme MetaGrad et ses différentes déclinaisons, dont celles présentées dans cette thèse, sur différents jeux de données réelles, à la fois pour juger des performances, mais aussi pour comparer les temps de calcul.

Comme indiqué, la section sur la conversion "online-to-batch" de MetaGrad se veut une première étape vers la construction, dans un cadre batch stochastique, d'un algorithme optimal sous l'hypothèse d'exp-concavité. Il serait ainsi intéressant de tenter de mener à son terme la résolution de ce problème. Le chapitre 5 trouve son origine dans une demande opérationnelle : lorsqu'on dispose d'observations jusqu'à l'instant présent T 0 , mais aussi de prévisions d'experts jusqu'à un futur assez lointain T , on souhaite pouvoir fournir des prévisions à long terme. L'incertitude plus grande due à l'éloignement des échéances, amène à proposer plutôt des régions de confiance, des faisceaux de prévision (dans notre cas, ces faisceaux seront des ensembles d'intervalles). Même si l'on va effectuer plusieurs prévisions, il s'agit d'une situation "batch" car on les effectue en une seule fois, sans retour d'expérience intermédiaire.

Formellement, on a donc accès :

• aux observations y t , t T 0 ;

• aux prévisions des experts f j,t à la fois pour t T 0 et pour T 0 < t T et on cherche à prévoir des intervalles de prévision [ y min t , y max t ] pour T 0 < t T . Cet objectif est relativement classique dans un cadre habituel de séries temporelles, où l'on peut obtenir des intervalles de confiance valables avec une probabilité donnée, par différentes méthodes (calcul direct, simulations de type Monte-Carlo,. . . ) : citons [START_REF]Estimating long-range dependence: finite sample properties and confidence intervals[END_REF] et [START_REF] Wan | Optimal prediction intervals of wind power generation[END_REF] (avec une utilisation de réseaux de neurones).

. . . abordée sous un angle original

Toutefois on cherche ici à conserver l'esprit "suites individuelles" en ne s'appuyant pas sur une modélisation stochastique, mais uniquement sur les prévisions des experts. En particulier, la notion de "niveau de confiance" (par exemple 95%) n'a pas forcément de sens.

On cumule ainsi les difficultés des suites individuelles (l'absence de modélisation), et celles de la prévision batch (pas de retour d'expérience). Ce problème est donc difficile ; et à notre connaissance il n'a pas encore été abordé dans le cadre des suites individuelles, dont les algorithmes ont crucialement besoin (généralement dans leur définition même) des retours d'expérience, des observations jusqu'à l'instant t -1 pour prévoir l'instant t. Or ici, en disposant des observations uniquement jusqu'à un instant T 0 (et, au-delà, seulement des prévision d'experts), on cherche à prévoir non seulement pour T 0 + 1, mais aussi T 0 + 2, T 0 + 3,. . .

Apports des travaux de cette thèse

Sur ce problème nouveau, l'apport des travaux de cette thèse est triple. Tout d'abord, est proposée une méthodologie pour adapter les algorithmes de suites individuelles à ce cadre. Cette méthodologie nécessitant une optimisation délicate, dépendant fortement de l'algorithme utilisé, on propose deux méthodes pour effectuer cette optimisation : une pour l'algorithme 1.5. Chapitre 5 : Faisceaux de prévision par agrégation séquentielle EWA (avec une extension pour l'algorithme "Fixed-Share EWA"), l'autre pour l'algorithme Ridge. Enfin, ces deux algorithmes sont mis en oeuvre (au chapitre 6) sur un jeu de données pétrolier d'IFP Energies nouvelles (avec l'introduction d'un indicateur de performance et d'un point de référence adéquats pour les évaluer).

Méthodologie : "Retour vers le futur" La méthodologie proposée repose sur l'idée suivante : "Les algorithmes de suites individuelles sont conçus pour être robustes et capables de gérer correctement tous les cas possibles. . . alors soumettons-leur tous les cas possibles et voyons leurs différentes prévisions".

La méthodologie consiste donc tout d'abord à définir un ensemble E de "scénarios possibles" :

E = {y T 0 +1 , y T 0 +2 , . . . , y T : [certaines conditions]}.
Cette ensemble doit être suffisamment large. Dans notre étude pratique, nous avons simplement limité les variations entre deux instants consécutifs, créant ainsi un "cône des scénarios possibles" :

E = {y T 0 +1 , y T 0 +2 , . . . , y T : ∀ j T -T 0 , y T 0 + j ∆ 1 y T 0 +j y T 0 + j ∆ 2 } (1.5.1)
avec ∆ 1 et ∆ 2 respectivement les bornes inférieure et supérieure des incréments. Il faut ensuite appliquer l'algorithme sur chaque scénario possible y T 0 +1 , y T 0 +2 , . . . , y T ∈ E, et noter les prévisions correspondantes : y T 0 +1 , y T 0 +2 , . . . , y T . Le faisceau de prévisions sera alors défini comme l'ensemble convexe des prévisions obtenues pour les différents scénarios.

On a donc, en quelque sorte, remplacé la "variabilité probabiliste" des approches traditionnelles par une "variabilité ensembliste".

Une optimisation délicate

La méthodologie présentée consiste non pas à calculer les réponses de l'algorithme à chacun des scénarios possibles (il y en a une infinité), mais à calculer pour chaque instant t = T 0 +k le minimum y min t et le maximum y max t que peut produire l'algorithme à l'instant t sur l'ensemble des scénarios possibles (entre T 0 + 1 et t -1).

Il s'agit donc d'une optimisation en dimension t -T 0 -1 (ce qui devient vite grand) d'une fonction souvent hautement non linéaire. Elle est non triviale : on se convaincra facilement de sa difficulté, par exemple sur l'algorithme EWA, en voyant qu'un scénario qui maximise la prévision à l'instant t est un scénario qui attribue un grand poids aux experts qui prévoient une haute valeur à l'instant t ; mais ces experts peuvent très bien avoir prévu des valeurs très différentes sur les instants précédents, ce qui empêche qu'ils aient tous été performants et obtiennent de grands poids. De même, un scénario qui prévoit des valeurs hautes d'observation jusqu'à l'instant t -1 ne va pas forcément aboutir à une prévision haute à l'instant t, cela dépend aussi des prévisions des experts.

La difficulté de cette optimisation dépend bien entendu de l'algorithme utilisé. Nous proposons dans le chapitre 5 un protocole de mise en oeuvre dans le cas de deux algorithmes, pour lesquels on dispose d'une forme explicite des poids :

1. Introduction générale

• pour l'algorithme Ridge, un calcul direct permet de résoudre exactement le problème d'optimisation ;

• pour l'algorithme EWA (et sa variante "Fixed-Share"), une méthode itérative aboutit à un intervalle proche de l'intervalle cherché, et qui le contient.

Voici l'expression des faisceaux obtenus pour les faisceaux tirés de l'algorithme Ridge (Lemme 5.2).

Lemma 1.4. On suppose que l'on dispose de bornes sur les observations : B s y s B s pour tout instant T 0 < s t. On note F t = (f s,j ) s t;j K la matrice des prévisions des experts jusqu'à l'instant t, et

V t+1 := (f 1,t+1 , . . . , f K,t+1 )(F t F t + λI K ) -1 F t ∈ R t .
Alors l'intervalle de prévisions [ y min t , y max t ] vérifie :

y max t = T 0 s=0 V t+1 s y s + t s=T 0 +1 V t+1 s (B s 1 V t+1 s 0 + B s 1 V t+1 s <0 ) et y min t = T 0 s=0 V t+1 s y s + t s=T 0 +1 V t+1 s (B s 1 V t+1 s 0 + B s 1 V t+1 s <0 ).
Comme indiqué en (1.5.1), dans le cadre de l'étude pratique, les bornes considérées seront

B t = y T 0 + (t -T 0 )∆ 1 et B t = y T 0 + (t -T 0 )∆ 2 ,
formant un "cône des scénarios possibles".

Mise en oeuvre pratique

Dans le chapitre 6, nous mettons en oeuvre la méthodologie et l'optimisation décrites ci-dessus sur un jeu de données d'IFP Energies nouvelles (cette étude est présentée plus en détails en Section 1.6). Plusieurs adaptations et améliorations pratiques sont utilisées afin d'améliorer la qualité des faisceaux. Par exemple, le niveau de bruit (on s'autorise ici une petite hypothèse de modélisation : on suppose que ce niveau n'augmente pas sur la période de prévision) doit être pris en compte et constituer une largeur minimale pour le faisceau de prévisions. Pour Ridge, une pré-sélection (automatisée) des experts permet également d'améliorer la prévision.

Le passage d'une évaluation qualitative "à l'oeil nu" à une évaluation quantitative nécessite de définir un indicateur de la qualité d'un faisceau. La simple proportion d'observations contenues dans le faisceau (qui correspondrait d'une certaine manière à un niveau de confiance : 90%, 85%. . . ) n'est pas pertinente : il suffirait de prendre un faisceau aussi large que possible pour remplir ce critère. La qualité d'un faisceau dépend donc aussi de sa "largeur". On définit donc au chapitre 6 un nouvel indicateur, "l'efficacité" ("efficiency"), comme le rapport du nombre d'observations contenues dans le faisceau, à l'aire du faisceau. Il s'agit d'un critère relatif, en ce qu'il n'a de sens que pour comparer des méthodes sur un jeu de données précis.

Une fois ce critère établi, juger de la qualité de nos méthodes nécessite un point de comparaison, un "benchmark" (banc d'essai).

Nous en proposons un (le meilleur faisceau d'experts parmi les choix d'experts "raisonnables", cf. chapitre 6). Nous sommes alors en mesure d'évaluer nos algorithmes. Les résultats obtenus 1.6. Chapitre 6 : Application de méthodes d'agrégation à la prévision pétrolière sont souvent satisfaisants pour Ridge : en dehors des propriétés chaotiques, on parvient généralement à faire aussi bien voir beaucoup mieux que le benchmark, et obtenir des faisceaux souvent précis, avec cependant une marge de progression sur la calibration des paramètres.

Pistes de recherche futures

Sur le plan pratique, améliorer et rendre plus robuste la calibration des paramètres des algorithmes (qui est l'une des principales difficultés rencontrées au chapitre 6) serait une avancée importante.

Un autre travail pourrait consister à chercher comment effectuer l'optimisation évoquée plus haut pour d'autres algorithmes, par exemple LASSO. En ce qui concerne EWA, une optimisation directe et exacte serait également une avancée. L'application de ces algorithmes de faisceaux sur d'autres jeux de données serait également un travail intéressant, permettant notamment de se faire une idée plus complète de leur précision mais aussi de leur robustesse.

Enfin, une piste conséquente de recherche serait d'essayer d'obtenir des garanties théoriques, absolues ou relatives, sur les faisceaux. Cela passera sans doute par la définition d'un critère adapté. En effet, le simple critère de proportion d'observations situées à l'intérieur du faisceau, ou le critère d'efficacité décrit plus haut, qui en découle, ne semblent pas pertinents (il suffit de laisser une "bande" d'observations possibles au-dessus ou au-dessus du faisceau pour s'exposer à voir toutes les observations dans cette "bande" et aucune dans le faisceau). Des pistes envisageables seraient de s'intéresser à des critères de taille du faisceau, de distance des observations par rapport au milieu ou aux limites du faisceau.

1.6. Chapitre 6 : Application de méthodes d'agrégation à la prévision pétrolière 1.6.1. Différentes approches pour la prévision de production pétrolière

Le chapitre 6 est consacré à l'application de méthodes d'agrégation en suites individuelles à la prévision de production pétrolière.

On s'intéresse à la prévision de trois quantités liées aux puits d'un champ pétrolifère (le jeu de données est présenté plus en détails Section 1.6.2) : le débit de pétrole, le débit d'eau et la pression en fond de puits.

Calage d'historique

La prévision de ces quantités s'appuie sur des modèles géophysiques qui tiennent compte notamment des caractérisitiques du sous-sol (perméabilité et porosité des différentes roches, etc), souvent en le subdivisant par une grille. Plusieurs types de données existent pour obtenir des informations sur le sous-sol : les données statiques (données diagraphiques, données sismiques. . . ), et les données dynamiques, qui portent notamment sur la composition et les flux des fluides présents. Les grandeurs que nous chercherons à prévoir font partie de ces données dynamiques.

Introduction générale

Un premier type de travail peut consister à utiliser les données statiques pour reconstituer certaines caractéristiques du sous-sol, ce qui permet ensuite d'effectuer un travail de prévision des données dynamiques. Mais il est évidemment souhaitable, pour améliorer la pertinence et la précision des modèles, de tenir compte des données dynamiques dont on a connaissance au fur et à mesure du temps pour améliorer les modèles. Ce problème inverse est nommé "calage d'historique" : il consiste à reconstituer certaines caractéristiques du sous-sol et des fluides présents, de manière à pouvoir expliquer au mieux les données dynamiques observées. C'est un problème très complexe, et coûteux en temps de calcul. De plus, il est "mal posé", au sens où plusieurs sous-sols différents (i.e., plusieurs jeux de paramètres différents pour les divers points de la grille de subdivision du sous-sol) peuvent aboutir aux mêmes grandeurs observées.

Un certain nombre de travaux récents se concentrent davantage sur la capacité prédictive des modèles que sur la "véracité" des paramètres sous-jacents.

Approche des travaux de la thèse L'utilisation des algorithmes d'agrégation présentée au chapitre 6 propose un compromis intéressant : calculer une fois pour toutes (à partir des données statiques) différents modèles de sous-sol crédibles et les prévisions correspondantes, et agréger ces différentes prévisions en tenant compte à chaque pas de temps des données dynamiques reçues entre-temps. Les algorithmes d'agrégation proposés ont un temps de calcul négligeable devant les calculs géophysiques de prévision, et permettent ainsi de tenir compte au fil du temps des informations fournies par les données dynamiques, sans avoir à recalculer les modèles géophysiques ou leurs prévisions. Il faut toutefois garder à l'esprit que, du fait de la non-linéarité des phénomènes étudiés, l'estimateur agrégé obtenu n'a pas d'interprétation immédiate en termes de composition du sous-sol (moyenner les paramètres de différents modèles de sous-sol considérés, n'est pas équivalent à moyenner les prévisions correspondantes).

La figure 1.4 résume les différentes approches évoquées.

Jeu de données étudié

La figure 1.5 résume les étapes de la construction des prévisions. Le jeu de données étudié, nommé "Brugge", est artificiel, il a été créé (cf. [START_REF] Peters | Results of the Brugge benchmark study for flooding optimization and history matching[END_REF]) pour servir d'étalon en préparation d'un "workshop" (à Bruges -Brugge en flamand et anglais, d'où son nom) et sert depuis de référence sur ce sujet. Il simule un champ pétrolifère ressemblant à ceux présents en mer du Nord, composé de 20 puits producteurs et de 10 puits injecteurs sur une surface d'une trentaine de km 2 , et porte sur des mesures mensuelles sur une période de 10 ans.

Les créateurs de ce jeu de données ont d'abord simulé un modèle très fin (non transmis) de sous-sol, à partir desquelles ils ont pu calculer les données statiques et dynamiques. Ils ont ensuite généré 104 modèles géologiques de sous-sol en adéquation avec les données statiques de ce champ pétrolier. Les caractéristiques de ces modèles, transmises par les créateurs de "Brugge", sont les faciès et divers paramètres (porosité, perméabilité, saturation en eau. . . ) en chaque bloc d'une subdivision du sous-sol par une grille d'environ 60 000 blocs (139 × 48 × 9). Ces caractéristiques ont été transmises par les créateurs de "Brugge", ainsi qu'une version bruitée des propriétés dynamiques à prévoir afin de pouvoir juger de l'efficacité des méthodes utilisées. Ces propriétés sont :

• la pression en fond de puits pour les puits injecteurs et producteurs ;

• le débit d'eau pour les puits producteurs ;

• le débit de pétrole pour les puits producteurs.

A partir de chacun de ces 104 modèles, les chercheurs d'IFP Energies nouvelles (IFPEN) Sébastien Da Veiga et Véronique Gervais-Couplet, ont calculé des prévisions mensuelles sur 10 ans pour ces propriétés dynamiques. Ce sont ces prévisions qui sont utilisées au chapitre 6, ainsi que les versions bruitées des données dynamiques.

Apports de la thèse

Prévision ponctuelle

Au chapitre 6, on applique trois algorithmes : EWA (voir Section 1.2.2), Ridge et LASSO (voir Section 1.3.4) , pour agréger sur chacune des propriétés étudiées, les prévisions liées aux 104 modèles de sous-sol, mois par mois. Les paramètres pour ces deux algorithmes sont choisis de manière empirique et opérationnelle (en particulier, on n'utilise pour la prévision à l'instant t que les données réellement disponibles avant l'instant t), suivant en cela la méthodologie de [START_REF] Devaine | Forecasting the electricity consumption by aggregation of specialized experts; application to Slovakian and French country-wide (half-)hourly predictions[END_REF].

Sur la majorité des propriétés, les prévisions proposées sont proches (pour EWA et Ridge) et même meilleures (LASSO) que la meilleure simulation proposée par l'IFPEN.

Introduction générale

Faisceaux de prévision

Les problèmes de calage d'historique sont généralement des problèmes mal posés. En effet, ils sont sur-paramétrés : même en imposant une cohérence géologique dans les paramètres des différents blocs de la grille, il reste encore beaucoup plus de modèles de sous-sol possibles que de mesures de données dynamiques, d'où une non-injectivité de l'application "paramètres du sous-sol → prévisions du modèle géophysique". Par conséquent, les travaux récents ont tendance à considérer une variabilité sur les modèles de sous-sol considérés, et partant, une variabilité sur leurs observations futures.

Dans la Section 6.5 du chapitre 6, on adopte cet état d'esprit en fournissant des faisceaux de prévision (c'est-à-dire des ensembles de prévisions, qui dans notre cas seront des intervalles) court-terme et long-terme. Le fait de passer de prévision mensuelles à des prévisions multiples et surtout plus éloignées dans le temps augmente a priori l'incertitude, ce qui fait un second motif pour prévoir des faisceaux plutôt que des valeurs uniques.

On applique la méthodologie générale présentée au chapitre 5, afin d'obtenir des faisceaux de prévision portant sur les dernières années de l'étude, à partir des algorithmes Ridge et EWA. On effectue certaines adaptations liées au problème étudié, en particulier vis-à-vis du bruit et d'éventuels biais ; le problème du choix des paramètres est également présenté, mais non entièrement résolu (on propose néanmoins deux choix pour cette calibration dans le cas de l'algorithme Ridge).

Un critère de mesure de performance (l'efficacité) est proposé, ainsi qu'un "benchmark" (un point de référence), ce qui permet d'évaluer les performances des faisceaux proposés par Ridge (voir Section 1.5.2), avec souvent des résultats satisfaisants.

Pistes de recherche futures

Une piste intéressante serait de tenter de réaliser l'objectif habituel du calage d'historique tel que présenté plus haut (c'est-à-dire une rétro-ingénierie des paramètres du modèle de sous-sol).

Alors que les travaux présentés traitent séparément les différentes propriétés et les différents puits, une première étape (qui constitue déjà en elle-même un problème de recherche) dans ce sens serait de tenter une approche globale sur l'ensemble des propriétés, aboutissant à des résultats cohérents. Par exemple, on pourrait chercher des poids permettant de prévoir correctement l'ensemble des propriétés. Or ces propriétés, et les pertes associées, sont liées à des quantités physiquement différentes, avec des ordres de grandeur inégaux (et dépendant des unités utilisées). Cela rend difficile la comparaison des pertes entre ces différentes propriétés ; par conséquent être capable de gérer de manière "équitable" ces pertes liées à des quantités physiquement différentes dans la détermination des poids d'agrégation sera un point-clé de cette approche (il ne semble par exemple pas pertinent de sommer directement ces pertes issues de grandeurs physiques différentes). Arriver à concevoir une mise à l'échelle pertinente et automatique des différentes propriétés étudiées, et parvenir à des garanties théoriques intéressantes à ce sujet dans un contexte de suites individuelles, est à notre connaissance un problème ouvert.

Au contraire, on pourrait aussi se concentrer uniquement sur les capacités prédictives des Bien évidemment, il serait également intéressant de tester les méthodologies présentées (en particulier, vu leur nouveauté, celles liées aux faisceaux de prévision) sur d'autres jeux de données pour juger de leur efficacité dans différents contextes.

Chapter 2

Mathematical introduction

This mathematical introduction describes the frameworks in which this thesis works take place, and presents mathematical tools used in the following chapters, for self-containment.

First, it introduces the setting of forecasting with expert advice, and two corresponding important frameworks: the "individual sequences" setting and the "batch" setting. Then, it presents several classical algorithms for sequential aggregation, and some bounds they guarantee. Afterwards, it tackles a generalization of sequential aggregation: online convex optimisation, with a focus on the importance of the losses at hand, and introduces "online-to-batch" framework conversions. The final part is devoted to a discussion about regularization, which will be at the heart of Chapter 3. Online prediction. The framework of online forecasting consists in predicting a timeevolving quantity y t in a convex space E (often R or R d ), the "observation", at several instants ("rounds of prediction") t = 1, 2, . . . The total number of rounds, T , can be known beforehand or not. Before each round, the statistician predicts a value y t , then the environment reveals the real observation value y t , and the statistician suffers a loss ( y t , y t ). The goal of the statistician is to minimize the cumulative loss:

L t := t s=1
( y s , y s ).

Classical loss functions include:

• the square loss: ( y t , y t ) = ( y ty t ) 2

• the absolute loss: ( y t , y t ) = | y ty t |

• the "0/1 loss": ( y t , y t ) = 1 yt =yt

Online prediction is the framework of a large number of problems, from weather forecasting [START_REF] Mauricette | Ozone ensemble forecast with machine learning algorithms[END_REF], [START_REF] Zhu | A method for improving the accuracy of weather forecasts based on a comprehensive statistical analysis of historical data for the contiguous United States[END_REF]) to traffic and travel times [START_REF] Shamas | Performance evaluation of an adaptive travel time prediction model[END_REF]), electricity consumption [START_REF] Devaine | Forecasting the electricity consumption by aggregation of specialized experts; application to Slovakian and French country-wide (half-)hourly predictions[END_REF]) or stock markets [START_REF] Gokcan | Forecasting volatility of emerging stock markets: linear versus non-linear GARCH models[END_REF]).

The case of experts aggregation. The experts aggregation setting consists in building the forecast y t by a combination (in this thesis, it will be a linear combination) of predictions of exterior forecasters, often called "experts". These can be algorithms, institutions, human beings. . . We assume that the number of experts is finite and denote it by K. The experts are indexed by k ∈ {1..K}, and release a forecast at the beginning of each round t. We denote by f k,t the forecast of the k-th expert for round t.

The forecast of an online aggregation algorithm at instant t is of the form:

y t = K k=1 u k,t f k,t .
The object of an aggregation algorithm is to choose the weights u k,t . The process of online prediction with expert advice is summarized in Setting 3. Some modified versions of this framework exist: for example, [START_REF] Blum | Empirical support for Winnow and Weighted-Majority algorithms: Results on a calendar scheduling domain[END_REF] and Freund et al. [1997] deal with specialized experts, that do not all provide forecasts at every round, and [START_REF] Gofer | Regret minimization for branching experts[END_REF] tackles the problem of so-called "branching experts".

One can notice that this framework covers, for example, the case of geometric (and more generally multiplicative) averages, since applying the log function transforms these averages into "additive" averages considered in our framework. Input: Decision space E, K experts. for t = 1, 2, . . .

1. Each expert k provides a forecast f k,t 2.
Based on all available data, the statistician chooses weights (u 1,t , . . . , u K,t ) and a forecast y t = K k=1 u k,t f k,t 3. The environment reveals a value y t 4. The statistician incurs the loss ( y t , y t ) and the experts incur losses (f k,t , y t ) Formally, expert aggregation is a particular case of linear regression, though with some specificities.

Firstly, if some (or even most) of the experts are assumed to perform well, then it makes sense to use convex algorithms (i.e., algorithm with nonnegative weights summing up to 1; we will then denote these weights by p k,t instead of u k,t ), whereas these algorithms may have less interest in more general linear regression setups. Yet, classical linear aggregation methods can also be used in the experts setting (cf. the Ridge and LASSO forecasters discussed in Section 2.2.3, and applied for instance in Chapter 6).

Secondly, the performance is generally not measured in some absolute manner, (e.g., through the cumulative loss):

L t := t s=1
( y s , y s ) but rather relatively to a reference benchmark linked to the experts. The cumulative loss of the k-th expert until round t will be denoted by:

L k,t = t s=1 (f k,s , y s ).
In this thesis, we will often focus on the following quantity:

R t := t s=1 ( y s , y s ) -min k=1..K t s=1 (f k,s , y s ) := L t -min k=1..K L k,t
It is called the (cumulative) regret, because it measures how much the statistician would have improved his or her performance if he or she had followed the best (in hindsight) expert (supposing that the real observations would have remained y 1 , . . . , y t ).

It is also possible to compare the algorithm to other benchmarks that the best expert, for example the best constant combination of experts:

t s=1 ( y s , y s ) - min (γ 1 ,...,γ K )∈Γ t s=1 K k=1 γ k f k,s , y s 2.
Mathematical introduction with Γ being R K (best linear combination), or the probabilistic simplex (best convex combination):

Γ = (γ 1 , . . . , γ K ) : ∀ k, 0 γ k 1 and K k=1 γ k = 1 .
This leads to various notions of regret.

Inequalities on the regret are sometimes described as "oracle inequalities" because they involve benchmarks (e.g., the minimum cumulative loss among the experts or their combinations) that depend on quantities that are not available to the learner during the forecasting period (the "best" expert and the "best" expert combination are only known in hindsight).

Individual sequences In the "individual sequences" setting, the data is assumed generated in an unknown deterministic way: no modeling (except boundedness) is performed on the data-generating process. It can take any value (within some range) at any instant. Therefore, theoretical results do not focus on the prediction of one precise round, rather they are often uniform bounds on the (cumulative) regret, valid for any individual sequence of values for the data (hence the name "individual sequences"):

∀ y 1 , . . . , y T , R T r T .
The bound r T on the regret depends on the algorithm and on the hypotheses on the loss. For a loss bounded by M , a trivial bound on the regret for any algorithm is the linear (in T ) quantity M T . Therefore, theoretical results aim at bounds that sublinear in T as for the regret:

r T = o(T )
Equivalently, they aim at a vanishing bound on the averaged regret: R T /T r T /T with lim

T →+∞ r T /T = 0.
The absence of major hypotheses prevents deducing links among the observations, and therefore limits the theoretical impact of a learning sample (preliminary sample given to the learner to help building the algorithm), which is usually not considered at all in the theorems. However, such a learning sample is often very useful in practical applications, allowing to "tune" the algorithm beforehand and avoid some chaotic first forecasts.

The absence of modeling and the uniform guarantees make these methods quite robust, that is why the term "robust aggregation" can be used to refer to the algorithms of expert aggregation in this individual sequences framework.

An important reference in this field is the monograph by [START_REF] Cesa | Prediction, Learning, and Games[END_REF]. One can also cite the papers by Freund and Schapire [1997], [START_REF] Littlestone | The weighted majority algorithm[END_REF], or [START_REF] Cesa-Bianchi | Improved second-order bounds for prediction with expert advice[END_REF].

2.1. Aggregation for individual sequences and in the batch setting

Batch setting

In the so-called "batch setting", the learner has immediately access to all the past observations and forecasts (the full "batch" of them, hence the name), and generally has to make only one forecast (or several forecasts, but without supplementary information or feedback).

Formally, the learner has observed N vectors x 1 , . . . , x N (the regressors), the corresponding N output values y 1 , . . . , y N and tries to forecast the output value y N +1 given x N +1 . The (x i , y i ) i=1...N form the learning sample. Contrary to the previous setup (individual sequences online learning), in the batch setting the learning sample plays a crucial role, as it is the main source of information for the "one-shot" forecast.

The process of batch prediction is summarized in Setting 4.

Setting 4 Batch forecasting process Input: Decision space E.

Learning sample:

The statistician has access to N vectors x 1 , . . . , x N and N corresponding values y 1 , . . . , y N Forecasting stage: In this context, the data is generally considered as random variables, upon which some stochastic assumptions are made (it is then called "stochastic batch setting"). In particular, to enable learning, the distribution of (x N +1 , y N +1 ) is generally linked to the distributions of the (x i , y i ) i N , for example through an assumption of stationarity, or even more strongly through an i.i.d. (independent and identically distributed) assumption.

1. A vector x N +1 is revealed
The performance of an algorithm is measured by the risk, which is the expected loss:

R y := E ( y(x N +1 ), y N +1 ) | (x 1 , y 1 , . . . , x N , y N )
In the same spirit as above, one usually seeks guarantees not directly on the risk, but relatively to the risk of some benchmark, either in expectation or with high probability with respect to (x 1 , y 1 ), . . . , (x N , y N ).

For instance, one can seek guarantees relatively to the best linear function with high probability:

P (x 1 ,y 1 ),...,(x N ,y N ) R y -min f linear E (f (x N +1 ), y N +1 ) ζ N 1 -δ N (2.1.1)
for some ζ N and δ N , where P (x 1 ,y 1 ),...,(x N ,y N ) denotes the probability with respect to (x 1 , y 1 ), . . . , (x N , y N ). Two main settings exist: the case where both the x i 's and the y i 's are random variables is called "random design"; the case where the x i 's are fixed (chosen or not by the learner) and only the y i 's are random variables is called "fixed design".

Mathematical introduction

One can notice that "classical" linear regression problems, such as finding an estimator β of a vector β, given a matrix X and the values

Y = Xβ + ε
where ε is a random noise vector, correspond to the batch setting. Similarly to the individual sequences forecasting, the term "aggregation" is usually used for situations where some individual predictors (the columns of X in the previous example) are assumed to be good and are used as benchmarks to be competed against. However, one should note that Tsybakov [2003] uses the term of "aggregation" for different problems, depending on the benchmark (which is the "best" vector in a set Γ) one wants to compete against. These problems are

• the model selection aggregation (Γ is the canonical basis of R d );

• the convex aggregation (Γ is the probabilistic simplex of R d );

• the linear aggregation (Γ = R d ).

In the batch setting one can derive results on the forecasts (cf. (2.1.1)), like in the individual sequences framework, but also on the "quality" of the learning (e.g. bounds on β -β in the previous example). These two objectives are known as the "prediction problem" and the "estimation problem".

Some comparisons between batch and individual sequences settings

One can see that in the batch setting, the N rounds taken into consideration are learning rounds, on which no predictions and no errors are made by the statistician, and which help to better know the data, so that the expected error at the final forecasting round tends to decrease when N grows. On the contrary, in the online setting, the T rounds taken into account are rounds of prediction, so the cumulative error tends to grow when T grows. However, the averaged regret R T /T tends (for "reasonable" algorithms!) to decrease; some links between averaging in online learning and batch results will be shown in the "online-to-batch" parts (Section 2.3.3 and Section 4.3). More generally, it is possible to "transfer" methods that are efficient in the online setting to the batch setting.

Conversely, we will make use in Chapters 5 and 6 of two algorithms, the Ridge and LASSO forecasters, that were initially developed for linear regression with fixed design, but that give interesting results for individual sequences predictions (cf. Section 2.2.3).

Algorithms for the forecasting of individual sequences 2.2.1. Introductory remarks

Failure of some naive algorithms. One of the simplest aggregation algorithm is the (arithmetic) average of the experts' predictions: y t = K k=1 f k,t /K. It is obvious that for most losses it can fail: even if some experts are good, some poorly-predicting experts may drive the average to suffer a large loss.

Algorithms for the forecasting of individual sequences

Another "natural" approach is to "follow the leader": allocate a weight 1 to the expert that has suffered the smallest cumulative loss until the instant to forecast (the "leader"), and a weight 0 to the other experts. This approach also fails: it suffices to consider a situation where, at each instant t, the "leader" until t -1 suffers the highest loss at time t, and then loses its leadership; such situations exist (in particular when two experts have a similar overall performance).

Necessity of hypotheses on the loss. Without hypotheses on the loss, it is not possible to guarantee a sublinear regret, whatever the algorithm. For instance, consider the loss (x, y) = 1 x =y , the set of predictions 0, 1 and two constant experts: f 0 always forecasting 0 and f 1 always forecasting 1. With this loss, for any sequence of predictions, there exist a sequence of observations in 0, 1 that leads to a loss of 1 at each round (y t = 1 when y t = 0, and y t = 0 when y t = 0), so the cumulative loss of this sequence of predictions is T . As for the experts f 0 and f 1 , at each round one of them receives a loss 0 and the other one receives a loss 1, so over T rounds one of them has a cumulative loss inferior (or equal) to T /2. As a consequence the regret is superior (or equal) to T /2, which is linear in T . We will see below that a sufficient assumption to get interesting results is the convexity of the losses (in their first argument).

In the following, we will only consider convex losses.

Two convex algorithms

We first focus on two "convex" algorithms, in the sense that they output nonnegative weights summing up to 1, which we denote by p k,t instead of our notation u k,t for general weights:

∀ t, ∀ k, p k,t 0 and

K k=1 p k,t = 1.
In particular, at each round, forecasts of convex algorithms are bounded between the uppest and lowest forecasts of the experts.

The Exponentially Weighted Average forecaster ("EWA") General presentation. The Exponentially Weighted Average forecaster (abbreviated in "EWA" in the following) is a convex algorithm with positive weights, relying on an exponential decrease of the weights with respect to the cumulative losses, and on a parameter (called "learning rate") chosen by the statistician.

The weights provided by EWA, with fixed learning rate η > 0, are as follows: p k,1 = 1/K, and for each round t 2,

p k,t = exp(-ηL k,t-1 ) K i=1 exp(-ηL i,t-1 )
(2.2.1)

A higher learning rate η leads to a faster learning (larger differences among the weights), a lower η leads to a weight vector closer to the uniform vector (which corresponds to the "limit" case η = 0).

Mathematical introduction

One can see that the weights output by EWA can be computed by a recursive update:

p k,t = p k,t-1 exp -η (f k,t-1 , y t-1 ) K i=1 p i,t-1 exp -η (f i,t-1 , y t-1 )
First theoretical bounds. The next theorem and its corollary show that EWA, with a correctly tuned η, can achieve a regret of order √ T .

Theorem 2.1. If all the losses of the experts lie in the [m, M ] interval (where m M are real numbers) then the EWA algorithm with fixed parameter η incurs a regret bounded by:

R T := L T -min k=1,...,K L k,T log(K) η + η (M -m) 2 T 8 (2.2.2)
Proof. It is a classical proof (see for instance [START_REF] Cesa | Prediction, Learning, and Games[END_REF]). Denote by w i,t = exp(-ηL i,t ) any "weight before normalization", and by W t := K i=1 w i,t the "total weight before normalization". The proof relies on upper and lower bounds on the quantity log W T W 0 . First, as a sum of positive terms is greater than the maximum of its terms, one has:

log W T W 0 = log K i=1 exp(-ηL i,T ) -log(K) log max i=1,...,K exp(-ηL i,T ) -log(K) = -η min i=1,...,K L i,T -log(K). ( 2

.2.3)

As for the upper bound we use the following lemma, due to Hoeffding.

Lemma 2.2. Let X be a random variable such that a X b. Then for any s ∈ R,

log E[exp(sX)] sE[X] + s 2 (b -a) 2 8
We apply this lemma to a random variable X such that

P X = (f i,t , y t ) = w i,t-1 / K k=1 w k,t-1 . Therefore: log W t W t-1 = log K i=1 w i,t-1 exp(-η (f i,t , y t )) K k=1 w k,t-1 -η K i=1 w i,t-1 (f i,t , y t ) K k=1 w k,t-1 + η 2 (M -m) 2 8 -η K i=1 w i,t-1 f i,t K k=1 w k,t-1 , y t + η 2 (M -m) 2 8 = -η ( y t , y t ) + η 2 (M -m) 2 8
2.2. Algorithms for the forecasting of individual sequences

The last inequality comes from the convexity of . A telescopic sum over t leads to: Combining (2.2.3) and(2.2.4) and dividing by η leads to the result.

log W T W 0 -η L T + T η 2 (M -m) 2 8 . (2.2.4)
If one has access to T , m and M , it is then possible to minimize the right-hand side of (2.2.2), by choosing η = 8 log(K)/((Mm) 2 T ), which guarantees the following bound.

Corollary 2.3. Under the hypothesis of Theorem 2.1, the EWA algorithm with parameter η = 8 log(K)/((Mm) 2 T ) satisfies:

R T (M -m) T log(K) 2
Tuning of the learning parameter. In many situations one does not have access to T or M . It is then useful to have an adaptive tuning of η, changing over time:

p k,t = exp(-η t L k,t-1 )/ K i=1 exp(-η t L i,t-1 )
, where η t is set based on past data. When m and M are known in advance, but not T , a possible approach is the so-called "doubling trick". It consists in dividing the rounds in "epochs", with epoch j containing the rounds {2 j , . . . , 2 j+1 -1}. At the beginning of each epoch, the weights are reset, and within epoch j, whose length is 2 j , the EWA algorithm is run with learning parameter η t = 8 log(K)/((Mm) 2 2 j ). Summing the regrets of all epochs give the following result.

Corollary 2.4. The EWA algorithm run using the "doubling trick" guarantees that:

R T (M -m) √ 2 √ 2 -1 T log(K) 2
Proof. The index of the last epoch is J = log 2 (T ) . Therefore, the sum of the regrets over all the epochs is bounded by:

J j=0 (M -m) 2 j log(K) 2 = √ 2 J+1 -1 √ 2 -1 (M -m) log(K) 2 (M -m) √ 2 √ 2 -1 T log(K) 2 .
If m and M are also unknown, one can apply variants that use a parameter η t independent of T , m and M . A possibility is provided in De [START_REF] De Rooij | Follow the leader if you can, hedge if you must[END_REF].

Proposition 2.5. If one defines t := k (f k,t , y t )/K and:

δ t = 1 η t K k=1 p k,t exp -η t (f k,t , y t ) -t
then the EWA algorithm run with learning parameter η t = log(K)/ t-1 s=1 δ s guarantees:

R T (M -m) T log(K) + (M -m) 2 +

Mathematical introduction

However, all these choices for η or η t , tuned to tackle worst-case scenarios, tend to be too conservative to get top performance with real data sets. In particular, the η t are often too small. One can rather use more "data-driven" choices for η t . In Chapter 6, we will use at round t the empirically best parameter for the rounds until t -1, i.e., the parameter η that would have led to the smallest cumulative loss on rounds until t -1 if one had run the EWA algorithm with this fixed parameter η. This choice is not associated with theoretical guarantees, but it often leads to a better practical performance, since it is more adapted to the data.

The Fixed-Share EWA forecaster

The previous algorithm EWA is based on the cumulative loss of each expert. It is therefore not carved to take advantage of changes in the performance of the experts (in particular, experts incurring a large loss during the first rounds are bound to have small weights for a long time). The following modified version of EWA, introduced by Herbster and Warmuth [1998], allows to benefit from recent improvements of any expert. It is called "Fixed-Share EWA" because it guarantees a minimal weight (the "fixed share") for each expert.

The weights p k,t of Fixed-Share EWA, with fixed parameters α (share) and η (learning rate) are given as follows: p k,1 = 1 K , and for each round t 2:

p k,t = α K + (1 -α) p k,t-1 exp -η (f k,t-1 , y t-1 ) K i=1 p i,t-1 exp -η (f i,t-1 , y t-1 ) (2.2.5)
See [START_REF] Herbster | Tracking the best expert[END_REF] for a "shifting bound" guaranteed by Fixed-Share EWA, i.e., a bound with respect to the minimal loss one can get by forecasting the same values that a precise expert, but with the possibility of shifting from one expert to another one a defined number of times.

Two non-convex regularized algorithms

We now focus on two algorithms that do not output convex weights but general linear weights (signed weights whose sum does not need to equal 1). They are "regularized" algorithms, i.e. the weights chosen for round t are minimizers of the quantity:

t-1 s=1 K k=1 u k f k,s , y s + Ψ(u 1 , . . . , u K )
with Ψ depending on the algorithm.

This notion of regularization will be at the heart of Section 2.4, where we will also explain its interest in terms of avoiding overfitting issues.

Algorithms for the forecasting of individual sequences

The Ridge regression forecaster

The Ridge regression forecaster is a regularized "least squares" estimator, introduced by [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]. It relies on a regularization parameter λ, and on a Euclidean regularization Ψ : u → u 2 2 . Consider some instant t, and denote:

-by f t the vector of all the experts' forecasts at time t:

f t = (f 1,t , . . . , f K,t ) , -by F t = (f s,k ) s t k K
the t × K matrix of the experts' forecasts up to time t (notice that we inverse the order of indices of the experts' forecasts), -by Y t = (y 1 , . . . , y t ) T the column-vector of the observations up to time t (. T meaning "transpose").

The weight vector output by Ridge u R t+1 :=

  u R 1,t+1 . . . u R K,t+1   is the minimizer of: u =   u 1,t+1 . . . u K,t+1   -→ Y t -F t u 2 2 + λ u 2 2 . (2.2.6)
The weight vector generated by Ridge and therefore the Ridge regression forecasts, are linear mappings of the observations, as is shown in Lemma 2.6.

Lemma 2.6. The weights and forecasts provided by the Ridge algorithm (with regularization parameter λ) are linear with respect to the observations vector Y t :

  u R 1,t+1 . . . u R K,t+1   = M t Y t with M t := (F T t F t + λI K ) -1 F T t thus y t+1 = V t+1 Y t with V t+1 := (f 1,t+1 , . . . , f K,t+1 )(F T t F t + λI K ) -1 F T t ∈ R t
Proof. The weight vector u R t+1 given by the Ridge regression forecaster is defined in (2.2.6) as the minimizer of a smooth convex function. Therefore, it is a zero of its gradient:

-2F T t Y t -F t u R t+1 + 2λu R t+1 = 0.
This leads directly to

F T t F t + λI K u R t+1 = F T t Y t .
For any λ > 0, the matrix F T t F t + λI K is invertible (symmetric definite positive), so one gets the first result:

u R t+1 = (F T t F t + λI K ) -1 F T t Y t .
These weights lead to the forecast (at time t + 1):

y t+1 = f t+1 u R t+1 = V t+1 Y t .

Mathematical introduction

One has the following bound on the regret of the Ridge forecaster. We will use for convenience the notation:

t (u) := K k=1 u k f k,t -y t 2 .
Theorem 2.7. Assume that the |f k,t | are bounded by F , and that the y t are bounded by Y .

Then, for all λ > 0, the Ridge forecaster run with regularization parameter λ guarantees, for all u = (u 1 , . . . , u K ) ∈ R K that:

T t=1 t (u R t ) - T t=1 t (u) λ u 2 2 + 4KY 2 1 + KF 2 T λ log 1 + F 2 T λ (2.2.7)
Proof. The proof is derived from Azoury and Warmuth [2001] and [START_REF] Vovk | Competitive on-line statistics[END_REF].

Firstly, an induction gives that:

∀u ∈ R K , T t=1 t (u R t+1 ) - T t=1 t (u) λ u 2 2 -u R 2 2 2 (2.2.8) It is true for T = 1 by definition of u R 2 . If it is true for some T 1, then by definition of u R T +2 , ∀u ∈ R K , T +1 t=1 t (u) + λ u 2 2 T +1 t=1 t (u R T +2 ) + λ u R T +2 2 2 T +1 (u R T +2 ) + T t=1 t (u R t+1 ) + λ u R 2 2 2 = T +1 t=1 t (u R t+1 ) + λ u R 2 2 2
so it is also true for T + 1. Secondly, let us prove that:

T t=1 t (u R t ) -t (u R t+1 ) 2 T t=1 f t F T t F t + λI K ) -1 f t × max t T t (u R t ) (2.2.9)
Using the fact that

F t F t = F t-1 F t-1 +f t f t , one can easily derive from the proof of Lemma 2.6 that: u R t+1 = F t F t + λI K -1 F t-1 Y t-1 + y t f t = F t F t + λI K -1 F t F t + λI K -f t f t u R t + y t f t = u R t -F t F t + λI K -1 (f t u R t ) -y t f t

Algorithms for the forecasting of individual sequences

Combined with the gradient convexity inequality, this gives:

t (u R t ) -t (u R t+1 ) ∇ t (u R t ) (u R t -u R t+1 ) = 2 f t u R t -y t f t F t F t + λI K -1 f t u R t -y t f t = 2f t F t F t + λI K -1 f t × t (u R t )
Summing over t gives Inequality (2.2.9). Thirdly, one can use the proof of Lemma 11 of [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF] (applied to u t = f t and ε = λ) to get that:

T t=1 f t F t F t + λI K -1 f t log det F t F t + λI K det (λI K ) Let us denote by λ 1 , • • • , λ K the eigenvalues of the symmetric positive matrix F t F t . Then det F t F t + λI K = K k=1 (λ + λ k
) and one has that:

T t=1 f t F t F t + λI K -1 f t K k=1 log 1 + λ k λ Since the f k,t are bounded by F , one has that K k=1 λ k = Tr(F t F t ) KT F 2 . Therefore, as x → log(1 + x/λ) is concave, one has: 1 K K k=1 log 1 + λ k λ log 1 + K k=1 λ k Kλ log 1 + T F 2 λ
So one has:

T t=1 f t F t F t + λI K -1 f t K log 1 + T F 2 λ (2.2.10)
Fourthly, it remains to bound max

t T t (u R t ). Using the inequality (a -b) 2 2(a 2 + b 2 ),
Cauchy-Schwarz inequality and the fact that f t 2 2

KF 2 , one has that:

t (u R t ) 2 (f t u R t ) 2 + y 2 t 2 KF 2 u R t 2 2 + Y 2
By definition of u R t , comparing it to the null vector gives, in terms of regularized cumulative losses:

λ u R t 2 2 + t-1 s=1 s (u R t ) 0 + t-1 s=1 (0 -y s ) 2 (t -1)Y 2 Therefore, u R t 2 2
T Y 2 /λ and:

max t T t (u R t ) 2Y 2 1 + KF 2 T λ (2.2.11)
Combining all the previous results, namely (2.2.8), (2.2.9), (2.2.10) and (2.2.11), concludes the proof.

Mathematical introduction

The LASSO forecaster

The Least Absolute Shrinkage and Selection Operator (LASSO) forecaster is, like the Ridge regression forecaster, a regularized "least squares" estimator. It has been introduced by [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]. It will be presented with more details in Section 2.4.3. With the same notations as above, the weight vector output by LASSO u L t+1 := (u L 1,t+1 , . . . , u L K,t+1 ) is defined as a minimizer of:

u =   u 1,t+1 . . . u K,t+1   -→ Y t -F t u 2 2 + λ u 1 , (2.2.12)
for some parameter λ tuned by the statistician.

If one denotes λ := λ/N , then it is clearly equivalent to define u L t+1 as a minimizer of:

u -→ Y t -F t u 2 2 /N + λ u 1
, as it will be done in Section 2.4. There are no known non-trivial bounds for the LASSO forecaster in the context of individual sequences.

Let us get back to the notation t (u) used in the proof of Theorem 2.7: it emphasizes the fact that the loss of an aggregation algorithm is not only linked to its forecast, but also to the weight vector used to output this forecast. Focusing on this weight vector and on its link with the loss is the spirit of the framework in the next section takes place.

Online convex optimization

Framework

In this section, and in Chapter 4, we will go beyond linear regression with convex losses and tackle a more general framework: online convex optimization. In this setup, we still provide at each step a vector w t and get then a loss function t that we want to minimize, but few assumptions will be made on the t (typically convexity).

Loss in the regression setup:

w t → w t x t → (w t x t , y t ) → t (w t x t , y t )
(when typically t will be a constant).

Loss in the online convex optimization setup:

w t → t (w t )
Thus, in this setting, t encompasses both the predictions x t and the observation y t . The domain of interest K ⊂ R d is convex; we will assume it is closed. It is generally fixed, but in some cases it can vary over time.

Similarly to Section 2.2, the number of rounds T may be known beforehand, or not (in Chapter 4 it will be known).

Online convex optimization

The t are deterministic and sequentially picked over time by the environment: they can be any convex function. The goal is then to minimize the following regret:

T t=1 t (w t ) -min u∈K T t=1 t (u)
.

We recall that the mere hypothesis of convexity (without any smoothness assumption) guarantees the existence of a non-empty subgradient in any point in the interior int(K) of K:

∀ x ∈ int(K), ∂ t (x) = ∅, where ∂ t (x) := {v : ∀ y ∈ K, t (y) t (x) + v (y -x)}.
Of course, other assumptions can be added on the losses t : boundedness, (sub)gradient boundedness, hypotheses about curvature (e.g., exp-concavity, strong convexity). Their impacts are presented in the next section.

We will focus on the case of first-order information, where one has access after time t only to ∇ t (w t ), and to t (w t ), but not to the entire function t . This framework is summarized in Setting 5.

Setting 5 Online convex optimization (with first-order information) framework for t = 1, 2, . . .

1. Play w t ∈ K 2.
The environment picks a convex loss function t : K → R 3. Increase cumulative loss by t (w t ) and observe (sub)gradient g t ∈ ∂ t (w t ) as well as t (w t )

2.3.2. Different losses lead to different regrets -and require different algorithms?

General convex losses

It is clear that online convex optimization algorithms are likely to use ideas from different fields of mathematics: statistics, optimization, geometry, etc. A standard algorithm in optimization (a fortiori, in convex optimization, which prevents local non-global minima), is gradient descent. Its online version has been introduced in [START_REF] Zinkevich | Online convex programming and generalized infinitesimal gradient ascent[END_REF] (see also [START_REF] Cesa-Bianchi | Analysis of two gradient-based algorithms for on-line regression[END_REF]). It is based on the following update:

w t+1 = w t -η t ∇ t (w t ).
The choice of the learning rate η t is crucial. In the case of bounded domain and bounded gradient (at any time and any point), online gradient descent with η t of order 1/ √ t achieves the optimal regret for the framework described above (general convex functions): O( √ T ) (cf. [START_REF] Zinkevich | Online convex programming and generalized infinitesimal gradient ascent[END_REF]; see also [START_REF] Hazan | Introduction to online convex optimization[END_REF]).

Mathematical introduction

Yet, this 1/ √ t learning rate might not be optimal in practice. To overcome part of the problem of tuning η t , [START_REF] Duchi | Adaptive subgradient methods for online learning and stochastic optimization[END_REF] introduce a new algorithm, called Adagrad. It runs a separate learning rate for each dimension, and (still after one choice of the scaling factor η by the statistician) automatically tunes these learning rates based on the previous observed gradients. Denoting w j t and g j t the j-th component, respectively, of the output and of the gradient, the update is given by:

w j t+1 = w j t - η t s=1 (g j s ) 2 g j t .
Even if it was created for convex problems, Adagrad has been successfully used for nonconvex situations (cf. [START_REF] Maya R Gupta | Training highly multiclass classifiers[END_REF]).

If a regret of Ω( √ T ) is unavoidable (cf. [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]) when dealing with general convex losses (which include linear functions, without curvature), supplementary assumptions on the loss allow improved bounds.

Strongly convex losses

If convexity lower bounds the function by a linear quantity (it is the definition of the subgradient), a stronger hypothesis is to lower bound the function by a quadratic quantity. This is called strong convexity. A function f :

K → R is λ-strongly convex if it is convex and: ∀ x, y ∈ K, ∀ ∇f (x) ∈ ∂f (x), f (y) f (x) + ∇f (x) (y -x) + λ 2 y -x 2 2 .
This strong curvature leads to improved regret guarantees: a regret of O(log(T )) can be achieved with online gradient descent, with parameter η t of order 1/t (cf [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]). It is natural to use a learning rate smaller than for general convex functions, since the extra curvature increases the changes in the function, even in small areas.

Exp-concave losses

An intermediate assumption is exp-concavity. A function f is α-exp-concave (with α > 0) if exp(-αf ) is concave. One can easily see that it implies convexity, and that it is implied by strong convexity. The following lemma (proved for instance in [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]) illustrates that intermediate situation between convexity and strong convexity: Lemma 2.8. Let f : K → R be an α-exp-concave function. Let us assume that K is bounded with diameter D, and that for any point x ∈ K, one has ∇f (x) 2 G. Then, for β min (1/(4GD), α)) /2, one has:

∀ x, y ∈ K, f (y) f (x) + ∇f (x) (y -x) + β ∇f (x) (y -x) 2
Using Cauchy-Schwarz inequality (and the assumption ∇f (x) 2 G), the inequality of the previous lemma also holds if f is λ-strongly convex, taking β = λ/(2G 2 ).

Online convex optimization

In the context of the first-order information framework, one does not have access to the second-order derivative of t (if ever it exists). Therefore, one cannot resort to the second-order Taylor expansion:

f (y) f (x) + ∇f (x) (y -x) + 1 2 (y -x) ∇ 2 f (x)(y -x)
where ∇ 2 f (x) stands for the Hessian matrix of f at the point x. But Lemma 2.8 gives another way to be more precise than with mere linear approximation: adding a quadratic term involving the gradient, proportional to ∇f

(x) (y -x) 2 .
This idea of "replacing the Hessian matrix with a quadratic gradient term" leads to an interesting variation of Newton's method. The starting point is that a possibility for minimizing a convex function is to look at the zeros of its gradient. This can be done via Newton's method. The "Online Newton Step" algorithm, introduced in [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF], modifies this Newton's method, especially by replacing the Hessian matrix by εI d + t s=1 g t g t (with ε a term depending in particular on the parameter of "exp-concavity").

Under the assumption of exp-concavity for the losses, [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF] guarantees a bound O(d log(T )) on the regret with the Online Newton Step method (cf. Lemma 4.8).

Adaptivity

We have introduced so far in this section several algorithms, with different tunings of their learning rates, each one particularly adapted to a specific category of losses. To be optimal, the tuning of their learning parameter often requires the knowledge of the degree of curvature (parameter of exp-concavity, etc.). A natural objective is then to build adaptive algorithms, i.e. algorithms that are able to be efficient on several categories of convex losses, without knowing the degree of curvature beforehand. The MetaGrad algorithm, presented in Chapter 4, is an important step in this direction.

From online setting results to batch setting results

General approach. It is possible to transfer performance bounds in the online setting to the batch setting, in particular for convex losses. A "classical" way of doing so is given in Theorem 2.9 below (cf. Littlestone [1989]). Here we leave the "first-order information" framework: we assume that we have access after round t to the whole loss t . For instance, in the case of linear regression, we have access to the vector x t and to the observation y t , and therefore to the loss w → (w x t , y t ).

If the training sample is ( 1 , . . . , T ), the general idea consists in treating this training sample, totally available from the start, as if it were discovered in an online fashion, applying an online algorithm and getting the sequence:

w 1 , w 2 = w 2 ( 1 ), w 3 = w 3 ( 1 , 2 ), . . . , w T = w T ( 1 , . . . , T -1 ).
Then, the forecast is built upon these intermediate forecasts: w = w (w 1 , . . . , w T -1 ). Twists can be added, using the actual availability of the data, as in Proposition 2.10, or allowing 2. Mathematical introduction intermediate restarts that might be suboptimal in terms of the cumulative loss (because here, the cumulative loss is of no direct importance) but are useful for the real one-shot prediction of interest, like in Chapter 4.

A classical online-to-batch conversion. The following theorem only relies on the online regret of algorithms, not on the way they build their forecasts, so it can be applied to aggregation algorithms but also to other kinds of methods.

Theorem 2.9. Consider a set S of convex losses, a subset K of K, and an online algorithm such that, for any sequence of losses ( ¯ 1 , . . . , ¯ T ) ∈ S T , the outputs (w 1 , . . . , w T ) satisfy:

T s=1 ¯ s (w s ) -inf u∈K T s=1 ¯ s (u) r T (2.3.1)
where r T is some scalar value. Consider an i.i.d. sequence of losses 1 , . . . , T +1 drawn in S, where the training sample 1 , . . . , T is given and T +1 must be minimized. Then, applying in an online fashion this algorithm to the training sample, as described above, and forecasting as the average of the outputs:

w = 1 T T s=1 w s ( 1 , . . . , s-1 )
gives the guarantee:

E T +1 (w) -inf u∈K E T +1 (u) r T T
Proof. We first use the convexity of the loss T +1 to write:

E [ T +1 (w)] = E T +1 1 T T s=1 w s 1 T E T s=1 T +1 (w s ) (2.3.2)
Then, we use the fact that w s is only based on ( 1 , . . . , s-1 ) so it is independent of s and

T +1 , since the t are i.i.d. Therefore, E [ s (w s )] = E [ T +1 (w s )]
. We can then use the "online guarantee hypothesis" (2.3.1), which is true for any individual sequence of losses, and therefore also in expectation:

E T s=1 s (w s ) E inf u∈K T s=1 s (u) + r T
It suffices then to write:

E inf u∈K T s=1 s (u) inf u∈K E T s=1 s (u)
and to divide by T to get:

E [ T +1 (w)] 1 T inf u∈K E T s=1 s (u) + r T T 54 2.3. Online convex optimization
Using again the fact that the s and T +1 are i.i.d., we have:

E [ s (u)] = E [ T +1 (u)], therefore inf u∈K E T s=1 s (u) = T inf u∈K E [ T +1 (u)] ,
which gives the desired result.

As noticed by [START_REF] Audibert | Fast learning rates in statistical inference through aggregation[END_REF], one can see that the construction of w does not use T , so if the "online guarantee hypothesis" (2.3.1) remains true one step further (i.e. replacing T by T + 1), then one can include the "last online forecast" w T +1 into w:

w = 1 T + 1 T +1 s=1 w s ( 1 , . . . , s-1 )
It guarantees a batch regret of at most r T +1 /(T + 1) instead of r T /T .

A randomized online-to-batch conversion, useful for non-convex losses. It is possible to apply the ideas of the previous proof to the case of a non-convex loss, by replacing the averaging of the forecasts by a random choice among them. It is also possible to take into account the elements of the learning sample in a different order, for example a reverse order to prioritize the latest values. Those two ideas lead to the following lemma, seen in lectures slides of [START_REF] Bartlett | Online prediction[END_REF].

Proposition 2.10. Consider the setting of Theorem 2.9, excepted that the t are not necessarily convex. Then picking randomly and uniformly an integer J in {0, 1, . . . , T -1} and outputting w = w T -J ( J+1 , . . . , T -1 ) gives the following guarantee:

E [ T +1 (w)] -inf u∈K E [ T +1 (u)]
r T T Remark: for J = T -1, the output is w 1 , which is independent of the k 's.

Proof. First, we use the "tower rule" on the independent randomization to transfer the mean outside the loss:

E T +1 (w) = E J E T +1 (w) J = E J E T +1 (w T -J ( J+1 , . . . , T -1 )) J = 1 T T -1 s=0 E [ T +1 (w T -s ( s+1 , . . . , T -1 ))]
Then, since the s are i.i.d., one can shift the variables into the expectation, and then reorganize the sum:

1 T T -1 s=0 E [ T +1 (w T -s ( s+1 , . . . , T -1 ))] = 1 T T -1 s=0 E [ T +1 (w T -s ( 1 , . . . , T -s-1 ))] = 1 T T s=1 E [ T +1 (w s ( 1 , . . . , s-1 ))] 55

Mathematical introduction

As the last expression corresponds to the one in (2.3.2), the remaining part of the proof is similar to the one of Theorem 2.9.

One can thus see that the use of a "reverse order" in the procedure -using w = w T -J ( J+1 , . . . , T ) instead of w J ( 1 , . . . , Z -1 )-does not change the results.

One can also notice that, similarly to the previous case, if one has online guarantees up to T + 1, one can slightly change the procedure to get a r T +1 /(T + 1) bound, instead of a r T /T bound: it suffices to pick T in {0, 1, . . . , T } and to output w = w T +1-J ( J+1 , . . . , T ).

The slides of [START_REF] Bartlett | Online prediction[END_REF] also use more sophisticated techniques to tackle the case where the t are stationary, but not i.i.d.

Being suboptimal in the online setting, and optimal in the batch setting. In an "online-to-batch conversion", the goal is to get a good predictor for the batch setting, not to control the cumulative error. This fact allows to use, in the process, parameters or starting points that would not give the best guarantees for an online cumulative error, but which will allow to secure enough information to get good guarantees for the batch forecast. It is the case of the "Epoch Gradient Descent" algorithm, presented in [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF], that uses the Gradient Descent methodology, but splits the training sample into growing-sized parts (the "epochs"), and makes a restart at the beginning of each epoch:

• changing the parameter;

• using as a starting point the average of the outputs in the previous epoch. This approach leads to better bounds in the setting of [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF] that the "classical conversion" seen previously. In Chapter 4 (Section 4.3), we will apply and adapt this approach to a couple of algorithms, namely Online Newton Step and MetaGrad.

After having seen in this section some ways of transferring algorithms from the online setting to the batch setting, we will focus totally on this latter framework in the next section.

Regularization in a stochastic batch setting 2.4.1. The framework

In this section, we consider a stochastic batch setting. The goal is to be able to link an output y ∈ R to an input x belonging to some probability space χ, by a mapping f included in a set of functions F . Contrary to the individual sequences framework, here we have some knowledge about the way the data is generated. We have access to a sample of N i.i.d. pairs (x i , y i ) i=1...N of random variables, with the same distribution as (x, y). Moreover, one sometimes has access to some additional a priori knowledge about the distribution of the data.

Contrary to the individual sequences setting, more focused on the output y, here the input x has to be taken into consideration.

The performance is still measured by a loss function: : (w, y) -→ (w, y). Looking at the data as random variables, the risk of any mapping f can be defined as:

R f := E x,y (f (x), y)
2.4. Regularization in a stochastic batch setting A first goal is then to construct from the sample (x i , y i ) i=1...N a random mapping f with a risk R f as small as possible: this is called the "prediction problem". We assume that the infimum of R f , f ∈ F is achieved at a function f * ("the oracle"). The "prediction problem" can then be re-written as trying to compute a mapping f such that the regret (also called excess risk) R f -R f * is small, either in expectation or with high probability. Even if some tools connect results with high probability to results in expectation (Markov's inequality) and conversely (expectation computed as the integral of the tail of the distribution), the results are not always the same, and some methods can be optimal only for results in expectation, or only for results with high probability. In Chapter 3, we focus on high-probability results.

As the best choice in F in terms of risk is f * , if the loss is continuous in its first argument, then a mapping f close to f * may have a small risk (although this depends on the setting and the data). This idea, or sometimes simply the need of further information about f * , leads to a second problem: the "estimation problem", which consists in computing an estimator f of f * (as close as possible in expectation or with high probability). Even if these two problems are generally different, in some setups they are equivalent (it is the case for the setup of Chapter 3).

Why a regularization is useful

As our aim is to minimize the expectation of the loss E x,y (f (x), y) , a natural method is to minimize the empirical average of the loss on the learning sample 1 N N i=1 (f (x i ), y i ). This approach is called the Empirical Risk Minimization (ERM) method (see, for example, [START_REF] Vapnik | Statistical learning theory[END_REF] and [START_REF] Koltchinskii | Oracle inequalities in empirical risk minimization and sparse recovery problems[END_REF]):

f ERM ∈ argmin f ∈F 1 N N i=1 (f (x i ), y i )
However, when the set F is large, then the ERM tends to "overfit", i.e., it sticks too much to the data, and thus it is driven by the data noise, wasting part of its predictive ability because of a lack of generalization. A way to tackle this issue is to add a term Ψ(f ), called the "regularization term", to the empirical quantity to minimize. This term is chosen to favour a subset of F , which has desirable properties. Formally, one can thus define the Regularized Empirical Risk Minimization (RERM) method:

f RERM ∈ argmin f ∈F 1 N N i=1 f (x i ), y i + Ψ(f )• (2.4.1)
This approach can be interesting for several reasons. First, pushing the prediction towards smaller subsets of F limits the overfitting issue. Second, a well-chosen Ψ, favouring suitable subsets of F in the choice of f , may also help. If some characteristics of f * are known (for instance, if one knows that f * is small for some given norm • ) then an adequate Ψ allows to focus on subsets of F with such characteristics. It can also be a way of "forcing" a desirable property of the estimator (e.g., sparsity) even if f * is not assumed to satisfy it.

Mathematical introduction

The influence of the regularization term Ψ can also be seen on the following property (that can not be taken as a definition, due to its recursive aspect): once defined by (2.4.1), f RERM satisfies:

f RERM ∈ argmin f :Ψ(f ) Ψ( f RERM ) 1 N N i=1 f (x i ), y i (2.4.2)
A possible analysis of this expression is that regularization does not change the nature of the minimization to be performed (it is still an empirical risk minimization), but it implicitly changes the set on which this minimization takes place. One of the key points of Chapter 3 will be to modify the LASSO algorithm to ensure that the set {f : Ψ(f ) Ψ( f RERM )}, unknown in advance, is a relevant set (actually, up to multiplicative constants, the minimal one containing f * ).

Sparsity and 1 norm

Let us focus on linear regression, in the finite-dimensional vectorial case: one wants to estimate

t * ∈ argmin t∈R d E x, t , y .
A desirable property of an estimator t is sparsity (i.e., the fact of having few non-zero components), at least when it is compatible with a good estimation of t * . It is particularly useful in high-dimensional setups, because it makes posterior computations easier. This property is also useful in variable selection and interpretation, separating the most impactful variables from those with lesser importance.

A natural way to proceed would be to use what is sometimes called the 0 "norm": the number of non-zero coefficients (it is not an actual norm, since homogeneity is lacking):

t 0 := # i : t i = 0
But . 0 is not a convex function, and the computation of the corresponding RERM is NP-hard (cf. Natarajan [1995]), thus computationally out of reach. So one can use a "convex relaxation" of this problem. Consider the " 0 ball" composed of vectors with only one nonzero component (and such that this component is bounded by, say, 1); then its convex hull is an 1 ball. 1 is the largest p norm (and therefore has the smallest unit ball) whose unit ball is convex (cf. Figure 2.1). The idea is then to use a multiple (selected by the statistician) of the 1 norm for the regularization: λ . 1 , where λ is to be chosen. In the following, we focus on the case of the square loss: (x, y) = (xy) 2 . In this case, the ERM is often called the "least squares estimator", and its 1 regularized version, already introduced in Section 2.2.3, is called the LASSO ("Least Absolute Shrinkage and Selection Operator"):

t LASSO ∈ argmin t∈R d 1 N N i=1 y i -x i , t 2 + λ t 1
As written above, dividing by N the empirical risk is only a question of convention. Property (2.4.2) becomes:

t LASSO ∈ argmin t∈ t LASSO 1 B d 1 1 N N i=1 y i -x i , t 2
where B d 1 is the unit ball of R d for the 1 norm. Based on this expression, one can see in Figure 2.2 a geometric heuristic explanation for the sparsity often induced by the LASSO estimator. The ellipses represent the level curves of the empirical risk t → N i=1 y ix i , t 2 /N , the ellipses in green contain at least one element in µB 2 1 , and the minimal value within these green level curves is attained at a vertex of µB 2 1 , which is therefore a sparse vector. The shape of other unit balls (for instance the 2 unit ball, which is "round") would make the estimators based on them much more unlikely to be sparse. The LASSO has been introduced in [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF] (as a constrained estimator instead of a RERM). Its good performance has made it used in many fields, e.g., genetics [START_REF] Lu | A Lasso regression model for the construction of microRNA-target regulatory networks[END_REF]), ecology [START_REF] Milns | Revealing ecological networks using Bayesian network inference algorithms[END_REF]), electro-chemistry [START_REF] Saccoccio | Optimal regularization in distribution of relaxation times applied to electrochemical impedance spectroscopy: Ridge and Lasso regression methods -a theoretical and experimental study[END_REF]), or econometrics [START_REF] Bai | Forecasting economic time series using targeted predictors[END_REF]).

t 1 t 2 ERM • RERM 1 ball: t : t 1 µ level curves of empirical risk
Other approaches than the ones studied in this thesis rely on the 1 norm, for instance as a way of selecting the vector with the smallest 1 norm among all vectors satisfying some 2. Mathematical introduction conditions. For example, in the case of an "overcomplete dictionary", with several vectors t satisfying y i = x i , t for all i, the "Basis pursuit" selects the one with the smallest 1 norm. This method, introduced in Chen et al. [2001], is particularly used in compressed sensing [START_REF] Foucart | A mathematical introduction to compressive sensing[END_REF]) and in fields related to signal decompositions. In a more classical statistical setup, a method called the "Dantzig selector" [START_REF] Candes | The Dantzig selector: Statistical estimation when p is much larger than n[END_REF]) selects some vectors based on the correlation between their residuals and the design matrix, and chooses eventually the selected vector with the smallest 1 norm.

From model selection to regularization: introducing complexity

We have seen that adding a regularization criterion is a way of favouring the elements of F that behave well with respect to this criterion. If one has defined a set {F m : m ∈ M} of models, all included in F , such that m∈M F m = F , one can apply a methodology more obviously "subset-oriented" in its spirit: Model Selection (cf. [START_REF] Massart | Concentration inequalities and model selection[END_REF]). It consists in computing an ERM inside each model F m :

f m ∈ argmin f ∈Fm 1 N N i=1 (f (x i ), y i )
and in attributing a penalty pen (F m ) to each model (this penalty generally increases with the complexity, and therefore often the size, of the model). The final model is selected by balancing the performance of the ERM and the penalty term:

m ∈ argmin m∈M 1 N N i=1 f m (x i ), y i + pen(F m ) (2.4.3)
The estimator is then the empirically best element in the chosen subset: f m . This two-step procedure can be re-written as a one-step regularization approach (see [START_REF] Lecué | Interplay between concentration, complexity and geometry in learning theory with applications to high dimensional data analysis[END_REF]). Define for each f ∈ F its minimal penalty: min pen(f ) := inf m:f ∈Fm pen(F m ). When this infimum is achieved at some model m, the corresponding model F m can be seen as the "best-fitted model" for f . Then, the next lemma [START_REF] Lecué | Interplay between concentration, complexity and geometry in learning theory with applications to high dimensional data analysis[END_REF]) shows that min pen plays the role of a regularization function: Lemma 2.11.

f m ∈ argmin f ∈F 1 N N i=1 (f (x i ), y i ) + min pen(f )
Proof. One can use a reductio ad absurdum and assume that there exists f 0 such that:

1 N N i=1 f m (x i ), y i + min pen( f m ) - 1 N N i=1 (f 0 (x i ), y i ) + min pen(f 0 ) > 0.
2.4. Regularization in a stochastic batch setting Denote by ∆ 0 this gap, and consider a model F m 0 such that f 0 ∈ F m 0 and pen(F m 0 )-min pen(f 0 ) < ∆ 0 . Then by definition of f m 0 :

1

N N i=1 f m 0 (x i ), y i 1 N N i=1 (f 0 (x i ), y i ) .
Moreover, one has min pen( f m ) = pen(F m ) (if not, considering m 1 such that f m ∈ F m 1 and pen(F m 1 ) < pen(F m ) would give an immediate contradiction with (2.4.3)). Therefore:

1

N N i=1 f m (x i ), y i + pen(F m ) - 1 N N i=1 f m 0 (x i ), y i + pen(F m 0 ) > 0.
This is a contradiction with the definition of m given in (2.4.3).

Conversely, any regularization approach can be seen as a model selection procedure. It suffices to take for models the inverse images of the intervals ] -∞, r], by the regularization function:

M = {m r , r ∈ R} with m r = Ψ -1 ] -∞, r] .
In other words, one has to define each model as the set of elements f such that Ψ(f ) is inferior to a given value. Then, attributing its canonical penalty pen(m r ) = r to any model m r and applying the selection model procedure described above, we recover f RERM (see Lemma 3.7.2 in [START_REF] Lecué | Interplay between concentration, complexity and geometry in learning theory with applications to high dimensional data analysis[END_REF] for more details).

A difference in the regularization approach and the selection model approach, is that regularization focuses on individual properties of the elements f ∈ F , whereas selection model allows to focus on global properties of entire sets. An important example (as far as overfitting is concerned) is the complexity of the models. There are various ways of measuring complexity of sets, the most suitable ways depending on the context and the data. For finite-dimensional subspaces, the complexity can be their dimension, leading to approaches like Mallows' C p [START_REF] Mallows | Some comments on Cp[END_REF], the AIC [START_REF] Akaike | A new look at the statistical model identification[END_REF]) or the BIC [START_REF] Schwarz | Estimating the dimension of a model[END_REF]). For bounded sets, we may cite the Sudakov complexity, the Talagrand γ-functional, or the Gaussian mean width. The latter, particularly adapted when facing Gaussian or subgaussian data, is defined, for any subset T of R d as:

* (T ) = E sup t∈T X, t
where X is a standard Gaussian random variable. Some links and comparisons can be drawn about these complexity measures (for instance, the majorizing measure theorem, or the Sudakov inequality, cf. [START_REF] Ledoux | Probability in Banach Spaces: isoperimetry and processes[END_REF]).

A mid-point between the global complexity of a set F , and individual properties of its elements, is the concept of localized complexity. It is defined as the complexity of the intersection of F and a ball (or a sphere) of a given norm and of arbitrary radius:

E sup t∈F ∩ρB X, t or E sup t∈F ∩ρS X, t
where B and S denote respectively the unit ball and the unit sphere for the norm of interest. These localized complexities may lead to a sharper analysis on the sets at stake. They will play a key role in Chapter 3, where we will choose the sets to be intersected with F in a way adapted and "tailored" to the problem at stake, allowing thus an optimal (in a sense that will be defined) regularization.

Chapter 3

Minimax regularization

This chapter is a joint work with Guillaume Lecué.

It has been submitted for publication.

Classical approach to regularization is to design norms enhancing smoothness or sparsity and then to use this norm or some power of this norm as a regularization function. The choice of the regularization function (for instance a power function) in terms of the norm is mostly dictated by computational purpose rather than theoretical considerations.

In this chapter, we design regularization functions that are motivated by theoretical arguments. To that end we introduce a concept of optimal regularization called "minimax regularization" and, as a proof of concept, we show how to construct such a regularization function for the d 1 norm for the random design setup. We develop a similar construction for the deterministic design setup. It appears that the resulting regularized procedures are different from the one used in the LASSO in both setups.

Introduction

Let (X , µ) be a probability space and (X, Y ) be a couple of random variables, in which X is distributed according to µ. One is given a sample of N independent couples (X i , Y i ) i=1..N distributed according to the joint law of (X, Y ). On the basis of this sample, one tries to link X and Y by a random mapping f with f (X) close (in L 2 ) to Y . This is the classical problem, in learning theory, of the prediction of an output Y from an input X given i.i.d. copies of the couple (X, Y ).

To that end, one is given a class F of functions from X to R and the aim in learning theory is to mimic the best element in F for the prediction of Y by a function of X in F . We assume that F is closed and convex in L 2 (µ) so that it exists a function f * that minimizes the square loss in F :

f * ∈ argmin f ∈F E (Y -f (X)) 2 . (3.1.1)
This function is usually called the oracle (cf. [START_REF] Nemirovski | Lectures on probability theory and statistics[END_REF]); it is the closest function in F to Y in L 2 . Now, the goal is to construct an estimator f whose L 2 (µ) distance to f * is as small as possible using the dataset {(X i , Y i ) : i = 1, . . . , N }. In the framework considered in this chapter, the excess risk of f , which is the difference µ) and so estimating f * is equivalent to predicting Y ; thus we fall back on the original prediction problem by estimating f * in L 2 (µ).

E(Y -f (X)) 2 -E(Y -f * (X)) 2 , is actually equal to f -f * 2 L 2 (
One may therefore try to bound the quadratic error f -f * L 2 (µ) either in expectation or in deviation with respect to the sample. In this work, we obtain upper bounds on the quadratic error that are valid in deviation, showing that the results are true for "most" samples rather than in average.

Given that we want to be close to a function f * minimizing f → E(Yf (X)) 2 over F , a natural candidate for this problem is the Empirical Risk Minimizer (ERM) also known as the "least squares estimator":

f ERM ∈ argmin f ∈F 1 N N i=1 (Y i -f (X i )) 2 .
(3.1.2)

Many works have been carried out for general classes (see, [START_REF] Koltchinskii | Oracle inequalities in empirical risk minimization and sparse recovery problems[END_REF], [START_REF] Massart | Concentration inequalities and model selection[END_REF], van [START_REF] Van De Geer | Applications of empirical process theory[END_REF], van der Vaart and Wellner [1996]) or on the vectorial case(see [START_REF] Stein | Inadmissibility of the usual estimator for the mean of a multivariate normal distribution[END_REF] for the famous Stein paradox, and [START_REF] Chatterjee | A new perspective on least squares under convex constraint[END_REF] for elements about the admissibility of the ERM). It appears that when F is too large (for instance the whole L 2 (µ) space), the ERM tends to "overfit". The understanding of this phenomenon has led to the introduction of "regularization methods" which were originally used to smooth estimators in order to overcome the "overfitting phenomena". Those procedures are nowadays used beyond their smoothing effect and, in particular, they are now extensively used in Statistics and learning theory for their "low-dimensional / sparsity inducing properties". At a high level description, those methods make a trade-of between an "adequation to the data term" and a "regularization term" and 3.1. Introduction their general form (for the quadratic loss) is

f ∈ argmin f ∈F 1 N N i=1 (Y i -f (X i )) 2 + Ψ(f ) (3.1.3)
where Ψ is a function usually called the regularization function.

The "adequation to the data term" can be constructed from any loss function; for the case of the quadratic loss, this term reads like

N -1 N i=1 (Y i -f (X i )) 2 .
As for the regularization term Ψ, several choices are possible, enabling to smooth the estimator, or to force a low-dimensional structure. It depends thus on the a priori knowledge one has on the data (and in particular on f * ), and on computational issues.

A first option is the Tikhonov / Ridge regularization:

f λ ∈ argmin f ∈H 1 N N i=1 (Y i -f (X i )) 2 + λ f 2 H (3.1.4)
where • H is a Hilbert norm. One expects in this case that f H reflects the smoothness of

f (for instance f H = f (0) + R |f (t)| 2 dt 1/2
) and that the oracle function f * has a small f * H norm.

In the finite but high dimensional vectorial case, one often wishes the estimator to be sparse, i.e., to have few non-zero components (for some well-designed basis, cf. [START_REF] Mallat | A wavelet tour of signal processing[END_REF]). This may come from the fact that the vector to be estimated is known in advance to be sparse; or that, in high-dimensional problems, it is computationally important not to have to manage a huge amount of non-zero coefficients.

Then, a natural way to address this question is to use a sparsity-inducing penalization: like the number of non-zero components of the vector, sometimes called the " 0 norm". Even though it is theoretically appealing (cf. [START_REF] Giraud | Introduction to high-dimensional statistics[END_REF]), it proves to be computationally intractable (actually NP-Hard, in general, cf. Natarajan [1995]). But for geometric reasons, another regularization is efficient to induce sparsity: the 1 norm (which can be see as the convex relaxation of the 0 norm on the unit ∞ -ball). The associated estimator is called the LASSO ("Least Absolute Shrinkage and Selection Operator", [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]):

t λ ∈ argmin t∈R d 1 N N i=1 (Y i -X i , t ) 2 + λ t 1 . (3.1.5)
In [START_REF] Emmanuel | Near-ideal model selection by 1 minimization[END_REF], it is emphasized that LASSO leads generally to sparsity, though giving some counter-examples in which it doesn't work well -in particular LASSO struggles when dealing with a data matrix with high correlations among the columns. To tackle this kind of issues, it is possible to "mix" regularizations: it is the principle of the "Elastic net method" [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]), which penalty is a combination of the 1 and 2 norms. All these methods rely on the choice of one (even two for the Elastic net) regularization parameter λ, fixed by the statistician on the basis of empirical methods such as cross-validation. It has to be chosen wisely in order to make the right trade-off between the two terms and thus to minimize the rate of convergence of the regularization procedure towards the oracle.

Regularization and Model Selection

At this point, it should be clear to the reader that choosing (or even designing) a specific regularization norm like • H or • 1 depends much on an a priori knowledge we have. But once this choice has been made, why would someone use the square of this norm in one case (as for the Tikhonov / Ridge regularization), or the norm itself (as for the LASSO) or some other power of this norm (cf. [START_REF] Rohde | Estimation of high-dimensional low-rank matrices[END_REF] for some examples) in other situations? In many cases, this choice is only made following some computational considerations.

In this work, we want to support the choice of regularization functions (given some norm) on theoretical arguments. To that end we will rely on the model selection theory and, in particular, on a key principle in model selection which is to design penalty functions that capture the "complexity" of a model in the most accurate way.

The right calibration of penalty functions has opened an important stream of researches since the work of [START_REF] Barron | Risk bounds for model selection via penalization[END_REF]. It has led many researchers to (re)think about the notion of complexity in statistics. In a nutshell, there are mainly three types of quantities that have been introduced to measure the statistical complexity of a statistical model: combinatorial (like the VC dimension, [START_REF] Vapnik | Statistical learning theory[END_REF]), metric (like the entropy) and random (like Gaussian mean width and Rademacher complexities, [START_REF] Bartlett | Empirical minimization[END_REF], [START_REF] Koltchinskii | Local Rademacher complexities and oracle inequalities in risk minimization[END_REF]). Penalty calibration has culminated with the notion of "minimal penalty" that are sharp penalty functions with exact constants (cf. for instance [START_REF] Birgé | Gaussian model selection[END_REF]) thanks to second order term analysis of the notion of complexity of a model.

In the present work, we want to put forward the idea that the "right" choice (from a theoretical point of view) of a regularization function may also follow from a careful study of the complexity of a specific family of models. To that end, the first argument is to look at regularization as a model selection problem for which one has to design a sharp penalty. This has been done for instance in Chapter 3.7 in [START_REF] Lecué | Interplay between concentration, complexity and geometry in learning theory with applications to high dimensional data analysis[END_REF]. In the particular case where one is given a norm • for regularization, then the associated regularized ERM is a penalized estimator associated to the sequence of embedded models (m r ) r 0 where for all r 0, m r = {f ∈ F : f r} and the right way to regularize is given by a function reg : f ∈ F → pen(m f ) where m f is the smallest model in (m r ) r 0 containing f (see [START_REF] Birgé | Gaussian model selection[END_REF]). This idea is a baseline of this work.

Before diving into further details about the way we suggest to construct a regularization function, let us precise what we expect from a good procedure, in particular how we evaluate that a regularization function is the "right" one, at least from a theoretical point of view. We therefore need to introduce a concept of optimality for regularized estimators. Once again we rely on the basics of model selection theory.

Model selection procedures have been used originally to construct adaptive estimators. For the model selection problem we want to solve, this adaptivity problem reads like selecting the smallest model in the family ({f ∈ F : f r}) r 0 containing f * which is obviously {f ∈ F : f f * }. Therefore, the adaptation problem we want to solve here is to construct a procedure which performance is as good as if we had been given the value f * in advance. In particular, an estimator achieving the minimax rate of convergence over the model {f ∈ F : f f * } would solve this adaptation problem. In what follows, we design regularization functions in order to meet this requirement, but before that, we clarify the 3.1. Introduction notion of minimax rate over a model for the type of deviation results we prove below.

To simplify the exposition, we will focus on a specific, though very classical and widelyused, framework: the vectorial case, i.e. when F = •, t : t ∈ T is a class of linear functionals from R d to R indexed by some subset T ⊂ R d , with Gaussian design, and Gaussian noise (with known variance σ 2 ). Definition 3.1.1. Let T ⊂ R d , X denote a standard Gaussian vector in R d and ξ be a centered real-valued Gaussian random variable with variance σ 2 , independent of X. For all t * ∈ T , define the random variable Y t * = X, t * + ξ and denote by Y T := {Y t * : t * ∈ T } the set of all such random variables.

Let t N be a statistics from

(R d × R) N to R d . Let 0 < δ N < 1 and ζ N > 0.
We say that t N performs with accuracy ζ N and confidence 1δ N relative to the set of targets

Y T , if for all Y t * ∈ Y T , with probability, w.r.t. to a sample D := {(X i , Y i ) : i = 1, • • • , N } of i.i.d. copies of (X, Y ), at least 1 -δ N , t N -t * 2 2 ζ N .
We say that R N is a minimax rate of convergence over T for the confidence 1δ N if the two following statements hold:

1. there exists a statistics t N which performs with accuracy R N and confidence 1δ N relative to the set of targets Y T 2. there exists an absolute constant g 0 > 0 such that if tN is a statistics which attains an accuracy ζ N with confidence 1δ N relative to the set of targets Y T then necessarily

ζ N g 0 R N .
In the following (cf. Theorem 3.1.3 below), we recall a result from [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] on the minimax rate of convergence over T = ρB d 1 , the unit d 1 -ball of radius ρ 0, for a constant confidence (i.e., for instance, δ N = 1/4). Note that classical minimax rates of convergence are usually given in expectation (cf. for instance Tsybakov [2009]). The main difference here with Definition 3.1.1 is that it is given for deviation results: the minimax rate R N may depend on the confidence parameter δ N (cf. [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF]).

In the present work, we are interested in procedures achieving the minimax rate of convergence over the model {t ∈ R d : t t * }. This provides a natural way to introduce a notion of optimality for the problem of designing regularization functions. Definition 3.1.2. Let • be a norm on R d , 0 < δ N < 1 and T ⊂ R d . Let us consider the following RERM for some function Ψ : R + → R:

t ∈ argmin t∈R d 1 N N i=1 (Y i -X i , t ) 2 + Ψ( t ) constructed from a sample D := {(X i , Y i ) : i = 1, • • • , N } of i.i.d. copies of (X, Y t * ) where Y t * = X, t * +ξ with X ∼ N (0, I d×d ), ξ ∼ N (0, σ 2 ) is independent of X and t * ∈ R d .
We say that Ψ is a minimax regularization function for the norm • and the confidence 1δ N over T , if there exists an absolute constant g 1 > 0 such that for all t * ∈ T , the RERM t is such that with P t * -probability at least

1 -δ N , t -t * 2 2 g 1 R t * 1 N , where R t * 1 N
is the minimax rate of convergence over {t ∈ R d : t t * } and P t * denotes the probability distribution of a N sample of i.i.d. copies of (X, Y t * ).

Minimax regularization

The aim of this work is to show that one can design minimax regularization functions by finding the right notion of complexity of the sequence of embedded models {t ∈ R d : t r} r 0 . Note however that there should be some situations where designing such an optimal regularization function would be impossible at some given confidence parameter δ N . In particular, such a situation should happen when the Empirical risk minimization (ERM) procedure over the "true model" {t ∈ R d : t t * } is not itself a minimax procedure over the model {t ∈ R d : t t * }. This happens for constant confidence bound (for instance, when δ N = 1/4) when there is a gap in Sudakov inequality (cf. [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] for more details). Nevertheless, in [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF], it is proved that for high confidence bounds (that is when δ N decays exponentially fast with the complexity of the model) ERM is always minimax over convex classes. It appears that for the case of d 1 -balls ERM is minimax for all confidence regime therefore this subtlety will not show up in this special case.

General approach provided in this chapter

Let us now present our approach. As we mentioned before, we want to construct a regularization function depending on the complexity of the models {t ∈ R d : t r} for all r 0. This leads to choose a RERM (in the vectorial case) having the following form:

t ∈ argmin t∈R d 1 N N i=1 (Y i -X i , t ) 2 + comp t B • (3.1.6)
where B • is the unit ball associated with the given regularization norm • and for all

t ∈ R d , t B • = {u ∈ R d : u t }.
The key feature in (3.1.6) is the "complexity function" r 0 → comp rB • which aims at measuring with the best possible accuracy the complexity of the models rB • for all r 0 from a statistical point if view. Or course, finding the right notion of complexity is paramount in this approach.

To aim at optimality, we advocate complexities that are tailored for the specific statistical problem at stake. It turns out that the "right" choice of complexity, and hence, of regularization, is linked to the behavior of two empirical processes. Those two empirical processes are ultimately connected to the two sources of statistical complexities in the considered problem. When estimating t * from the data (X i , Y i ) N i=1 there are two statistical issues:

• 1)(an inverse problem) t * is observed only through X, where X ∈ R N ×d is the operator whose rows vectors are given by the X i 's;

• 2)(noisy data) the observations have been corrupted by some noise ξ.

The action of the operator X on the models rB • for all r 0 plays a prominent role in our analysis. In particular, the size of the intersection of its kernel with the model is a natural minimax lower bound for any estimator since any two vectors in the kernel of X and the model are indistinguishable. The effect of the "distortion" of the operator X does not show up for small models (i.e. small values of r) because of the presence of the noise which blurs everything at small scales. But passing beyond some threshold for the signal-to-noise ratio r/σ, only the 3.1. Introduction distortion of X matters from a statistical point of view. This phenomenon occurs only when N d, because that is the regime where X has a non trivial kernel. On the contrary, in the low dimensional setup d N , X is well conditioned with high probability and therefore, there is no statistical complexity coming from the distortion of X since in that regime there is no such distortion. Controlling the distortion of X is a key issue in high-dimensional statistics. It is behind all classical properties like RIP (cf. [START_REF] Candès | The power of convex relaxation: near-optimal matrix completion[END_REF]) or REC (cf. Bickel et al. [2009]) and it will play equally a key role in our analysis. In particular, the d 2 diameter of the intersection of X with the model rB • will appear explicitly in the optimal regularization. Given that X is a standard Gaussian random matrix, this diameter will be the Gelfand width of rB • in our example (cf. [START_REF] Pinkus | n-widths in approximation theory[END_REF], Chapter 2 in [START_REF] Chafaï | Interactions between compressed sensing random matrices and high dimensional geometry[END_REF] or [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] for more details on Gelfand widths and their role in signal processing and learning theory).

Overview of the chapter; main results

As a proof of concept we present an example of the construction of a minimax regularization function in the popular set-up of regularization by the where X ∼ N (0, I d×d ) and ξ ∼ N (0, σ 2 ) are independent, centered Gaussian variables in R d and R respectively. As written before, a dataset (X i , Y i ) N i=1 of N i.i.d. copies of the couple (X, Y ) is provided and one wants to use it to estimate t * .

Note that we choose a Gaussian random design to make the exposition as simple as possible. The results can be extended to more general sub-Gaussian designs. Nonetheless, our goal is not to provide general results but to show that the approach we present allows to achieve minimax regularization in some classical set-up. Moreover, we want to see the effect of the random design on the construction of a minimax regularization.

In the supplementary material 3.4, we consider a fixed design setup, in which one still has for all i: Y i = X i , t * + ξ i with ξ i ∼ N (0, σ 2 ) i.i.d., but with X i deterministic, satisfying an "isomorphic property" on "compressible vectors", equivalent to the RIP from [START_REF] Candès | The power of convex relaxation: near-optimal matrix completion[END_REF]. We will see that under this property, the arguments and the results will be quite similar to the random design case.

We will not be interested in getting optimal or sharp numerical constants, and some of the inequalities and coefficients in the arguments will be rather loose from this point of view, but what actually matters is that the quantity will have the right order of magnitude w.r.t. N, d, σ and t * 1 .

As for our choice to consider regularizations that are functions of the d 1 norm, its motivation is that the d 1 -norm has been one of the most studied regularization norm since the beginning of high-dimensional statistics, in particular for the reasons presented in this Introduction. Moreover, as mentioned previously, the ERM over d 1 -balls is minimax for every confidence 1δ N (cf. [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF]), this makes the construction of minimax regularization possible for different deviation parameters and this makes the exposition also simpler.

Minimax regularization

Let the choice of the d 1 -norm as a regularization norm be made once and for all. Now, the problem we want to solve is to construct a regularization function Ψ : R → R such that the regularized procedure

t Ψ ∈ argmin t∈R d 1 N N i=1 (Y i -X i , t ) 2 + Ψ( t 1 ) (3.1.8)
achieves the minimax rate of convergence over t * 1 B d 1 given N i.i.d. data (X i , Y i ), i = 1, . . . , N distributed according to (3.1.7). And, we want t Ψ to satisfy that property whatever t * ∈ R d is.

Now, let us explain the strategy we use to design a minimax regularization function. We denote for all ρ 0 and r 0,

ρB d 1 = {t ∈ R d : t 1 ρ}, and rB d 2 = {t ∈ R d : t 2 r}.
The starting point to our approach is that t Ψ minimizes t → P N L Ψ t over R d , where, for every t ∈ R d ,

P N L Ψ t := 1 N N i=1 (Y i -X i , t ) 2 + Ψ( t 1 ) - 1 N N i=1 (Y i -X i , t * ) 2 + Ψ( t * 1 ) , in particular, P N L Ψ t P N L Ψ t * = 0. So if one shows that P N L Ψ t > 0 for all t 1
t * 1 then this will prove that t Ψ 1 t * 1 , proving that t Ψ belongs to the right model. This will be essentially the main step since for the correct choice of Ψ, we will show that the regularization has no effect within the right model and that the RERM t Ψ has essentially the same statistical behavior as the ERM in t * 1 B d 1 which is known to be minimax. We will therefore conclude that t Ψ can learn t * at the minimax rate of convergence within the model t * 1 B d 1 without knowing in advance the radius t * 1 .

Using the quadratic / multiplier decomposition as in [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF], [START_REF] Saumard | Optimal upper and lower bounds for the true and empirical excess risks in heteroscedastic least-squares regression[END_REF][START_REF] Saumard | A concentration inequality for the excess risk in least-squares regression with random design and heteroscedastic noise[END_REF], one can write P N L Ψ t as the sum of three terms:

P N L Ψ t = P N Q t-t * + P N M t-t * + R t,t * where • P N Q t-t * := N i=1 X i , t * -X i , t
2 /N is the "quadratic process"

• P N M t-t * := 2 N i=1 Y i -X i , t * X i , t * -X i , t /N is the "multiplier process" • R t,t * := Ψ( t 1 ) -Ψ( t * 1 ) is the regularization part.
The definition of our complexity will thus be a consequence of the study of the behavior of the quadratic and multiplier empirical processes indexed by t ∈ ρB d 1 for all ρ 0. The two processes are associated to the two statistical complexities previously discussed:

• 1) the quadratic process can be written as

P N Q t-t * = X(t -t * ) 2
2 and is well behaved (i.e. of the order of tt * 2

2 ) when X is well conditioned;

• 2) the multiplier process is measuring the statistical complexity coming from the noise ξ = Y -X, t * , P N M t-t * is the empirical correlation between the noise and the model shifted by •, t * .

Introduction

All the game is now to identify regions of the space R d where the statistical complexity comes from the distortion of X or from the noise. This drives the construction of the optimal regularization function Ψ.

In order to identify those regions, note that for every fixed t ∈ R d , the distribution of these two processes depend on tt * only by its d 2 -norm tt * 2 , in two different ways: P N Q t-t * in a quadratic way, P N M t-t * in a linear way. So it is natural to partition the model ρB d

1 into vectors with "small" 2 norm -i.e. the intersection of ρB d 1 ∩ rB d 2 for an adequate radius r -and vectors of ρB d 1 with d 2 -norm larger than r. We will see that outside rB d 2 , with high probability the two processes are "well-behaved" and regularization is unnecessary; but inside rB d 2 it is not the case, the operator X may have a kernel and the noise is making the estimation hard: hence, this is where the regularization will be needed to keep control of the situation and this is precisely where the regularization function is designed. In that case, either the statistical complexity comes from the size of the intersection of the kernel of X with ρB d 1 and therefore one needs to take Ψ(ρ) of the order of this diameter (which appears to be equal to the Gelfand width of ρB d 1 to the square) or the statistical complexity comes from the noise and then Ψ(ρ) is of the order of the oscillations of the multiplier process inside

ρB d 1 ∩ rB d 2 .
The choice of the "adequate radius" r is of course paramount in our approach. It results from the right understanding of the two previously discussed sources of statistical complexities: the bigger these complexities, the bigger this radius (since, as we mentioned, outside rB d 2 the processes are well-behaved). First, we want to identify the smallest d 2 radius r Q (ρ) above which X is well-behaved in ρB d 1 , i.e. such that for every

t ∈ ρB d 1 , if t -t * 2 r Q (ρ), then P N Q t-t * = X(t -t * ) 2 2 ∼ t -t * 2 2
. Then, we need to identify the smallest d 2 -radius r M (ρ) above which the effect of the noise is below the signal intensity that is above which one can clearly identify if t = t * when tt * 2 r M (ρ). To that end we want to make the oscillations of the multiplier process smaller than the one of the quadratic process, which is of the order of tt * 2 2 when tt * 2 r Q (ρ). It will appear that, in our framework, the two radii obtained from the above trade-offs are solution of fixed point equations for all ρ 0: for some absolute constants Q and η (to be chosen later):

• the "quadratic fixed point" is r Q (ρ) := inf r > 0 : * (ρB d 1 ∩ rB d 2 ) = Qr √ N • the "multiplier fixed point" is r M (ρ) := inf r > 0 : σ * (ρB d 1 ∩ rB d 2 ) = ηr 2 √ N where * (ρB d 1 ∩ rB d 2 ) is the Gaussian mean width of the localized set ρB d 1 ∩ rB d 2 defined as * (ρB d 1 ∩ rB d 2 ) = E sup t∈ρB d 1 ∩rB d 2 G, t
where G is a standard Gaussian vector in R d . As our framework involves "Gaussian randomness" in both the design and the noise, it is not surprising that the Gaussian mean width arise when dealing with the control of the two processes. However, Gaussian mean widths appear in learning theory, statistics and signal processing way beyond the "full Gaussian framework" as considered here (see, for instance, [START_REF] Lecué | Sparse recovery under weak moment assumptions[END_REF]).

Minimax regularization

These two fixed points have been introduced in Lecué and [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] and used later in [START_REF] Lecué | Regularization and the small-ball method I: sparse recovery[END_REF] for the study or ERM and RERM. As their names suggest, the quadratic fixed point will be used to control the quadratic process, and the multiplier fixed point to control the multiplier process. Their general definitions use an inequality rather than an equality inside the infimum: r Q (ρ) is defined as inf r > 0 :

* (ρB d 1 ∩ rB d 2 ) Qr √ N and r M (ρ) as inf r > 0 : σ * (ρB d 1 ∩rB d 2 ) ηr 2 √
N . This allows to deal with infinite-dimensional set-ups in which the mapping r → * (F ∩ rB 2 ) is not necessarily continuous. But in our case, this mapping is continuous and the infimum is attained in a point for which there is exact equality.

It appears that one can provide an explicit formulation for the two fixed point r Q (ρ) and r M (ρ) in many situations and, in particular, in the case of the d 1 -norm (cf. [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF]): for some absolute constants C

(1)

M , C (2)
M , C

(1)

Q , C (2)
Q and ζ < 1 < ζ , for all ρ, there exists C M ∈ [C (1) M , C (2)
M ] such that:

r 2 M (ρ) = C M                  σ 2 d N if ρ 2 N σ 2 d 2 ρσ 1 N log eσd ρ √ N if σ 2 log d ρ 2 N σ 2 d 2 ρσ log(ed) N if ρ 2 N σ 2 log d.
(3.1.9) and there exists

C Q ∈ [C (1) Q , C (2)
Q ] such that

r 2 Q (ρ) = C Q 0 if N ζ d ρ 2 N log ed N if N ζd. (3.1.10)
Note that when ζd N ζ d, r Q (ρ) decays from (ρ 2 /N ) log(ed/N ) to 0 and one only has an upper estimate on r Q (ρ) given by C Q ρ 2 /N . We will therefore not consider this case in the following since it involves to deal with sharp estimates on the spectra of squared or approximatively squared Gaussian random matrices. Note also that C M and C Q may depend on ρ, N, d, σ but they are both controlled from above and below by absolute constants (independent of ρ, N, d and σ).

Now that we have a way to measure the statistical complexity of a model we need one more thing before turning to the effective construction of a minimax regularization for the d 1 -norm: we need to know the minimax rate of convergence over d 1 -ball ρB d 1 for all ρ 0. We will see below that one way to measure the statistical complexity of a model is closely related to its minimax rate. To that end, we summarize the main results in the constant deviation case δ N = 1/4 from section 4.1 in [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] in the following theorem.

         ρ 2 if ρ 2 N σ 2 log d, ρσ 1 N log ed 2 σ 2 ρ 2 N if σ 2 log d ρ 2 N σ 2 N 2 log(ed/N ) , ρ 2 N log ed N if ρ 2 N σ 2 N 2 log(ed/N ) 3. when N ζ d, by          ρ 2 if ρ 2 N σ 2 log d, ρσ 1 N log ed 2 σ 2 ρ 2 N if σ 2 log d ρ 2 N σ 2 d 2 , σ 2 d N if ρ 2 N σ 2 d 2 .
In other words, for all ρ 0 and t Note that we have decided to present the result in the constant deviation result (that is for δ N = 1/4) whereas it is actually true with a much better probability estimate in section 4.1 in [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF]. We will also obtain our main results with an exponentially large deviation below.

* ∈ ρB d 1 , the ERM t ERM ρ ∈ argmin t∈ρB d 1 N i=1 (Y i -X i , t ) 2 , is such that, with probability at least 3/4, t ERM ρ -t * 2
As mentioned previously, when N ∈ [ζd, ζ d], we only have an upper bound on (r Q (ρ)) 2 that does not match the minimax lower bound. As a consequence, the N ∼ d regime is not considered in Theorem 3.1.3. Notable is that the rate ρ 2 is the trivial rate obtained by taking the d 2 diameter of the model ρB d 1 which is simply 2ρ. Therefore, any statistics tN (like the ERM t ERM ρ ) taking its values in ρB d 1 satisfies with probability 1, tNt * 2 2 4ρ 2 for all t * ∈ ρB d 1 . This is a trivial bound that one can get for free as long as the radius ρ is known. However, for the construction of an optimal regularization function which can be seen as an adaptation to the radius t * 1 , which is therefore not known, this trivial bound is not available. This will be an issue for designing a minimax regularization function when t * 1 is unknown and small (actually smaller than σ log(ed)/N ). Somehow the "signal-to-noise ratio" is too small for the models ρB d 1 with small ρ's. Therefore, the trivial upper bound ρ 2 is optimal when ρ is known but in the other case we will have to pay the price due to the noise and there will be no way to achieve the trivial optimal ρ 2 bound for small ρ's (except for the trivial estimator t 0 = 0, see the discussion after Proposition 3.1.6). That is the reason why we will not be able to construct a minimax regularization function over the entire space R d but only for t * such that t * 1 σ/ log(ed)/N . We will also show that such a construction of an optimal regularization function over the entire space R d is actually not possible at all later in Proposition 3.1.6.

Minimax regularization

Finally let us turn to the construction of a minimax regularization function for the d 1norm. To that end we will use the function ρ 0 → r 2 (ρ) = max r 2 Q (ρ), r 2 M (ρ) as a sharp way to measure the complexity of the model ρB d

1 . The main result of this article is that this function is a minimax regularization function as introduced in Definition 3.1.2. where

r(ρ) = max r Q (ρ), r M (ρ) for r Q (ρ) = inf r > 0 : * (ρB d 1 ∩ rB d 2 ) = Qr √ N and r M (ρ) = inf r > 0 : σ * (ρB d 1 ∩ rB d 2 ) = ηr 2 √ N denoting by * (ρB d 1 ∩ rB d 2 )
the Gaussian mean width of the localized sets

ρB d 1 ∩ rB d 2 .
In that case, the rate achieved by the RERM t Ψ is the minimax rate r 2 ( t * 1 ) when t * 1 ∆ 0 σ log(ed)/N .

The shape of the minimax regularization function ρ → Ψ(ρ) = c 0 r 2 (ρ) is given in Figure 3.1 in the two cases N ζd ("high-dimensional statistics") and N ζ d ("classical or lowdimensional statistics"). The only difference between the two cases (ζ d N or ζd N ) appears for large radii ρ. The reason for that lies in the statistical complexity coming (or not) from the distortion of the operator X. In the low-dimensional case, X is such that (with high probability), Xt 2 ∼ t 2 for all t ∈ R d . There is no distortion coming from X. Somehow observing Xt * is the same as 3.1. Introduction observing t * itself, one just has to invert X -this can be done because X acts like an isomorphy on the entire space R d . Therefore, there is no statistical complexity coming from X and so its associated complexity parameter r Q (•) does not show up in the final complexity parameter r(•) = max(r M (•), r Q (•)). We therefore end up with r(•) = r M (•) in the low-dimensional case. In particular, for large radii ρ (for which, one has

σ log(d)/N σ N/log(d/N) ρ r 2 (ρ) ρσ log(ed) N ρσ 1 N log eσd ρ N ρ 2 N log ed N N ζd σ log(d)/N σd/ N ρ r 2 (ρ) ρσ log(ed) N ρσ 1 N log eσd ρ N σ 2 d/N N ≥ ζ d
ρB d 1 ∩ r M (ρ)B d 2 = r M (ρ)B d 2 )
, we pay the worst rate of convergence in R d , which is σ 2 d/N because learning over ρB d 1 for large values of ρ is as hard as learning over the entire space R d and the price for the latter is the rate σ 2 d/N .

The situation is totally different in the high-dimensional setup because in that case the operator X ∈ R N ×d has a non-trivial kernel; therefore, observing Xt * is totally different from observing t * (for instance, imaging that t * ∈ ker X). This adds to the statistical complexity of the problem of estimating t * . In this regime, both the noise and the distortion effect of X appear in the statistical complexity of the estimation problem; this means that both complexity parameter r Q (•) and r M (•) appear in the total complexity parameter r(•) and therefore in the ultimately designed minimax regularization function. For small values of ρ, the effect of the noise is predominant but for large values of ρ this is the effect of X which is the main responsible of the statistical complexity. In particular, the d 2 -diameter of ker

X∩ t * 1 B d 1 is important because there is of course no way to distinguish t * from t * + h for all h ∈ R d such that Xt * = X(t * + h) that is for all h ∈ ker X such that t * + h ∈ t * 1 B d 1 .
Hence, estimating t * is at least as hard as estimating any point in (t * + ker X) ∩ t * B d 1 and therefore, no estimator t can estimate t * at a rate better than diam ker

X ∩ t * 1 B d 1 , d 2 2
. The latter quantity is itself lower bounded by the Gelfand's N -width of t * B d 1 defined as

c N ( t * 1 B d 1 ) := inf diam(ker Γ ∩ t * 1 B d 1 ) : Γ ∈ R N ×d ∼ t * min 1,
log(ed/N ) N (3.1.12) the latter result is due to Garanaev and Gluskin Garnaev and Gluskin [1984]. It appears that the Gelfand's N -width of t * B d 1 are achieved (up to absolute constants and with high probability) by the kernel of standard N × d Gaussian matrices, which is exactly the case of the design matrix X. Therefore, with high probability, diam ker

X ∩ t * 1 B d 1 , d 2 2 ∼ c 2 N ( t * 1 B d 1 ) ∼ t * 2 1 N log ed N ∼ r 2 Q ( t * 1 ) (3.1.13)
This is exactly the price we pay in r Q (ρ) when ρ σ N/ log(ed/N ). That is the reason why we take the regularization function Ψ(ρ) of the order of the Gelfand's N -width of ρB d 1 (to the square) for large radii ρ: it is the right concept of statistical complexity that shows up in this part of the space R d , where the statistical complexity coming from the distortion of X becomes more important than the one due to the noise.

Minimax regularization

This complexity is obtained by simply removing the localization (i.e. the intersection with rB d 2 ) in the multiplier process when computing r M (•), and does not take r Q (•) into account. This means that the distortion of the operator X is assumed to have no effect on the statistical complexity of the problem. This is why estimation results for the LASSO deal only with the reconstruction of vectors which are sparse or almost sparse, i.e. for vectors belonging to the cone appearing in the RE or CC conditions, cf. [START_REF] Peter | Simultaneous analysis of Lasso and Dantzig selector[END_REF]. Over this cone, the quadratic process behaves nicely (that is, the isomorphic property from Proposition 3.2.1 holds on this cone) or in other words, the operator X is well-conditioned on the set of vectors we want to reconstruct, so that there is no statistical complexity coming from the distortion of this operator. So, as long as estimation of sparse or approximately sparse vectors is concerned, there is no need for the complexity function r Q (•). That is why the regularization function used for the LASSO take into account only the fixed point r M (•) associated to the statistical complexity due to the noise and not the one from the inverse problem. On the contrary, by taking r M (•) and r Q (•) into account, our regularization function allows us to deal with the full space R d (except for a small d 1 -ball centered in 0, cf. Proposition 3.1.6) and not only a cone. Moreover, as said before, the way the regularization function is designed in (3.1.14) is suboptimal because it uses a trivial upper bound on r M (•) instead of using the exact formulation of r M (•) as in (3.1.9). Contrary to the LASSO, this latter exact formulation takes into account, thanks to the localization, the fact that the regularization is not needed on the whole space -in some areas the random processes behave nicely whatever. The suboptimal approach for the LASSO is likely to be responsible for a loss in the rate of convergence achieved by the LASSO, which is sσ 2 log(ed)/N whereas the minimax rate is sσ 2 log(ed/s)/N (cf. [START_REF] Bellec | Slope meets Lasso: improved oracle bounds and optimality[END_REF]). This is not a big loss, especially when d >> s, but from a purely theoretical point of view the right way to regularize for the reconstruction of sparse vectors should be using r 2 M ( t 1 ) instead of σ t 1 log(ed)/N as it is the case for the LASSO. However, the resulting regularization function would be concave (cf. the right-hand side plot in Figure 3.1). Therefore, the small price paid from a theoretical point of view by using the trivial upper bound in (3.1.14) seems to be worth the computational gain obtained by using a convex regularization as does the LASSO.

Let us now turn to the adaptation problem in the ball ρB d 1 for ρ ∼ σ log(ed)/N . We want to answer the following question: is it possible to construct a regularization function Ψ(•) so that the associated regularized procedure t Ψ is adaptive on the entire space R d ? Or (even stronger) is there any statistic that can be adaptive (in the sense that it achieves the rate of the ERM on t * 1 B d 1 without knowing t * 1 beforehand) on the entire space R d (this statistics may not be a regularized procedure)? It appears that the answer to this question is negative, which we prove in the following proposition.

However, one needs to be cautious with the next statement because there is a trivial estimator t 0 = 0 such that for every t * ∈ R d , with probability 1, t 0t * 2 2 = t * 2 2 and therefore t 0 is adaptive on ρB d 1 as long as the minimax rate over ρB d 1 is ρ 2 , which is the case for any ρ σ log(ed)/N . Therefore, there exists a procedure adaptive on ρB d 1 when ρ ∼ σ log(ed)/N . Moreover, according to Theorem 3.1.4, there exists a procedure adaptive on R d \ρB d

1 . But the question concerns the adaptation on the entire space R d at the same time.

The following statement shows that if t is a procedure adaptive on R d \ρB d 1 then it cannot 3.2. Proof of Theorem 3.1.4 be adaptive on ρB d 1 for ρ ∼ σ log(ed)/N . Moreover, it also proves that adaptation on the entire space R d is not possible and that Theorem 3.1.4 is optimal given that the range of radii [∆ 0 σ log(ed)/N , +∞) on which it is adaptive cannot be inflated (up to absolute constants). Before turning to the statement let us denote by P t * the probability distribution of a N -sample (X i , Y i ) N i=1 of i.i.d. copies of (X, Y ) when (X, Y ) is distributed according to (3.1.7).

Proposition 3.1.6. Assume that 2d exp(544/225) and that there exists an absolute constant χ 1 such that the following holds. Let ρ 2σ (log(2d))/(96N ) be such that 16χ 1 r 2 (ρ) ρ 2 and denote by (e j ) d j=1 the canonical basis of R d . Assume that t is an estimator such that for every t * ∈ {±ρe 1 , . . . , ±ρe d },

P t * t -t * 2 2 χ 1 r 2 (ρ) 3 4 .
Then, for every t * ∈ (ρ/2)B d 1 ,

P t * t -t * 2 2 ρ 2 /16 1 2 . (3.1.15)
The proof of Proposition 3.1.6 is given in Section 3.3. Note that the only property of the design X used to prove Proposition 3.1.6 is isotropicity. Since isotropicity does not tell much on the distortion properties of the design matrix X, it means that Proposition 3.1.6 is only based on the statistical complexity coming from the noise. This is not a surprise given that Proposition 3.1.6 is a result for very small radii less than ∼ σ log(ed)/N . At that scale, even if ker X is in the worst possible position, i.e. diam(ker

X ∩ ρB 2 1 , d 2 ) 2 = diam(ρB 2 1 , d 2 ) 2 = ρ 2 , we still have ρ 2 r 2
M (ρ). Hence, the distortion of X does not play any role at this very small scale and therefore that is not a surprise that Proposition 3.1.6 is true for any isotropic design X.

Finally, let us rephrase Proposition 3.1.6 in other words. Proposition 3.1.6 shows that if a procedure can learn all vectors in {±ρe 1 , . . . , ±ρe d } at the minimax rate r 2 (ρ) then this estimator cannot learn any t * ∈ (ρ/2)B d 1 at the optimal minimax rate ρ 2 for confidence 1/4. For instance, given that the result (3.1.15) holds for any t * ∈ (ρ/2)B d 1 , in particular, for t * = 0, it tells that t cannot estimate t * = 0 at a rate better than ρ 2 ∼ σ 2 log(ed)/N whereas the minimax rate over ρ * B d 1 for ρ * = 0 is obviously 0. Finally, note that the condition 16χ 1 r 2 (ρ) ρ 2 implies that ρ σ log(ed)/N so that the phase transition radius above which adaptation is possible but not below is of the order of σ log(ed)/N which is the radius we have found in Theorem 3.1.4.

Proof of Theorem 3.1.4

Most of the proof consists in showing that with high probability, t belongs to t * + ρ * B d 1 where

ρ * = max 10, 8 (C (2) M ) 2 (C (1) M ) 2 η + 1 t * 1 .
(3.2.1)

Minimax regularization

(where the value of η will be fixed later). Once this goal is achieved, it is straightforward to show (again with high probability) that tt * 2 2 is less than the minimax rate of convergence over ρ * B d 1 . To do so, we will prove that with high probability, any t outside t * + ρ * B d 1 satisfies P N L Ψ t > 0 (whereas

P N L Ψ t 0). We partition R d \(t * + ρ * B d 1 ) into shelves of the form t * + (2 j+1 ρ * B d 1 \(2 j ρ * )B d 1 )
, in which the regularization function remains mostly constant. We only need to study the smallest shells, i.e. for k = 1, . . . , K 0 for some well-chosen K 0 (K 0 is the smallest integer so that 2

K 0 -1 ρ * B d 1 ∩ r(2 K 0 -1 ρ * )B d 2 = r(2 K 0 -1 ρ * )B d 2 )
, the part of R d for which t 1 2 K 0 ρ * will be treated by an homogeneity argument.

On each of the smallest shelves, the argument is roughly and heuristically the following: we place ourselves on a high probability event on which random processes (P N M, P N Q, their supremum, their infimum,. . . ) "behave nicely" (i.e. they both scale in a favourable way with respect to tt * 2

2 ). Then, as P N M t-t * is the only possibly negative term, it suffices to identify zones where P N Q t-t * > P N M t-t * (then directly P N L Ψ t > 0), and compensate |P N M t-t * | on the other part by using a penalty that is close to the supremum (on this other part) of P N M t-t * . As P N Q t-t * grows quicker than P N M t-t * with respect to tt * 2 , the big zone where P N Q t-t * > P N M t-t * will be the exterior of t * + rB d 2 for an adequate r (cf. Figure 3.2). This r must be such that any t in the exterior of this ball satisfies

P N Q t-t * t -t * 2 2
P N M t-t * , and we will see that the first inequality amounts to r r Q (ρ), and the second one to r r M (ρ). Next, the supremum of

P N M t-t * on ρB d 1 ∩ r(ρ)B d 2 is less than r M (ρ)r(ρ) r 2 (ρ) for r(ρ) = max(r M (ρ), r Q (ρ))
. We therefore set the regularization function Ψ(ρ) at level ρ to be proportional to the quantity r 2 (ρ) because it is this quantity measuring the amplitude of the oscillation of the multiplier process in ρB d 1 ∩ r(ρ)B d 2 . As for its presentation, the proof of Theorem 3.1.4 is divided into two parts. The first part (Section 3.2.1) defines the event on which the two processes "behave nicely" and computes a lower bound on its probability. In the second part (Section 3.2.2) we will place ourselves on this event and carry out the deterministic geometric part of the argument.

Probabilistic control of the processes

Instead of controlling the two processes on shelves, we will control them on the full 1 balls, because it does not change the complexity, up to constants, and the very last step of the proof requires a control on the two processes on the full 1 ball ρ * B d 1 .

Control of the quadratic process

This first section provides the classical analysis of the quadratic process based upon its isomorphic properties on the set of "almost sparse vectors". Such a property holds in the optimal regime of observation (or the optimal size of the cone of "almost sparse vectors"), only in the sub-Gaussian case. It is the case we are considering here since we assumed that the design is a standard Gaussian random variable. This analysis borrows some ideas from the "isomorphic method" from [START_REF] Bartlett | Empirical minimization[END_REF] or the Restricted Isometry Property from [START_REF] Candès | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[END_REF] in the sub-Gaussian case. For the sake of completeness we recall here the argument from [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF].

3.2. Proof of Theorem 3.1.4

Proposition 3.2.1. There are absolute constants C 1 and C 1 such that the following holds. Let X 1 , . . . , X N be N i.i.d. standard Gaussian vectors in R d . Denote by Ω * the event on which: for every ρ ρ * and all

t ∈ t * + ρB d 1 , if t -t * 2 r Q (ρ) then 1 2 t -t * 2 2 1 N N i=1 X i , t -t * 2 3 2 t -t * 2 2 . (3.2.2)
Then, one has

P[Ω * ] 1 -2 exp(-C 1 Q 2 N ) as long as Q C 1 .
Proof. First note that for all ρ > 0, r Q (ρ) = ρr Q (1). Indeed, we have

* (ρB d 1 ∩ rB d 2 ) = * ρ(B d 1 ∩ (r/ρ)B d 2 ) = ρ * B d 1 ∩ (r/ρ)B d 2 and so r Q (ρ) = inf{r > 0 : * (ρB d 1 ∩ rB d 2 ) = Q √ N r} = inf{r > 0 : * (B d 1 ∩ (r/ρ)B d 2 ) = Q √ N (r/ρ)} = ρ inf{r > 0 : l * (B d 1 ∩ rB d 2 ) = Q √ N r} = ρr Q (1). (3.2.3)
For all ρ > 0, define the event Ω(ρ) on which one has for all

t ∈ t * + ρB d 1 , if t -t * 2 r Q (ρ) then 1 2 t -t * 2 2 1 N N i=1 X i , t -t * 2 3 2 t -t * 2 2 .
Let us show that if Ω(ρ * ) holds then for any ρ ρ * , Ω(ρ) holds as well. Assume that Ω(ρ * ) holds. Consider t ∈ t * + ρB d 1 such that tt * 2 > r Q (ρ) and define

t := t * + (ρ * /ρ)(t -t * ) ∈ t * + ρ * B d 1 . It follows from (3.2.3) that r Q (ρ * ) = (ρ * /ρ)r Q (ρ). Thus t -t * 2 = (ρ * /ρ) t -t * 2 > (ρ * /ρ)r Q (ρ) = r Q (ρ * ), and since Ω(ρ * ) holds, it follows that t -t * 2 2 /2 N i=1 X i , t - t * 2 /N 3 t -t * 2 2 /2. This implies that t -t * 2 2 /2 N i=1 X i , t -t * 2 /N 3 t -t * 2 2 /2 so Ω(ρ) holds.
As a conclusion, Ω * = Ω(ρ * ) and we can now lower bound the probability that this event holds.

Let us consider the class of linear functions

F = •, t -t * , t ∈ t * + ρ * B d 1 ∩ r Q (ρ * )S d-1 2 = •, t , t ∈ ρ * B d 1 ∩ r Q (ρ * )S d-1 2 .
We assume that F is non empty (if F = ∅ then the theorem is trivially satisfied). It follows from Theorem 1.12 in [START_REF] Mendelson | Upper bounds on product and multiplier empirical processes[END_REF] that for any x > 0, with probability at least 1

-2 exp -C 1 min(x 2 , x √ N ) , sup f ∈F 1 N N i=1 f 2 (X i ) -Ef 2 (X) C 2 ∆γ √ N + γ 2 N + x∆ 2 √ N
where ∆ is the diameter in ψ 2 of F and γ is Talagrand's γ 2 functional of F w.r.t. ψ 2 . Note that since X is a standard Gaussian variable in R d , for any t ∈ R d , X, t ψ 2 = C 3 t 2 for some absolute constant C 3 . It follows that

∆ = 2 sup t∈ρ * B d 1 ∩r Q (ρ * )S d-1 2 C 3 t 2 = 2C 3 r Q (ρ * ) and γ = γ 2 (ρ * B d 1 ∩ r Q (ρ * )S d-1 2 , d 2 ).
Moreover, it follows from the Majorizing measure theorem (cf. Chapter 1 in [START_REF] Talagrand | Upper and lower bounds for stochastic processes[END_REF])

that γ 2 (ρ * B d 1 ∩ r Q (ρ * )S d-1 2 , d 2 ) C 4 * (ρ * B d 1 ∩ r Q (ρ * )S d-1 2 )
Since F is non-empty, by Lemma 3.3.1 the right-hand side is equal to

C 4 * (ρ * B d 1 ∩r Q (ρ * )B d-1 2 ) and so by definition of r Q (ρ * ), one has γ C 4 Qr Q (ρ * ) √ N . Since X is isotropic (i.e. for any t ∈ R d , E X, t 2 = t 2 2 ), we obtain for x = Q √ N that, with probability greater than 1-2 exp(-C 1 min(Q, Q 2 )N ), for any t ∈ t * +ρ * B d 1 ∩r Q (ρ * )S d-1 2 , 1 N N i=1 X i , t -t * 2 -t -t * 2 2 (2C 2 C 3 C 4 Q + C 2 C 2 4 Q 2 + 4C 2 QC 2 3 )r 2 Q (ρ * )
So, as long as long as:

Q C 1 := min{1, (12C 2 C 3 C 4 ) -1 , ( √ 6C 2 C 4 ) -1 , (24C 2 C 2 3 ) -1 }, one has, with probability greater than 1 -2 exp(-C 1 Q 2 N ), for all t ∈ t * + ρ * B d 1 ∩ r Q (ρ * )S d-1 2 , 1 N N i=1 X i , t -t * 2 -t -t * 2 2 r 2 Q (ρ * ) 2 = 1 2 t -t * 2 2 . (3.2.4)
In other words, the quadratic process satisfies an isomorphic property on the set

t * + (ρ * B d 1 ∩ r Q (ρ * )S d-1 2
). Now, it remains to extend this result to the set of vectors t ∈ t * + ρ * B d 1 such that tt * 2 r Q (ρ * ). Let t be such a vector and define t :

= t * + (r Q (ρ * )/ t -t * )(t -t * ). Since t ∈ t * + (ρ * B d 1 ∩ r Q (ρ * )S d-1 2
), it satisfies the isomorphic property from (3.2.4) and so

1 2 t -t * 2 2 1 N N i=1 X i , t -t * 2 3 2 t -t * 2 2
which corresponds exactly to the event Ω(ρ * ). Therefore,

P[Ω * ] = P[Ω(ρ * )] 1-2 exp(-C 1 Q 2 N ).

Control of the multiplier process

In this section, we provide a control of the multiplier process on several shelves of the space

R d . Define r 0 as the non-null solution to σ * (rB d 2 ) = ηr 2 √ N -i.e. r 0 = σ * (B d 2 )/(η √ N ) = (σ/η) d/N . Let ρ 0 be the smallest ρ such that ρB d 1 contains r 0 B d 2 -i.e., ρ 0 = r 0 √ d.
We can see that ρ 0 is such that r M (ρ) = r M (ρ 0 ) for all ρ ρ 0 . Indeed, one first sees that r M (ρ

0 ) = r 0 , since σ * (r 0 B d 2 ∩ ρ 0 B d 1 ) = σ * (r 0 B d 2 ) = ηr 2 0 √ N , and σ * (rB d 2 ∩ ρ 0 B d 1 ) = σr * (B d 2 ) > ηr 2 √ N 3.2. Proof of Theorem 3.1.4
for all r < r 0 and for r > r 0 , σ * (rB d 2 ∩ ρ 0 B d 1 ) σr * (B d 2 ) ηr 2 √ N . This last argument also holds for ρ ρ 0 . In the latter case, if r > r 0 then

σ * (rB d 2 ∩ ρB d 1 ) r r 0 σ * (r 0 B d 2 ∩ ρB d 1 ) = r r 0 σ * (r 0 B d 2 ∩ ρ 0 B d 1 ) = r r 0 ηr 2 0 √ N < ηr 2 √ N
which means that r M (ρ) r 0 . And as ρ ρ 0 , r M (ρ) r M (ρ 0 ) = r 0 . Therefore, for ρ ρ 0 , r M (ρ) is constant, equal to r 0 . And on [0, ρ 0 ], r M is non-decreasing: let ρ ρ ρ 0 , then

σ * (r M (ρ )B d 2 ∩ ρ B d 1 ) σ * (r M (ρ )B d 2 ∩ ρ B d 1 ) = ηr M (ρ ) 2 √
N so r M (ρ ) r M (ρ ). We denote K 0 = min{k ∈ N : 2 k ρ * 2ρ 0 }: we will see later that K 0 is defined that way to be the number of the first "shell" such that r M (2

K 0 -1 ρ * )B d 2 ⊂ 2 K 0 -1 ρ * B d 1 .
Proposition 3.2.2. There exists an absolute constant C 5 such that the following holds. Let X 1 , . . . , X N be N i.i.d. standard Gaussian vectors in R d and ξ 1 , . . . , ξ N be N standard realvalued Gaussian variables independent of the X i 's. For all k = 0, . . . , K 0 , denote by A k the event on which, for every t ∈ R d such that tt * 1 2 k ρ * :

|P N M t-t * | 1 4 max r M (2 k ρ * ) 2 , t -t * 2 2 .
(3.2.5)

Then, for η = 1/(16 √ 2), one has

P K 0 k=0 A k 1 -2 exp (-C 5 N ) -40 exp - C (1) M N r M (ρ * ) 2 ) 1024C (2)
M σ 2 when ρ * 4096 log(2)σ/ C

(1) M √ N .

Proof.

We first work conditionally to the ξ i , i = 1, . . . , N . Let ρ > 0 and define

T (ρ) := t * +ρB d 1 ∩r M (ρ)B d 2 .
It follows from the Gaussian concentration inequality (cf. Borell's inequality in [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]) that, for all x > 0, with probability greater than 1-2 exp(-x 2 /2),

sup t∈T (ρ) N i=1 ξ i X i , t -t * -E sup t∈T (ρ) N i=1 ξ i X i , t -t * xσ(T (ρ))
where

σ(T (ρ)) = sup t∈T (ρ) E N i=1 ξ i X i , t -t * 2 .
Conditionally to the ξ i 's, the Gaussian process

N i=1 ξ i X i , t -t * t∈T (ρ)
has the same distribution as the Gaussian process σ N X 1 , tt * t∈T (ρ) for σ N := N i=1 ξ 2 i . This yields

E sup t∈T (ρ) N i=1 ξ i X i , t -t * = σ N * (T (ρ) -t * ) = σ N * (ρB d 1 ∩ r M (ρ)B d 2 ) = σ N η √ N r 2 M (ρ) σ 3. Minimax regularization and σ(T (ρ)) = sup t∈T (ρ) σ N E[ X, t -t * 2 ] = σ N sup t∈T (ρ) t -t * 2 σ N r M (ρ).
So, conditionally on (ξ i ) N i=1 , for all x > 0, one has, with probability at least 1-2 exp(-x 2 /2),

sup t∈T (ρ) N i=1 ξ i X i , t -t * σ N η √ N r 2 M (ρ) σ + x σ N r M (ρ)
Thus, taking x = η √ N r M (ρ)/σ in the previous statement, one gets, on an event which probability is at least 1 -2 exp(-

η 2 N r 2 M (ρ)/(2σ 2 )), sup t∈t * +ρB d 1 ∩r M (ρ)B d 2 1 N N i=1 ξ i X i , t -t * 2η σ N σ √ N r 2 M (ρ). (3.2.6)
It remains to prove, on the same event, the result for all t ∈ t * + ρB d 1 such that tt * 2 > r M (ρ). Define t := t * + r M (ρ)/ tt * 2 (tt * ). Since t ∈ T (ρ), it follows from (3.2.6) that (on the same event):

1 N N i=1 ξ i X i , r M (ρ) t -t * 2 (t -t * ) = 1 N N i=1 ξ i X i , t -t * 2η σ N σ √ N r M (ρ) 2
and since r M (ρ)

t -t * 2 one gets 1 N N i=1 ξ i X i , t -t * 2η σ N σ √ N t -t * r M (ρ) 2η σ N σ √ N t -t * 2 .
Hence, with probability (conditionally to the ξ i ) at least 1 -2 exp -η 2 N r M (ρ) 2 )/(2σ 2 ) , the multiplier process is controlled such that sup

t∈t * +ρB d 1 |P N M t-t * | 4η σ N σ √ N max r M (ρ) 2 , t -t * 2 2 .
(3.2.7)

A control of the probability measure of the event A k follows by applying the previous result to ρ = 2 k ρ * when η 1/(16 √ 2) together with a control of the term σ N . It follows from an union bound that, conditionally to the ξ i , (3.2.7) is satisfied for all ρ = 2 k ρ * , k = 0, • • • , K 0 on an event whose probability measure is larger than

1 -2 K 0 k=0 exp -η 2 N r M (2 k ρ * ) 2 )/(2σ 2 ) .
(3.2.8)

We handle the last term below thanks to Lemma 3.3.2. Now, we handle the random variables ξ 1 , . . . , ξ N . It appears that only a control of the empirical variance term σ N / √ N is needed to get a fully deterministic upper bound in the right-hand term of (3.2.7). It follows from Bernstein inequality for subexponential variables (cf. Theorem 1.2.7 in [START_REF] Chafaï | Interactions between compressed sensing random matrices and high dimensional geometry[END_REF]) that with probability greater than 1-2 exp(-

C 5 N ), 1 N N i=1 ξ 2 i -σ 2 σ 2 , 3.2. Proof of Theorem 3.1.4 which implies σ N / √ N √ 2σ.
Therefore, for η = 1/(16 √ 2) we have

P 4η σ N /( √ N σ) 1/4 1 -2 exp (-C 5 N ) .
(3.2.9)

Binding together (3.2.7), (3.2.8) and (3.2.9) gives

P K 0 k=0 A k 1 -2 exp (-C 5 N ) -2 K 0 k=0 exp - N r M (2 k ρ * ) 2 1024σ 2 ) .
Finally, Lemma 3.3.2 yields the following bound

K 0 k=0 exp - N r M (2 k ρ * ) 2 ) 1024σ 2 10 1 -exp - C (1) M √ N ρ * 4096σ exp - C (1) M C (2) M N r M ρ * 2 1024σ 2
and the result follows when ρ * 4096 log(2)σ/(C

(1) M √ N ), which implies that the denominator of the right-hand side is greater than 1/2.

Conclusion: construction of the event Ω 0

We define the event

Ω 0 = Ω * ∩ K 0 k=0 A k .
It follows from Proposition 3.2.1 and Proposition 3.2.2 (as well as Lemma 3.2.4 below), that, as long as ρ * 4096 log(2)σ/(C

(1) M √ N ), P[Ω 0 ] 1 -4 exp(-C 6 N ) -40 exp - C 6 N r M ( t * 1 ) 2 ) σ 2
where C 6 and C 6 are absolute constants.

Deterministic part of the proof

We first start with some few lemmas on the growth of r M (•) and r Q (•). We then construct a partition of R d depending on the behavior of function r 2 (•) (in particular, its concavity for intermediate values is an issue; we solve it thanks to a peeling argument). We then turn to the main deterministic argument showing that t belongs to a d 1 -ball of radius ρ * around t * . The latter holds on the event Ω 0 introduced in Section 3.2.1.

Minimax regularization

Two Lemmas on the growth of r M and r Q Lemma 3.2.3. Let ρ > 0 and φ = 4(C

(2) M ) 2 /(C (1) M ) 2 . If φρ ρ 0 min(1, η), then for any ρ φρ, r 2 M ρ > 2r 2 M (ρ) and r 2 ρ > 2r 2 (ρ).
Proof. Since r M (•), r Q (•) and r(•) are non-decreasing, we only have to prove the result for ρ = φρ. Recall that C

(2) M

C

(1) M (so φ 4). First note that if N ζ d then r Q (ρ) = 0 and so the second claim follows from the first one since r(ρ) = r M (ρ) in this case. And when N ζd, one has

r 2 Q (φρ) = φ 2 r 2 Q (ρ) 16r 2 Q (ρ) > 2r 2 Q (ρ) (because in that case r Q (ρ) 2 > 0)
. Therefore the second claim is a straightforward consequence of the first one. So it only remains to study the behavior of r 2 M (•). For ρ < φρ ρ 0 , r M is given by one of the two last expressions of (3.1.9).

First assume that (φρ) 2 N σ 2 log d then

r 2 M (φρ) C (1)
M φρσ log(ed/N ) > 2C

(2)

M ρσ log(ed/N ) 2r 2 M (ρ). Now, assume that σ 2 log d ρ 2 N (φρ) 2 N then r 2 M (φρ) = C M φρσ 1 N log eσd φρ √ N for some C M ∈ [C (1) M , C (2)
M ]. One has that for all x φe, log(x/φ) > log(x)/φ, and, since ρ 0 = σd/(η √ N ), the assumption φρ ρ 0 min(1, η) guarantees that eσd/ρ √ N φe. Therefore, we have:

r 2 M (φρ) > C (1) M φρσ 1 φN log eσd ρ √ N 2C (2)
M ρσ

1 N log eσd ρ √ N 2r 2 M (ρ).
Finally, when ρ 2 N σ 2 log d (φρ) 2 N , since r M is increasing, it is clear considering the two previous cases that one has again r 2 M (φρ) > ( √ φC

(1)

M /C (2) M )r 2 M (ρ), so r 2 M (φρ) > 2r 2 M (ρ). Lemma 3.2.4. Let ν > 0. If ν 1 then r M (νρ) √ νr M (ρ) and r(νρ) νr(ρ). If ν 1 then r M (νρ)
√ νr M (ρ) and r(νρ) νr(ρ).

Proof. It is clear that r Q (νρ) = νr Q (ρ), because * (νρB d 1 ∩ νr Q (ρ)B d 2 ) = ν * (ρB d 1 ∩ r Q (ρ)B d 2 ). As for r M (νρ), for ν 1, one has σ * (νρB d 1 ∩ √ νr M (ρ)B d 2 ) σν * (ρB d 1 ∩ r M (ρ)B d 2 ) = νη √ N r M (ρ) 2 = η √ N ( √ νr M (ρ)) 2 . So r M (νρ) √ νr M (ρ).
As for the case ν 1, then 1/ν 1, and it suffices to write that r M (ρ) = r M ((1/ν)νρ) (1/ν)r M (νρ) to get the result (and still r Q (ρ/ν) = r Q (ρ)/ν).

Minimax regularization

Locating t in the central zone on the event Ω 0 To show that, on the event Ω 0 , t belongs to the central zone, it is enough to show that any t outside this area satisfies P N L Ψ t > 0 where we recall that in our case,

P N L Ψ t := 1 N N i=1 (Y i -X i , t ) 2 + c 0 r 2 ( t 1 ) - 1 N N i=1 (Y i -X i , t * ) 2 + c 0 r 2 ( t * 1 ) .
This will indeed prove that t -

t * 1 ρ * since P N L Ψ t 0 = P N L Ψ t * . Let t ∈ R d be outside the central zone -i.e. t -t * 1 > ρ * . First note that t satisfies t 1 t-t * 1 -t * 1 ρ * -t * 1 9 t * 1 . Therefore, we have R t,t * = Ψ( t 1 )-Ψ( t * 1 ) = c 0 r 2 ( t 1 ) -c 0 r 2 ( t * 1
) > 0 and (8/9) t 1 tt * 1 (10/9) t 1 . From now on, we place ourselves on the event Ω 0 introduced in Section 3.2.1. First assume that ρ * < 2ρ 0 , then both the "intermediate peeling zone" and the "exterior zone" must be considered. For t in any of these two areas, one has R t ,t * 0.

Let us begin by the "peeling zone". Consider t and k 1 such that

2 k-1 ρ * < t -t * 1 2 k ρ * and t -t * 1 2 K 0 ρ * .
We recall that 2 K 0 -1 ρ * < 2ρ 0 2 K 0 ρ * and ρ 0 is the smallest radius such that r M (ρ) = r M (ρ 0 ) for all ρ ρ 0 . One possibility is that

t -t * 2 2 r 2 (2 k ρ * ), then on Ω * one has P N Q t-t * t -t * 2 2 /2 and, on A k , one has |P N M t-t * | t -t * 2 2 /4. Hence, on Ω 0 , P N L Ψ t t -t * 2 2 /4 + R t ,t * > 0.
Let us tackle now the other case: tt * 2 2 < r 2 (2 k ρ * ). We will show that in this situation, if c 0 is large enough, R t,t * /4 > |P N M t-t * |.

As

t 1 > 4 t * 1 (C (2) M ) 2 /(C (1)
M ) 2 (t is not in the "central zone") and 4 t * 1 (C

(2)

M ) 2 /(C (1) M ) 2 ηρ 0 (since 2ρ 0 ρ * > 8 t * 1 (C (2) M ) 2 /((C (1)
M ) 2 η)), by Lemma 3.2.3 one has that c 0 r 2 ( t 1 ) 2c 0 r 2 ( t * 1 ). Thus, R t,t * c 0 r 2 ( t 1 )/2. So, thanks to Lemma 3.2.4, and since

t 1 2 k ρ * = t -t * 1 2 k ρ * t 1 t -t * 1 > 1 2 • 9 10 > 2 5 , one has R t,t * 1 2 c 0 r 2 ( t 1 ) 1 2 • t 1 2 k ρ * 2 c 0 r 2 (2 k ρ * ) > (2/25)c 0 r 2 (2 k ρ * ).
In addition, we have t

-t * 1 2 k ρ * and t -t * 2 2 < r 2 (2 k ρ * ), hence, on the event A k , |P N M t-t * | r 2 (2 k ρ * )/4. As a consequence, for c 0 > 13 one has R t,t * 4 -|P N M t-t * | > 2 25 • c 0 r 2 (2 k ρ * ) • 1 4 - 1 4 r 2 (2 k ρ * ) > 0. (3.2.10) A fortiori, R t ,t * -|P N M t-t * | > 0 and as P N Q t-t * 0, this implies in particular that P N L Ψ t > 0.
To sum up, we proved that for all t in the peeling zone, we have Let us now study the exterior zone. We will mainly use homogeneity arguments. Let t ∈ R d be outside the ball t

P N L Ψ t P N Q t-t * -|P N M t-t * | + R t ,
* + 2 K 0 ρ * B d 1 . Define t ∈ t * + 2 K 0 ρ * S d-1
1 such that tt * = α t (tt * ) for some α t 1. In particular, tt * 2 2r 0 > r M (2 K 0 ρ * ) so by Proposition 3.2.2, on the event A K 0 , one has |P N M t -t * | tt * 2 2 /4. We consider now two cases. If tt * 2 r Q ( tt * 1 ), then by Lemma 3.2.1, on the event Ω * , one has

P N Q t -t * t -t * 2 2 /2. So on the event Ω 0 , P N Q t -t * -|P N M t -t * | > 0 and P N Q t-t * -|P N M t-t * | = α 2 t P N Q t -t * -α t |P N M t -t * | α t (P N Q t -t * -|P N M t -t * |) > 0 therefore P N L Ψ t > 0. On the contrary, if t -t * 2 < r Q ( t -t * 1 ), as t -t * 2 2r 0 , then r Q ( t 1 ) 9r Q ( t -t * 1 )/10 > 9r 0 /5 = 9r M ( t 1 )/5 > r M ( t 1 ) so r( t 1 ) = r Q ( t 1 ). One has t 1 = t * + α t (t -t * ) 1 (4α t /5) t 1 since α t 1 and t -t * 1 ρ * . So r 2 ( t 1 ) r 2 Q (α t 4 t 1 /5) = 4 2 α 2 t r 2 ( t 1 )/5 2 .
We have seen before that r Q ( t 1 ) 9r 0 /5 9r M ( t * 1 )/5, and r Q ( t 1 ) 9r Q ( t * 1 ), so r 2 ( t 1 ) 3r 2 ( t * 1 ). As a consequence,

R t,t * = c 0 r 2 ( t 1 ) -c 0 r 2 ( t * 1 ) c 0 α 2 t 4 2 5 2 r 2 ( t 1 ) -(1/3)r 2 ( t 1 ) α t c 0 4 r 2 ( t 1 ) α t R t ,t * /4. Moreover, since t -t * = α t (t -t * ), one has P N Q t-t * = α 2 t P N Q t -t * and P N M t-t * = α t P N M t -t * . So, in the case t -t * 2 < r Q ( t -t * 1 ), by (3.2.11) applied to t , P N L Ψ t α t (P N Q t -t * -|P N M t -t * | + R t ,t * /4) > 0.
Let us now consider the case ρ * > 2ρ 0 . In this situation, there is no need for the intermediate "peeling" zone. Let t ∈ R d be outside the ball t * + ρ * B d 1 and set t ∈ t * + ρ * S d-1 1 such that tt * = α t (tt * ) with α t 1. Then one can apply arguments similar to the peeling case (with tt * 1 = ρ * , but this time k = K 0 = 0), on the event A 0 ∩ Ω * , to t . If tt * 2 r Q ( tt * 1 ), then by Proposition 3.2.1, on the event Ω 0 one has

P N Q t -t * |P N M t -t * |/2. Conversely, if t -t * 2 < r Q ( t -t * 1 ), then r 2 ( t 1 )
3r 2 ( t * 1 ) so for c 0 big enough, in the same spirit as (3.2.10), one gets that R

t ,t * /4 -|P N M t -t * | > 0. So in both cases, P N Q t-t * -|P N M t-t * | + R t ,t * /4 > 0.
The same argument as previously for the exterior zone, shows that

P N L Ψ t α t (P N Q t -t * -|P N M t -t * | + R t ,t * /4), so P N L Ψ t > 0. As a conclusion, in the case ρ * > 2ρ 0 , we have for all t ∈ R d satisfying t -t * 1 ρ * > 2ρ that P N L Ψ t > 0.
To sum up, on the event Ω 0 , any t outside the central zone satisfies P N L Ψ t > 0. Therefore, given that t satisfies P N L Ψ t 0, we conclude that t belongs to the central zone.

Conclusion of the proof of Theorem 3.1.4

On the event Ω 0 , t ∈ (t * + ρ * B d 1 ). Hence, either tt * 2 2 r 2 (ρ * ) and the proof is over or tt * 2 2 > r 2 (ρ * ). In the latter case, one has

P N Q t-t * > 1 2 t -t * 2 2 and |P N M t-t * | < 1 4 t -t * 2 2 87
3. Minimax regularization and so 0

P N L Ψ t 1 4 t -t * 2 2 + c 0 r 2 ( t 1 ) -c 0 r 2 ( t * 1 )
which implies tt * 2 2 4c 0 r 2 ( t * 1 ).

Thus, taking θ 0 = max 100, (8(C

(2)

M ) 2 /(C (1)
M ) 2 + 1) 2 , 4c 0 , with probability at least

P[Ω 0 ] 1 -4 exp (-C 6 N ) -40 exp -C 6 N r M ( t * 1 ) 2 /σ 2 one gets that in both cases, R( t) -R(t * ) = t -t * 2 2
θ 0 r 2 ( t * 1 ). Moreover, for t * 1 ∆ 0 σ log(ed)/N for ∆ 0 an absolute constant large enough, we have P[Ω 0 ] 3/4. Given that r 2 ( t * 1 ) is the minimax rate of convergence over t * 1 B d 1 (cf. Theorem 3.1.3), we conclude that Ψ(ρ) = c 0 r 2 (ρ) is indeed a minimax regularization function.

3.3. Technical material and proof of Proposition 3.1.6

Localization with balls and spheres

The next lemma shows that when the intersection is not trivial, localizing by intersecting an 1 ball with an 2 sphere or the corresponding full 2 ball is equivalent.

Lemma 3.3.1. If ρ r, then * (ρB d 1 ∩rS d-1 2 ) = * (ρB d 1 ∩rB d 2 ). If ρ < r, then * (ρB d 1 ∩rB d 2 ) = * (ρB d 1 ) and * (ρB d 1 ∩ rS d-1 2 ) = 0.
Proof. Since for any set T ⊂ R d , * (T ) = * (conv(T )) (with conv(T ) denoting the convex hull of T ), the result is a direct consequence of the fact that for ρ < r, conv(ρB

d 1 ∩ rB d 2 ) = ρB d 1 and conv(ρB d 1 ∩ rS d-1
2 ) = ∅ (which are two obvious statements), and that if ρ r, then conv(ρB

d 1 ∩ rS d-1 2 ) = conv(ρB d 1 ∩ rB d 2 )
, which we prove now. One inclusion is immediate, it remains to prove that conv(ρB

d 1 ∩rB d 2 ) ⊂ conv(ρB d 1 ∩rS d-1 2 ). First, conv(ρB d 1 ∩ rB d 2 ) = ρB d 1 ∩ rB d 2 = conv{t ∈ R d : max( t 1 /ρ, t 2 /r) = 1} so it only remains to show that {t ∈ R d : max( t 1 /ρ, t 2 /r) = 1} ⊂ conv(ρB d 1 ∩ rS d-1 2 ). First, remark that {t ∈ R d : t 1 /ρ 1, t 2 /r = 1} is included in ρB d 1 ∩ rS d-1 2 .
Let us now consider the set {t ∈ R d : t 1 /ρ = 1, t 2 /r < 1} and consider an element t in it. We denote by e 1 , . . . , e d the canonical basis of R d and we recall that each face of ρB d 1 is the convex hull of its vertices. So, since t ∈ ρS d-1

1 , there exist b 1 ∈ {ρe 1 , -ρe 1 }, b 2 ∈ {ρe 2 , -ρe 2 }, . . . , b d ∈ {ρe d , -ρe d } such that t is in the convex hull of {b 1 . . . , b d }: there exist non-negative coefficients µ 1 , . . . , µ d such that d j=1 µ j (b j -t) = 0.
Consider for each j ∈ {1, . . . , d} the mapping

f j : x ∈ [0, 1] → t + x(b j -t) 2 /r -t + x(b j -t) 1 /ρ This mapping is continuous, f j (0) < 0 (t is in ρS d-1 1 and rB d 2 but not in rS d-1 2
) , and f j (1) 0 (because b j 2 = b j 1 = ρ and ρ r). So there exists x j in (0,1] such that f (x j ) = 0, which means that t

+ x j (b j -t) ∈ ρS d-1 1 ∩ rS d-1 2 . One has that d j=1 µ j x j (t + x j (b i -t) -t) = d j=1 µ j x j x j (b j -t) = d j=1 µ j (b j -t) = 0.
3.3. Technical material and proof of Proposition 3.1.6

As a consequence, t is in the convex hull of the vectors t

+ x j (b j -t) ∈ ρS d-1 1 ∩ rS d-1 2
, j ∈ {1, . . . , d} which achieves the proof. 2). For all ν > 0, we have

Control of the probability estimate

Lemma 3.3.2. Set η = 1/(16 √
K 0 k=0 exp -νr M 2 k ρ * 2 10 1 -exp -νC (1) M σ 4 √ N ρ * exp - C (1) M C (2) M νr M ρ * 2 .
Proof. First, the terms of the sum are non-increasing (remember that r M is a nondecreasing function). So skipping the last terms will not change the order of magnitude:

K 0 k=0 exp -νr M 2 k ρ * 2 max 10 exp -νr M ρ * 2 , 10 K 0 -9 k=0 exp -νr M 2 k ρ * 2 .
One has 10 exp -νr M ρ * 2 > 10 K 0 -9 k=0 exp -νr M 2 k ρ * 2 when K 0 8. Let us now assume that K 0 9. We study the sum K 0 -9 k=0 exp -νr M 2 k ρ * 2 . In order to get rid of the "range" [C

(1) M , C (2)
M ] in the definition of r M (•), notice that

K 0 -9 k=0 exp -νr M 2 k ρ * 2 K 0 -9 k=0 a k with: a k :=        exp -νC (1) M 2 k ρ * σ log(ed) N if 2 k ρ * 2 N σ 2 log(d) exp -νC (1) M 2 k ρ * σ 1 N log eσd 2 k ρ * √ N
otherwise.

We emphasize that the sum goes only up to 2 K 0 ρ * , which excludes the constant third form of r M (•).

Applying a second time the "range" [C

(1)

M , C (2)
M ] in the bounds on r M allows to bound a 0 in terms of r M ρ * 2 : a 0 exp -( C(1)

M /C (2)
M )νr M ρ * 2 . Therefore, in the following we will bound K 0 -9 k=0 a k with respect to a 0 , and then get back to r M ρ * 2 . We now prove that there exists α independent on k (but dependent on the other parameters) such that for any k K

0 -9, a k+1 /a k α < 1. If (2 k+1 ρ * ) 2 σ 2 log(d)/N then a k+1 a k = exp -νC (1) M 2 k+1 ρ * σ log(ed) N + νC (1) M 2 k ρ * σ log(ed) N = exp -νC (1) M 2 k ρ * σ log(ed) N exp(-νC (1) M ρ * β 1 )
with β 1 = σ log(ed)/N σ/ √ N .

Minimax regularization

As for the case (2 k ρ * ) 2 σ 2 log(d)/N < (2 k+1 ρ * ) 2 (which can only occur when d > 1), then one has 2 k+1 ρ * 2σ √ log d/ √ N and so

a k+1 a k = exp -νC (1) M 2 k+1 ρ * σ 1 N log eσd 2 k+1 ρ * √ N + νC (1) M 2 k ρ * σ log(ed) N exp -νC (1) M 2 k ρ * σ √ N 2 log ed 2 √ log d -log ed exp -νC (1) M 2 k ρ * σ 2 √ N exp -νC (1) M ρ * σ 2 √ N
The second inequality relies on a straightforward analysis fact:

∀ d 2, 2 log ed 2 √ log d -log ed > 1 2
Let us tackle now the case (2 k ρ * ) 2 > σ 2 log(d)/N . We have a k+1 /a k = exp(b k νC

(1)

M 2 k ρ * σ/ √ N ) where b k := -2 log eσd 2 k+1 ρ * √ N + log eσd 2 k+1 ρ * √ N + log(2). Since √ x + y √ x +
√ y for all x, y 0, one has (still for k K 0 -9):

b k log(2) -log eσd 2 k+1 ρ * √ N log(2) -log eσd 2 K 0 -8 ρ * √ N (1 - √ 2) log(2) because 2 K 0 -8 ρ * 2 -6 ρ 0 2 -6 σd/(η √ N ) σd/(2 √ N ) (we recall that ρ 0 = σd/(η √ N ) and η = 1/(16 √ 2)). It follows that a k+1 a k = exp νC (1) M 2 k ρ * σ √ N b k exp -νC (1) M ρ * σ √ N (-b k ) exp(-νC (1) M ρ * β 2 ) with β 2 = ( √ 2 -1) log(2)σ/ √ N σ/(4 √ N )
. Then, we conclude that for all k K 0 -9,

a k+1 /a k exp -νC (1) M σ 4 √ N ρ * and so K 0 -9 k=0 a k a 0 K 0 -9 k=0 exp -kνC (1) M σ 4 √ N ρ * = a 0 1 -exp -(K 0 -8)νC (1) M σ 4 √ N ρ * 1 -exp -νC (1) M σ 4 √ N ρ * .
Finally, the result follows, since a 0 exp -(C

(1) The proof of Proposition 3.1.6 relies on key ideas developed in minimax theory. We refer the reader to Tsybakov [2009] for a state of the art in minimax theory.

M /C (2) M )νr M ρ * 2 and 1 -exp -(K 0 -8)νC (1) M σ 4 √ N ρ * 1 -exp -νC (1) M σ 4 √ N ρ * 1 1 -exp -νC
Let t satisfy the properties of Proposition 3.1.6, i.e. a procedure adaptive on the finite set

{±ρe 1 , • • • , ±ρe d } and let t * ∈ (ρ/2)B d 1 . Denote Λ = {t * , ±ρe 1 , . . . , ±ρe d } = {t * 0 , t * 1 , • • • , t * 2d } so that t * 0 = t * . It is straightforward to check that Λ is a ρ/2-
P t * j dP t * 0 dP t * j τ = P t * j dP t * j dP t * 0 1 τ = 1 -P t * j log dP t * j dP t * 0 > log(1/τ ) 1 - 1 log(1/τ ) log dP t * j dP t * 0 + dP t * j 1 - 1 log(1/τ ) K(P t * j , P t * 0 ) + 2K(P t * j , P t * 0 )
where K(P t * j , P t * 0 ) denotes the Kullback-Leibler divergence between P t * j and P t * 0 . Since the noise is Gaussian and independent of X in (3.1.7) and X is isotropic, we have

K(P t * j , P t * 0 ) = N t * j -t * 0 2 2 /(2σ

Minimax regularization

Using the latter result together with (3.3.2) in (3.3.1) for the values τ = 1/(2d), we obtain that

P t * 0 t -t * 0 2 ρ/4 1 2 .

Minimax regularization function in the fixed design setup

In this section, we consider the Gaussian regression model with fixed design. We are therefore given a deterministic N × d design matrix X, whose i-th row vector is denoted by X i ∈ R d , and we observe N noisy projections of a vector

t * ∈ R d , Y i = X i , t * + ξ i , i = 1, . . . , N ,
where the noises ξ 1 , . . . , ξ N are independent centered Gaussian variables with variances σ 2 . The data (Y i , X i ) N i=1 are used to construct estimators of t * and the only difference with the previous random design setup is that the X i 's are deterministic vectors whereas so far they were random vectors. We will use most of the time the matrix form of the data:

Y = Xt + ξ where Y = (Y i ) N
i=1 and ξ ∼ N (0, σ 2 I N ) where I N is the N × N identity matrix. Note that the fixed design setup is usually considered in signal processing over a finite grid or in experiences where the statistician can chose in advance the design of an experience and then observed an output.

In order to design a minimax regularization function in this setup, we need to adapt the definitions introduced in Section 3.1 to the fixed design case. We first start with the definition of a minimax rate of convergence over a subset of R d . We use the empirical (or normalized) Euclidean inner product and norm: for all u, v ∈ R N ,

u, v L N 2 = 1 N N i=1 u i v i and u L 2 N = 1 N N i=1 u 2 i
and the associated unit ball

B L 2 N = {u ∈ R N : u L 2 N 1}.
Definition 3.4.1. Let T ⊂ R d , X denote a (deterministic) N × d design matrix and ξ be a centered Gaussian random vector in R N with covariance matrix σ 2 I N . For all t * ∈ T , define the random vector Y t * = Xt * + ξ and denote by Y T := {Y t * : t * ∈ T } the set of all such random vectors. Let t N be a statistics from R N to R d . Let 0 < δ N < 1 and ζ N > 0. We say that t N performs with accuracy ζ N and confidence 1δ N relative to the set of targets Y T , if for all t * ∈ T , with probability, w.r.t. to a vector Y distributed as

Y t * , at least 1 -δ N , X( t N (Y ) -t * ) 2 L 2 N ζ N .
We say that R N is a minimax rate of convergence over T for the confidence 1δ N if the two following hold:

1. there exists a statistics t N which performs with accuracy R N and confidence 1δ N relative to the set of targets Y T 3.4. Minimax regularization function in the fixed design setup 2. there exists an absolute constant g 0 > 0 such that if tN is a statistics which performs with accuracy ζ N and confidence 1δ N relative to the set of targets Y T then necessarily ζ N g 0 R N .

Note that we use the empirical

L N 2 -metric X • L 2 N
(to the square) with respect to the design X as a measure of performances of estimators in Definition 3.4.1. The reason we do so is that it is the natural counterpart to the random design case -that is when X is a standard Gaussian matrix then E Xt 2

L 2 N = t 2
2 and the d 2 -norm is the metric used to measure the performance of estimators in the random design setup -and that it is the natural metric associated to the prediction of Y problem given that if

R(t) = E Y -Xt 2 L 2 N is the risk of t for all t ∈ R d then we have for any estimator t N , R( t N ) = R(t * ) + X( t N -t * ) 2 L 2 N
. So that predicting Y is the same problem as estimating t * with respect to the empirical

X • L 2 N metric.
Now, we adapt the definition of a minimax regularization function to the fixed design setup in the next definition.

Definition 3.4.2. Let • be a norm on R d , T ⊂ R d and 0 < δ N < 1. Let us consider the following RERM for some given function Ψ : R + → R:

t ∈ argmin t∈R d Y -Xt 2 L 2 N + Ψ( t )
constructed from a N × d deterministic matrix X and a random vector Y = Xt * + ξ, with ξ ∼ σN (0, I N ). We say that Ψ is a minimax regularization function for the norm • over T for the confidence 1δ N , if there exists an absolute constant g 1 > 0 such that for all t * ∈ T , the RERM t is such that with probability at least

1 -δ N , t -t * 2 L 2 N g 1 R N ,
where R N is the minimax rate of convergence over {t ∈ R d : t t * }.

The statistical bounds one can prove in the fixed design setup depend generally on the property of the design matrix X. Many different assumptions have been introduced during the last two decades in high-dimensional statistics and we refer to [START_REF] Van | On the conditions used to prove oracle results for the Lasso[END_REF] for some of them. In particular, norm preserving properties like the RIP or weaker assumption on the restricted eigenvalues like the REC or CC have played an important role in statistics (cf. Bickel et al. [2009], [START_REF] Bühlmann | Statistics for high-dimensional data[END_REF], [START_REF] Candès | The power of convex relaxation: near-optimal matrix completion[END_REF], [START_REF] Giraud | Introduction to high-dimensional statistics[END_REF], van de Geer [2007], [START_REF] Van | On the conditions used to prove oracle results for the Lasso[END_REF]). In this chapter, we assume that X satisfies the "Restricted Isometry Property". It appears that this condition is equivalent (up to constants) to the property satisfied by a standard Gaussian matrix as in Lemma 3.2.1. Assumption 3.4.1 (RIP(s)). If N < d and N/ log(ed/N ) > 1 then we set s = N/ log(ed/N ). We assume that all t in Σ s := {x ∈ R d : x 0 s} is such that

1 2 t 2 Xt L 2 N 3 2 t 2 (3.4.1)
where x 0 is the size of the support of x. If N d, we assume that (3.4.1) is satisfied for all t ∈ R d .

Minimax regularization

Note that in the high-dimensional case, i.e. d > N , only the situation N/ log(ed/N ) > 1 is considered in Assumption 3.4.1 to avoid the ultra-high dimensional phenomena discovered in [START_REF] Verzelen | Minimax risks for sparse regressions: ultra-high dimensional phenomenons[END_REF]. RIP was introduced in Candes and Tao [2005] and it has been widely used and discussed (cf. for example [START_REF] Mark | Analysis of orthogonal matching pursuit using the restricted isometry property[END_REF], [START_REF] Baraniuk | A simple proof of the restricted isometry property for random matrices[END_REF] or [START_REF] Garg | Gradient descent with sparsification: an iterative algorithm for sparse recovery with restricted isometry property[END_REF]), in particular in the field of Compressed Sensing. From our perspective, we use this result for two reasons:

• 1) the minimax results over d 1 balls we need to develop for our proof of minimax regularization function has been obtained in the fixed design under this condition (or an equivalent one) in [START_REF] Rigollet | Exponential screening and optimal rates of sparse estimation[END_REF];

• 2) the complexity parameter that we will be using in the fixed design setup have been computed under very general design matrix X in [START_REF] Bellec | Localized Gaussian width of M -convex hulls with applications to Lasso and convex aggregation[END_REF] but they were only proved to be optimal under the RIP assumption.

Our main result covers more general design matrix than the one satisfying RIP but it turns out that those results do not allow to conclude on the minimax optimality of the associated regularization function beyond the RIP case; moreover and to our knowledge no sharp closed form are available for the computation of this regularization function beyond the RIP case.

The multiplier process and its associated fixed point

Our analysis is based upon the study of the same regularized excess empirical risk quantity as in the random design section: for all t ∈ R d ,

P N L Ψ t := Y -Xt 2 L 2 N + Ψ(t) -Y -Xt * 2 L 2 N + Ψ(t * ) .
We use the same quadratic / multiplier decomposition as in the random design case: for all

t ∈ R d , P N L Ψ t = P N Q t-t * + P N M t-t * + R t-t * where • P N Q t-t * := X(t -t * ) 2 L 2 N
is the quadratic part

• P N M t-t * := 2 ξ, X(t * -t) is the multiplier part • R t,t * := Ψ(t) -Ψ(t * ) is the regularization part.
Contrary to the random design case, in the fixed design setup the only source of randomness is the Gaussian noise ξ, in particular the quadratic process is fully deterministic. Therefore, there is no need to define a fixed point similar to r Q for a control on the quadratic process.

The only fixed point we introduce is a version of the previous multiplier fixed point r M (•) adapted to the fixed design setup: for η = 1/8, let

r X (ρ) := inf r > 0 : σ * ρ √ N XB d 1 ∩ rB N 2 η r 2 √ N (3.4.2)
where

XB d 1 := {Xt : t ∈ B d 1 } and B N 2 is the unit ball in N 2 .
3. Minimax regularization Proposition 3.4.4 [START_REF] Rigollet | Exponential screening and optimal rates of sparse estimation[END_REF]). Let X ∈ R N ×d be a matrix satisfying RIP(2s). For all ρ 0, the minimax rate of convergence over ρB d 1 in the Gaussian linear model with fixed design X in expectation is given by min

σ 2 Rank(X) N , ρσ 1 N log eσd ρ √ N , ρ 2 . (3.4.5)
The latter result holds in expectation whereas we are interested in deviation results. Even though the minimax rate of convergence in deviation over ρB d 1 in the Gaussian linear model under RIP has not been established, we believe that this rate of convergence in deviation is identical to the one given in Proposition 3.4.4. Note that a proof of this fact follows from Section 3 in [START_REF] Lecué | Learning subgaussian classes: Upper and minimax bounds[END_REF] for the minimax lower bound in deviation and from the quadratic / multiplier decomposition of the excess loss together with Lemma 3.4.6 below to show that the ERM over ρB N 1 achieves the minimax bound in deviation. We do not provide the proof here but we will use (3.4.5) as a benchmark for our regularization function.

Theorem 3.4.5. Let X ∈ R N ×d be such that the column vectors of X are in B N 2 . Consider the following regularization function:

ρ 0 → Ψ(ρ) = c 0 r X 2 (ρ)
where r X (ρ) is defined in (3.4.4) and c 0 2 is an absolute constant. Then there exist absolute constants κ 1 , κ 2 and κ 3 such that for any t * ∈ R d the RERM t constructed from the data

Y = Xt * + ξ: t ∈ argmin t∈R d Y -Xt 2 L 2 N + Ψ( t 1 )
satisfies with probability greater than 1 -

κ 1 exp -κ 2 N r X ( t * 1 ) 2 /σ 2 , X( t -t * ) 2 L N 2 κ 3 min σ 2 Rank(X) N , t * 1 σ √ N log edσ t * 1 √ N , t * 1 σ log(ed) N , .
Note that the probability estimate 1κ 1 exp -κ 2 N r X ( t * 1 ) 2 /σ 2 3/4 only when t * 1 ∆ 0 σ log(ed)/N for some absolute constant ∆ 0 large enough. Therefore, if (3.4.5) is indeed the minimax rate of convergence over ρB d 1 for the deviation 1δ N = 3/4 under RIP then Theorem 3.4.5 proves that ρ 0 → Ψ(ρ) = c 0 r X 2 (ρ) is a minimax regularization function for the d 1 -norm over R d \∆ 0 σ log(ed)/N B d 1 for the constant confidence regime.

Proof of Theorem 3.4.5

The proof is split into a probabilist part used to identify a high probability event Ω 0 on which the multiplier process is well controlled on the entire space R d and a deterministic part where it is proved that, on the event Ω 0 ,

P N L Ψ t > 0 if X(t -t * ) 2 L 2 N r X ( t * 1 ) 2 .

Minimax regularization

To prove Theorem 3.4.5, we use the same argument as in the proof of Theorem 3.1.4 for the random design. Let ρ * = 10 t * 1 /η and split R d into three zones:

• the "central zone" t * + ρ * B d 1 ,
• the intermediate "peeling zone":

{t ∈ R d : ρ * < t -t * 1 2 K 0 ρ * } -to be considered only when K 0 1. This part of R d is itself partitioned into K 0 shelves: for k = 1, . . . , K 0 , {t ∈ R d : 2 k-1 ρ * < t -t * 1 2 k ρ * },
• the "exterior zone":

{t ∈ R d : t -t * 1 > 2 K 0 ρ * } on which r X is constant equal to r 0 .
For all k = 0, . . . , K 0 , we denote by A k the event on which for all

t ∈ t * + 2 k ρ * B d 1 , |P N M t-t * | 1 2 max r X (2 k ρ * ) 2 , X(t -t * ) 2 L 2 N .
We consider the event

Ω 0 = A 0 ∩ • • • ∩ A K 0
. It follows from Lemma 3.4.6 and an argument similar to the one in Lemma 3.3.2 that for some absolute constants κ 1 , κ 2 and κ 4 ,

P Ω 0 1 -κ 1 exp -κ 2 N r X ( t * 1 ) 2 /σ 2
as long as t * 1 κ 4 σ/ √ N (which is the case when t * 1 ∆ 0 σ log(ed)/N for ∆ 0 κ 4 ). Let us now assume for the remaining of the proof that Ω 0 holds. Note that unlike in the random design case, there is no event such as Ω * in Ω 0 on which the quadratic process is controlled, because, in the deterministic design case this process is deterministic.

Our strategy is to show that t belongs to the "central zone". To that end it is enough to prove that P N L Ψ t > 0 for every t ∈ R d such that tt * 1 > ρ * because by definition P N L Ψ t 0. Let t be in the intermediate peeling zone (which can happen only if ρ * 2ρ 0 ), say in the k-th shell for some k ∈ {0, . . . , K 0 }:

2 k-1 ρ * < t -t * 1 2 k ρ * . In particular t 1 > t * 1 and R t,t * > 0. Therefore, if X(t -t * ) L 2 N r X (2 k ρ * ) then by Lemma 3.4.6, |P N M t-t * | X(t -t * ) 2 L 2 N = P N Q t-t * and so P N L Ψ t > 0. Now, if X(t -t * ) L 2 N r X (2 k ρ * ) then by
Lemmas 3.4.6 and 3.4.8, for c 0 2,

|P N M t-t * | 1 2 r X (2 k ρ * ) 2 < c 0 2 r X 2 ( t -t * 1 )
and since

t 1 t -t * 1 -t * 1 4 t * 1 , and 4 t * 1 η ρ * /2 η ρ 0 , by Lemma 3.4.7 , one has c 0 r X 2 ( t 1 ) 2c 0 r X 2 ( t * 1 ). As a consequence, R t,t * c 0 r X 2 ( t 1 )/2 > |P N M t-t * | and so P N L Ψ t > 0.
Let us now tackle the exterior zone in both cases ρ * 2ρ 0 and ρ

* > 2ρ 0 . Let t ∈ R d be such that t -t * 1 > 2 K 0 ρ * . We have R t,t * 0 because Ψ( t 1 ) = c 0 r X 2 ( t 1 ) = c 0 r 2 0 c 0 r X 2 ( t * 1 ) = Ψ( t * 1 ). Let t = t * + α t (t -t * ) for some 0 < α t < 1 be such that t -t * 1 = 2 K 0 ρ * . By definition of K 0 and ρ 0 , we have X(t -t * ) L 2 N r 0 . Therefore, since A K 0 ⊂ Ω 0 , we have |P N M t -t * | (1/2) X(t -t * ) 2 L 2
N which implies that P N Q t -t * + P N M t -t * > 0 and therefore by an homogeneity argument that P N Q t-t * + P N M t-t * > 0. Finally, given that R t,t * 0 we conclude that P N L Ψ t > 0.

Minimax regularization function in the fixed design setup

This proves that t lies in the central zone in both cases ρ * 2ρ 0 and ρ * > 2ρ 0 . But, now given that A 0 ⊂ Ω 0 , we have

|P N M t-t * | 1 2 max r X (ρ * ) 2 , X( t -t * ) 2 L 2 N . If X( t -t * ) 2 L 2 N r X (ρ * ) 2 the proof is over and otherwise |P N M t-t * | (1/2) X( t -t * ) 2 L 2 N which implies that t -t * 2 L 2 N 2Ψ( t * 1 ) 2c 0 r X (ρ * ) 2 because 0 P N L Ψ t 1 2 t -t * 2 L 2 N + Ψ( t 1 ) -Ψ( t * 1 ).
This proves, on the event Ω 0 , that tt * 2

L 2 N 2c 0 r X (ρ * ) 2 .
Chapter 4

Improvements on an online convex optimization algorithm: MetaGrad

This chapter comes from joint work with Tim van Erven and Dirk van der Hoeven.

In this chapter, we tackle the Online Convex Optimization framework: the losses can be any convex function, and the goal is to minimize the (cumulative) regret. The methods to use, and the optimal performance, depend on the curvature of the losses: additional assumptions on it (such as strong convexity) allow a better performance; however, the algorithms that achieve optimal bounds are often specific and need to be tuned with a priori knowledge on the convexity parameters.

We focus on a recent algorithm, MetaGrad, introduced by van Erven and Koolen [2016], that, interestingly, is adaptive to the level of curvature: it provides optimal or nearly-optimal bounds for various types of convexities, without requiring the user to tune it beforehand using a priori knowledge on the curvature.

Our contributions are twofold. First, we present modifications of MetaGrad that reduce computation time, and exhibit some corresponding theoretical bounds. Secondly, we tackle another framework, the batch setting, where i.i.d. losses are suffered, and we modify the classical "online-to-batch" adaptation of MetaGrad (and of another algorithm, Online Newton Step) to improve the bounds and take benefits of the adaptivity properties of MetaGrad, in the case of strongly convex losses.

Online Convex Optimization. In this chapter, we will go beyond linear regression with convex losses to tackle a more general framework: online convex optimization. In this setup, we still provide round after round a vector w t in a domain K ∈ R d and then get a loss f t (w t ), that we want to minimize, but the only specification about this loss is that is is convex.

In the particular setup of linear regression using the square loss, f t (w t ) = (w t x ty t ) 2 , where y t is the observation, and x t is the vector of explanatory variables.

The domain of interest K ⊂ R d is convex; we will assume it is closed. It is generally fixed, but in some cases it can vary over time (cf. Section 4.2.2 for an example).

We will assume in this chapter that the number of rounds T is known beforehand. We will first focus on the case where the f t are deterministic and unknown in advance: they can be any convex functions. The goal is then to minimize the following regret:

T t=1 f t (w t ) -min u∈K T t=1 f t (u).
Another framework in Section 4.3.

We will see in Section 4.3 a batch stochastic framework, where the f t are random functions, i.i.d., drawn from an unknown distribution P. Then, the goal will be (leaning on a learning sample known in advance) to build an algorithm which outputs a vector w T with a small expected regret, i.e., which tries to minimize:

E f,f 1 ,...,f T ∼P [f ( w T )] -min u∈K E f ∼P [f (u)]
where the first expectation is defined with respect to f, f 1 , . . . , f T .

The MetaGrad algorithm

First-order information. We recall that the mere hypothesis of convexity (without any smoothness assumption) guarantees the existence of a non-empty subgradient at any point in the interior K of K:

∀ x ∈ K, ∂f t (x) = ∅, where ∂f t (x) := v : ∀ y ∈ K, f t (y) f t (x) + v (y -x) .
We will focus on the case of first-order information, where one has access after time t to an element of the subgradient of the loss: g t ∈ ∂f t (w t ) (we will assume that this subgradient will be non-empty even if w t / ∈ K, which is the case for instance if the f t are differentiable). The process of online convex optimization with first-order information is summarized in Setting 6.

Setting 6 Online convex optimization (with first-order information) framework for t = 1, 2, . . . , T :

1. Play w t ∈ K t 2. The environment picks a convex loss function f t : K t → R 3. Incur cumulated loss by f t (w t ) and observe (sub)gradient g t ∈ ∂f t (w t ) end for Impact of the loss functions. Let us recall some elements about the losses (see Section 2.3.2 of Chapter 2 for more details). The loss functions, and in particular their curvature, impact strongly the results one can get, and the efficiency of algorithms. For general convex losses, the "Online Gradient Descent" algorithm, introduced by [START_REF] Zinkevich | Online convex programming and generalized infinitesimal gradient ascent[END_REF] and based on the following update: w t+1 = w tη t ∇f t (w t ), achieves the optimal regret O( √ T ) when the parameter η t is of order 1/ √ t. In the case of strongly convex losses, the same Online Gradient Descent algorithm, with the parameter η t chosen of order 1/t, achieves a regret of order O(log T ). As for the intermediate case of exp-concave losses, the "Online Newton Step" algorithm, introduced by [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF], guarantees a bound O(d log T ) on the regret. As this algorithm will play a key role in this chapter, we detail in Algorithm 7 a version of Online Newton Step adapted to the context of our works.

The question of adaptivity, with respect to the degree of curvature (type of convexity, but also for instance strong convexity parameter or exp-concavity parameter) is therefore very important, as much for theoretical problems as for practical applications (for which this degree of curvature is often unknown); that is precisely one of the key features of the algorithm we focus on in this chapter: the MetaGrad algorithm.

The MetaGrad algorithm. The MetaGrad algorithm is an online convex optimization algorithm introduced in van Erven and Koolen [2016]. It aims at adaptivity on two aspects: the type of curvature (discussed in the previous paragraph) and the optimal learning rate. To do so, it relies on two algorithmic layers. The first layer is composed of several "parallel" algorithms, nicknamed the "slaves", that only differ by their learning rate, and are close in spirit to Online Newton Step. The second layer is composed of a meta-algorithm, nicknamed

Initialization:

Get the starting point w 1 and the initial covariance matrix inverse A 0 = εI d .

for t = 1, 2, . . .

1. Observe (sub)gradient g t ∈ ∂f t (w t ) 2. A t = A t-1 + mg t g t 3. w t+1 = w t -(A t ) -1 ηg t 4. w t+1 = argmin u∈K (u -w t+1 )A t (u -w t+1 )
the "master", that will perform an aggregation (using modified exponential weights) upon the slaves' forecasts, and thus will be able, in some sense, to "learn the learning rate". MetaGrad is detailed in Algorithm 8 and Algorithm 9. It uses in the master algorithm a surrogate loss, which depends on the output of the algorithm w t :

η t (u) = η(u -w t ) g t + η(u -w t ) g t 2 .
(4.1.1)

Several remarks can be made on the algorithms. First, the slave update is very close to the Online Newton Step update, but here the gap of the gradient measurement point w t and the slave output w η t leads to the addition of an extra term 2η 2 (A η t ) -1 g t g t (w η tw t ) to try to counterbalance this gap. Also notice that the projection in Step 5 of Algorithm 9 is not the usual Euclidian projection, but uses instead a Mahalanobis norm that depends on the data and on the previous choices of w t , and is a metric more adapted to the situation (in particular it will be useful in the analysis). The matrix corresponding to the dot product of this norm, A η t , is denoted by Σ η t+1 -1 in the original paper van Erven and Koolen [2016].

As for the master algorithm, its update uses "tilted" exponential weights (in the sense that they are multiplied by their learning rate), giving larger weights to the largest η; this will be motivated by the analysis (but one could also add that in practice, favouring a bit the large learning rate is often a good idea, the smallest learning rate being generally too conservative, at least for well-behaved data).

Analysis

We recall in this Section a major theorem of [START_REF] Van Erven | MetaGrad: Multiple Learning Rates in Online Learning[END_REF], and afterwards some analyses, and intermediate results leading to it, upon which we will build the arguments of the next sections.

g t 2 .
Starting point w 1 . Initial covariance matrix inverse A 0 = εI d .

Initialization:

1. Define the grid of learning rates: η i = 2 -i /(5DG) for i = 0, 1, . . . , log 2 (T )/2 with prior weights

π η i 1 = 1+1/(1+ log 2 (T )/2 ) (i+1)(i+2)
. 2. Launch the corresponding MetaGrad Slaves algorithms (Algorithm 9). 3. Send the starting point w 1 and the initial covariance matrix inverse A 0 = εI d to the slaves.

for t = 1, 2, . . .:

1. Get prediction w η t ∈ K of slave η (Algorithm 9) for each η.

Play w

t = η π η t ηw η t η π η t η ∈ K. 3. Observe gradient g t ∈ ∂f t (w t ). 4. Update π η t+1 = π η t e -η t (w η t ) ν π ν t e -ν t (w ν t )
for all η where the surrogate loss η t is defined in (4.1.1)

Algorithm 9 MetaGrad Slave Input: Learning rate 0 < η 1 5DG . Domain K.

Initialization:

Get the starting point w η 1 := w 1 and the initial covariance matrix inverse A η 0 := A 0 = εI d from MetaGrad Master (Algorithm 8).

for t = 1, 2, . . .

1. Send w η t to MetaGrad Master (Algorithm 8) 2. Observe (sub)gradient g t ∈ ∂f t (w t ) (at master point w t ) ### Update: 3. A η t = εI d + 2η 2 t s=1 g s g s 4. w η t+1 = w η t -(A η t ) -1 ηg t + 2η 2 g t g t (w η t -w t ) 5. w η t+1 = argmin u∈K (u -w η t+1 )A η t (u -w η t+1 )
4. Improvements on an online convex optimization algorithm: MetaGrad A bound on the MetaGrad algorithm . Let us recall Theorem 1 of van Erven and Koolen [2016].

Theorem 4.1. Let g t = ∇f t (w t ) and, for any

u in K, V u T = T t=1 (u -w t ) g t 2 .
Then the regret of MetaGrad is simultaneously bounded by O T log log(T ) and by

T t=1 f t (w t ) - T t=1 f t (u) T t=1 (w t -u) g t = O V u T d log(T ) + d log(T )
The inequality in this theorem is a direct consequence of the convexity and of the subgradient definition. We recall in the remaining of this section some of the steps to upper bound for any u the quantity Ru t := T t=1 g t (w tu).

A useful lemma for the master algorithm analysis.

Lemma 4.2.

Define Φ t := η π η 1 e -t s=1 η s (w η s ) . When η ∈ [0, 2 3DG ], the master algorithm guarantees 1 = Φ 0 Φ 1 . . . Φ T . Proof. Recall that η s (u) = η(u -w s ) g s + η(u -w s ) g s 2 .
The Cauchy-Schwarz inequality gives:

|(u -w s ) g s | u -w s 2 g s 2 DG.
Applying the inequality e x-x 2 1 + x, true for any x -2/3, we have that

e -η s (w η s ) 1 + η(w s -w η s ) g s for any η ∈ [0, 2 3DG
].

This shows that the potential is non-increasing:

Φ t+1 -Φ t = η π η 1 e -t s=1 η s (w η s ) e -η t+1 (w η t+1 ) -1 η π η 1 e -t s=1 η s (w η s ) η(w t+1 -w η t+1 ) g t+1 = 0,
where the final equality is due to the definition of w t+1 . The initialization Φ 0 = 1 is immediate since

η π η 1 = 1.
Slaves algorithms. Now, following [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF] (see Lemma 4.8 below and its proof) and van Erven and Koolen [2016], let us analyse the slaves performance.

For readability convenience, in all the following, we will write A t instead of A η t when there is no ambiguity. Lemma 4.3. For η ∈ (0, 1 5DG ] we have that the regret of the slave algorithm is bounded by:

T t=1 η t (w η t ) T t=1 η t (u) + ε 2 u -w η 1 2 2 + 1 2 log det I d + 2η 2 ε T t=1 g t g t (4.1.2)
for any u ∈ K.

The MetaGrad algorithm

Proof. In the first part of the proof, we will obtain a useful form for T t=1 η t (w η t )-T t=1 η t (u), using the updates of the algorithm.

We denote M t = g t g t .

η t (w η t ) -η t (u) = -η(w t -w η t ) g t + η 2 (w t -w η t ) M t (w t -w η t )+ η(w t -u) g t -η 2 (w t -u) M t (w t -u) = η(w η t -u) g t + η 2 w η t M t w η t -2η 2 w η t M t w t -η 2 u M t u + 2η 2 u M t w t = η(w η t -u) g t + η 2 2w η t M t w η t -2w t M t w η t + 2u M t w t -2u M t w η t - η 2 2w η t M t w η t + u M t u -2u M t w η t = η(w η t -u) g t + 2η 2 (w η t -u) M t (w η t -w t ) -η 2 (u -w η t ) M t (u -w η t ) = η(w η t -u) gt -η 2 (u -w η t ) M t (u -w η t )
, where gt = 1 + 2ηg t (w η tw t ) g t . One has wη t+1 = w η t -ηA -1 t gt . The following argument is then similar to the Online Newton Step analysis (cf. proof of Lemma 4.8). One can use the following classical property of projections:

(u -w η t+1 ) A t (u -w η t+1 ) (u -wη t+1 ) A t (u -wη t+1 ) = (u -w η t + ηA -1 t gt ) A t (u -w η t + ηA -1 t gt ) = η 2 g t A -1 t gt + (u -w η t ) A t (u -w η t ) -2η(w η t -u)
gt . Thus, one gets:

η(w η t -u) gt 1 2 η 2 g t A -1 t gt + (u -w η t ) A t (u -w η t ) -(u -w η t+1 ) A t (u -w η t+1 ) . Therefore, we have η t (w η t ) -η t (u) η 2 2 g t A -1 t gt + 1 2 (u -w η t ) A t (u -w η t ) - 1 2 (u -w η t+1 ) A t (u -w η t+1 ) -η 2 (u -w η t ) M t (u -w η t ) (4.1.3)
This bound does not make use of the real values and construction of the A s (it is true for any symmetric positive matrix A s ): this will be the key element for comparisons in the "Sketching" Section 4.2.2.

One can divide the right-hand side of the "cumulated" version of (4.1.3) into two parts:

T t=1 η t (w η t ) - T t=1 η t (u) R G + R D with R G = T t=1 η 2 2 g t A -1 t gt and R D = T t=1 1 2 (u -w η t ) A t (u -w η t ) - 1 2 (u -w η t+1 ) A t (u -w η t+1 ) -η 2 (u -w η t ) M t (u -w η t ) . (4.1.4)
4. Improvements on an online convex optimization algorithm: MetaGrad

The notations R G and R D , taken from [START_REF] Luo | Efficient second order online learning by sketching[END_REF], stand respectively for "gradient (part of the) regret" and "diameter (part of the) regret".

In the second part of the proof, we study and bound more precisely R G and R D to get the desired result.

Recalling that A t+1 = A t + 2η 2 M t+1 and A 0 = εI d , one gets the following telescopic sum:

R D = T t=1 1 2 (u -w η t ) A t (u -w η t ) - 1 2 (u -w η t+1 ) A t+1 (u -w η t+1 )+ η 2 (u -w η t+1 ) M t+1 (u -w η t+1 ) -η 2 (u -w η t ) M t (u -w η t ) = T t=1 (u -w η 1 ) ( 1 2 A 1 -η 2 M 1 )(u -w η 1 ) - 1 2 (u -w η T +1 ) A T (u -w η T +1 ) Therefore: R D ε 2 u -w η 1 2 2 (4.1.5)
Notice that w η T +1 is only an artifact of computation that does not need to be known, since (uw η T +1 ) A T (uw η T +1 )/2 is simply lower bounded by 0 in the computations. For η ∈ (0, 1 5GD ], the Cauchy-Schwarz inequality yields:

1 + 2ηg t (w η t -w t ) 1 + 2η g t 2 w η t -w t 2 1 + 2ηGD 7 5 ,
and similarly 1 + 2ηg t (w η tw t ) 3/5. So for all t,

η 2 2 g t A -1 t gt 7 5 2 η 2 2 g t A -1 t g t η 2 g t A -1 t g t .
Let us recall Lemma 11 of [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF].

Lemma 4.4. Let u t ∈ R d for t = 1, . . . , T such that for some r > 0, u t 2 r. Define V t = t s=1 u s u s + εI d (and V 0 = εI d ). Then:

T t=1 u t V -1 t u t log det(V T ) det(V 0 ) d log r 2 T ε + 1 .
Applying this lemma with u t = √ 2ηg t , one gets that:

R G := T t=1 η 2 2 g t A -1 t gt 1 2 T t=1 2η 2 g t A -1 t g t 1 2 log det I d + 2η 2 ε T t=1 g t g t (4.1.6)
The result follows from (4.1.4), (4.1.5) and ( 4.1.6).

Speeding MetaGrad up

Combined analyses. Those results on the master algorithm and on the slaves algorithms allow to bound for any u the quantity Ru t := T t=1 g t (w tu) and therefore, by convexity,

T t=1 f t (w t ) -f t (u).
Lemma 4.5. Starting from an arbitrary point w 1 ∈ K, apply T iterations of MetaGrad, using as covariance matrix initialization A η 0 = εI d . Then, for any η ∈ (0, 1 5DG ], and for any u ∈ K:

Ru T η T t=1 (u -w t ) g t 2 + 1 η ε 2 w 1 -u 2 2 -log(π η 1 ) + 1 2 log det I d + 2 η 2 ε T t=1 g t g t
Proof. By Lemma 4.2, one has log(Φ T ) 0. Moreover, by Lemma 4.3,

log(Φ T ) log (π η 1 ) - T t=1 η t (w η t ) log (π η 1 ) - T t=1 η t (u) - ε 2 u -w 1 2 2 - 1 2 log det I d + 2η 2 ε T t=1 g t g t
Then, combining the two previous inequalities and using the definition of η t (•) leads to:

η Ru T ε 2 u -w 1 2 2 -log (π η 1 ) + η 2 T t=1 (u -w t ) g t 2 + 1 2 log det I d + 2η 2 ε T t=1 g t g t
and dividing by η gives the result.

It remains then to choose correctly the grid of the η's to obtain guarantees on the regret, as it is shown in Theorem 7 of van Erven and Koolen [2016], which leads to Theorem 4.1, and in Section 4.3.3.

Speeding MetaGrad up

Motivation. One limitation of MetaGrad is computational: it is its speed. In high dimension d, the matrices d × d are heavy to manage (in particular as far as the projection step is concerned). It is therefore natural to try and modify the original MetaGrad (called "full MetaGrad" in van Erven and Koolen [2016]) to accelerate it, if possible without (or at least without too much) spoiling its theoretical guarantees and practical performance.

A previous attempt. A first attempt is made in van Erven and [START_REF] Van Erven | MetaGrad: Multiple Learning Rates in Online Learning[END_REF]: "Diagonal MetaGrad". It consists in replacing the covariance matrices Σ t+1 = A -1 t by diagonal matrices, which decreases strongly the computing time. The paper provides an analysis for "Diagonal MetaGrad" quite similar to the full version, though it does not provide all the same theoretical bounds. Some experiments that we made on real datasets (not detailed in this thesis) show some losses of accuracy in the case of "Diagonal MetaGrad" compared to the full version.

Improvements on an online convex optimization algorithm: MetaGrad

Our work. To speed-up MetaGrad, we suggest exploiting the similarity between the Meta-Grad slave algorithm and the Online Newton Step algorithm (introduced in [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]) to adapt two ideas introduced for Online Newton Step in [START_REF] Luo | Efficient second order online learning by sketching[END_REF]. The first one is to use a data-dependent evolving domain, to make the projection easier. The second idea is to use sketching techniques to reduce the dimension of matrices at stake. Thus, in this section, we adapt some of the corresponding proofs and results of [START_REF] Luo | Efficient second order online learning by sketching[END_REF] to MetaGrad.

We focus on losses based on linear regression: f t : w t → t (w t x t ) with known feature vector x t , where the t : R → R are convex and differentiable.

Changing the domain to accelerate the projection

Speed key step: the projection. The bottleneck of MetaGrad computations is often the projection step:

w η t+1 = argmin u∈K (u -w η t+1 )A η t (u -w η t+1 ).
This projection is not bound to occur at each step (if wη t+1 ∈ K, then the projection step is just skipped: w η t+1 = wη t+1 ); possibly, if K is large enough, it may just never occur. But when it is needed, it is the most complex step of the algorithm.

A possible improvement. An acceleration can be obtained if one defines a data-dependent domain that evolves along time. We assume that an upper bound L is known on t (z) for all z ∈ R. The data-dependent domain will depend on a size C that must be chosen beforehand such that C 5DG/3L. Then, for the t-th forecast, given the feature vector x t , one works in the domain

K t := {w : |w x t | C}.
The interest of this domain change is the following quick-to-compute, closed-form expression provided by [START_REF] Luo | Efficient second order online learning by sketching[END_REF] to project on K t :

w η t+1 = min w∈K t+1 (w -wη t+1 ) A t (w -wη t+1 ) = wη t+1 - τ C (x t+1 wη t+1 ) x t+1 A -1 t x t+1 A -1 t x t+1
where τ C (y) = sign(y) max{|y| -C, 0}.

As the algorithm gives guarantees at time t compared to vectors of K t , it is natural then to seek guarantees comparing the algorithm to the vectors that belong to all the K t . Thus, the comparison K set will be the intersection of all these domains:

K = t=1..T K t = {w : ∀ t = 1..T, |w x t | C}
Using the Cauchy-Schwarz inequality, one can see that K contains a minima the Euclidean ball {w : w 2 C/(max t ( x t 2 ))}.

Speeding MetaGrad up

One can check that Lemma 4.2 still holds for η ∈ [0, 1/5DG]. This comes from the fact that w t g t = w t x t t (w t x t ), which leads for any s to:

|η(w s -w η s ) g s | |ηL(w s -w η s ) x s | 2ηLC.
Since C 5DG/3L and η 1/5DG one has η(w sw η x ) g s -2/3. Therefore one can still apply e -η s (w η s )

1 + η(w sw η s ) g s , which leads to the lemma. One can easily check that the other parts of the analysis of MetaGrad remain unchanged by this change of domain.

Sketching to deal with smaller matrices

In this section, we consider modifications for the slaves algorithm, the master algorithm (which is fast enough to compute) being unchanged.

Idea of sketching: let us contract time!

The computational complexity of the algorithm increases substantially with the dimension d. The largest matrices to maintain are the d × d covariance matrices Σ η t+1 = A -1 t . In high dimension, computations involving A -1 t are costly. To speed the computations up, first, notice that A t can be written as:

A t = εI d + G t G t
where the t×d matrix G t is such that its i-th row is √ 2ηg i . One can then write the following, as a special case of the Woodbury formula:

A -1 t = 1 ε I d -G t εI t + G t G t -1 G t .
This computation involves the inverse of the t × t square matrix εI t + G t G t . The idea of sketching is then to approximate G t G t by a matrix S t S t where S t is a m × d matrix, with m < t and, as much as possible, m d. We want to contract time, in some sense. . . Then, replacing A t by Ãt = εI d + S t S t , and denoting H t = (εI m + S t S t ) -1 (which is only an m × m matrix), one can apply again the Woodbury formula:

Ã-1 t = 1 ε I d -S t H t S t
If one uses the projection seen in Section 4.2.1, and denotes:

γ t = τ C (x t+1 wη t+1 )/(x t+1 x t+1 -x t+1 S t H t S t x t+1
) then (with gt = 1 + 2ηg t (w η tw t ) g t ) the slave updates now become:

wη t+1 =w t - η ε g t (I d -S t H t S t ) w t+1 = wη t+1 -γ t (x t+1 -S t H t S t x t+1 ).
4. Improvements on an online convex optimization algorithm: MetaGrad Operations involving S t gt or S t x t+1 have a computational cost of O(md), and the operations with H t have a cost of O(m 2 ). Hence, as long as we can maintain H t and S t efficiently, this new "Sketched MetaGrad" is then (much) faster than the original MetaGrad.

We show, in the following, ways of obtaining possible S t matrices (and we introduce the corresponding modified versions of the matrix A t in the MetaGrad slaves algorithm), based on random projections, and on "frequent directions", and study the impact of this sketching on the bounds of MetaGrad. Other approaches exist, for instance sparsifying matrices (cf. [START_REF] Arora | A fast random sampling algorithm for sparsifying matrices[END_REF] and [START_REF] Achlioptas | Fast computation of low-rank matrix approximations[END_REF]), or using only a subset of the columns or rows -"(sub)sampling"-of the matrices (cf. [START_REF] Boutsidis | An improved approximation algorithm for the column subset selection problem[END_REF] and [START_REF] Boutsidis | Near-optimal column-based matrix reconstruction[END_REF]).

Random Sketching

General principles. The first approach we adapt to MetaGrad is a "Random Sketching" presented in [START_REF] Luo | Efficient second order online learning by sketching[END_REF], which is based on a multiplication by a random matrix. This idea of random dimensionality reduction can be found in [START_REF] Achlioptas | Database-friendly random projections[END_REF], or in [START_REF] Har-Peled | Approximate nearest neighbor: Towards removing the curse of dimensionality[END_REF] for the approximate nearest neighbours problem.

Following [START_REF] Luo | Efficient second order online learning by sketching[END_REF], instead of appending a new row √ 2ηg t to G t-1 to update it into G t , here we add to S t-1 the outer product √ 2ηr t g t of √ 2ηg t with a Gaussian random vector r t of dimension m. The idea behind that approach is that with high probability, it achieves some kind of approximate "isometry" (cf. Theorem 2.3 of Woodruff [2014], and (4.2.1) below). We define the following update for S t : S t = S t-1 + √ 2ηr t g t with r t ∼ N (0, (1/ √ m)I m ). Then one can check that it is possible to update H -1 t by two rank-one updates:

H -1 t = H -1
t-1 +q t r t +r t q t where q t := √ 2ηS t g tη 2 g t 2 2 r t . Therefore, using the Woodbury formula, this leads to a O(md) update of S t and H t .

The sketching algorithm. The Random Sketching algorithm is described in Algorithm 10.

Algorithm 10 Random projection sketch

Input: dimension m. Initialization:

1. Set S 0 = 0 m×d . 2. Set H 0 = 1 ε I m . Sketch update: 1. Draw r t ∼ N 0, 1 √ m I m and update S t = S t-1 + √ 2ηr t g t . 2. Compute q t = √ 2ηS t g t -η 2 g t 2 2 r t . 3. Update Ht-1 = H t-1 - H t-1 qtr T t H t-1 1+r T t H t-1 qt and H t = Ht-1 -Ht-1 rtq T t Ht-1 1+q T t Ht-1 rt .
Bounds obtained for MetaGrad. One has the following bounds for the η slave of the "Random Sketched MetaGrad". We recall that gt := 1 + 2ηg t (w η tw t ) g t , and use an 4.2. Speeding MetaGrad up adapted version of the quantities R G and R D defined in (4.1.4). We use a modified version of A t that we denote by A t,r (r standing for "random projection").

Theorem 4.6. Let A t,r = εI d + S t S t and η ∈ (0, 1 5GD ]. For any δ ∈ (0, 1) and any ρ ∈ (0, 1), if the sketch size m is such that m = Ω((rank(G T ) + log(T /δ))ρ -2 ) then the following holds. First, with probability greater than 1δ, for any vector u ∈ K:

R G := T t=1 η 2 2 gT t A -1 t,r gt 1 2(1 -ρ) log det I d + 2η 2 ε T t=1 g t g t
Secondly, for any vector u ∈ K:

R D : = E T t=1 1 2 (u -w η t ) A t,r (u -w η t ) - 1 2 (u -w η t+1 ) A t,r (u -w η t+1 ) -η 2 (u -w η t ) M t (u -w η t ) ε 2 u -w 1 2 2
Quite naturally, one has to make a trade-off between on the one hand the regret bound (ρ parameter) and the probability (δ parameter) of the algorithm, on the other hand, the size of the sketch m (which depends on δ and ρ). But if m if chosen correctly (i.e. large enough), then one can see by a comparison of Theorem 4.6 with (4.1.5) and (4.1.6), that the bounds of the original MetaGrad are preserved (exactly for R D , up to a multiplicative constant and with probability 1δ for R G ).

One can easily see that the same bound remains valid if one replaces the domain (0, 1/(5DG)) by (0, 1/(10LC)) (where L and C are defined in the framework of Section 4.2.1), because one keeps then the bounds 3/5 1 + 2ηg t (w η tw t ) 7/5. The fact that the bound on R G holds with probability 1δ whereas for R D we have a bound on an expectation, prevents uniting them to get a global expression on the regret (cf. decomposition (4.1.4)). In the following section, we will present another algorithm, "Frequent Directions", for which we will derive a complete regret bound: see (4.2.4).

Proof. The starting point is the following property of the random projection method (cf. Theorem 2.3 of Woodruff [2014]): since the sketch size m is such that m = Ω (rank(G t ) + log(T /δ)) ρ -2 , one has on an event of probability greater than 1δ:

∀ t = 1..T, (1 -ρ)G t G t S t S t (1 + ρ)G t G t (4.2.1)
(where A B means that the matrix B -A is positive semi-definite).

On this event, one has:

A -1 t,r = εI d + S t S t -1 1 1 -ρ εI d + G t G t -1 = 1 1 -ρ A -1 t . (4.2.2)
Combining the end of the proof of Lemma 4.3 with (4.2.2) leads to:

R G := T t=1 η 2 2 g t A -1 t,r gt 1 2 T t=1 2η 2 g t A -1 t,r g t 1 2(1 -ρ) log det I d + 2η 2 ε T t=1 g t g t 4.
Improvements on an online convex optimization algorithm: MetaGrad Let us tackle R D now. Since for t 1, A t,r = εI d + S t S t = εI d + S t-1 + √ 2ηr t g t S t-1 + √ 2ηr t g t , one has that for t 1:

A t,r -A t-1,r = √ 2ηS t-1 r t g t + √ 2ηg t r t S t-1 + 2η 2 r t 2 2 g t g t .
As r t is independent of S t-1 and g t , and r t ∼ N (0, (1/ √ m)I m ), one has E S t-1 r t g t = 0,

E g t r t S t-1 = 0 and E r t 2 2 g t g t = E g t g t = E [M t ]. So E [A t,r -A t-1,r ] = 2η 2 E [M t ]
. But r t is also independent of w η t , so one has for any u ∈ K:

E (u -w η t ) (A t,r -A t-1,r ) (u -w η t ) = 2η 2 E (u -w η t ) M t (u -w η t )
Therefore, R D is a telescopic sum, and (using as in (4.1.5) w T +1 as an artifact of computations that does not need to be known):

R D = E 1 2 (u -w η 1 ) A 1,r (u -w η 1 ) -η 2 (u -w η 1 ) M 1 (u -w η 1 ) - 1 2 (u -w η T +1 ) A T,r (u -w η T +1 ) (u -w η 1 ) E 1 2 A 1,r -η 2 M 1 (u -w η 1 ) = ε 2 u -w 1 2 2

Frequent directions algorithm

General principles. The algorithm we address in this section is deterministic. It is called "Frequent Directions" and is presented in [START_REF] Ghashami | Frequent directions: Simple and deterministic matrix sketching[END_REF] and [START_REF] Liberty | Simple and deterministic matrix sketching[END_REF]. We transfer some results and bounds obtained by [START_REF] Luo | Efficient second order online learning by sketching[END_REF] (Theorem 3) for Online Newton

Step, to MetaGrad. The approach is based on a SVD decomposition of the sketch, followed by a deletion of its last row, replaced later by the new data (the vector √ 2ηg t ). The SVD decomposition allows the procedure to lead to a matrix H t that is diagonal and therefore easy to deal with. The idea of deletion was originally used to compute item frequencies (cf. [START_REF] Misra | Finding repeated elements[END_REF]). A collection of items can be represented as a Boolean matrix, whose columns form a dictionary and whose rows are indicator vectors. Then matrix sketching is the equivalent of"smart" count of items in [START_REF] Misra | Finding repeated elements[END_REF]. The idea of deleting items so that only the most frequent ones remain, transposes as shrinking rows (and even making null at least one row at each step). Thus, one keeps only the most "frequent directions". This argument is presented in [START_REF] Ghashami | Frequent directions: Simple and deterministic matrix sketching[END_REF].

The sketching algorithm. The "Frequent Directions" sketching algorithm is described in Algorithm 11.

As before, the matrices H t and S t built in the algorithm will be used to compute a modified version of A t (and its inverse), that we will denote by A t,fd (fd standing for "frequent directions") and that will replace A t in the slaves algorithm of MetaGrad.

Speeding MetaGrad up

Algorithm 11 Frequent Directions sketch

Input: dimension m. Initialization:

1. Set S 0 = 0 m×d . 2. Set H 0 = 1 ε I m . Sketch update: 1. Replace the last row of S t-1 by √ 2ηg t . 2. Compute eigendecomposition S t-1 S t-1 = V t ΣV t , with Σ diagonal of size m × m, and V t of size m × d satisfying: V t V t = I m . 3. Set S t = (Σ -ρ t I m ) 1 2 V t , where ρ t = Σ m,m . 4. Set H t = diag 1 ε+Σ 1,1 -ρt , 1 ε+Σ 2,2 -ρt . . . , 1 ε ).
One can notice that in Algorithm 11, the last row of S t is null, before being replaced by √ 2ηg t in Step 1 of the sketch update. As for the last diagonal term of H t , it is 1/(ε+Σ m,m -ρ t ) which is 1/ε by definition of ρ t .

To help understanding, we recall here the dimensions of some of the vector and matrices at stake:

g t : d × 1; S t : m × d (so S t S t : d × d); G t : t × d; H t : m × m; V t : m × d; Σ: m × m
(which is actually a bloc extracted from the full diagonal d × d matrix that appears in the eigendecomposition, dropping the last dm null eigenvalues).

Bounds obtained for MetaGrad. We use the same notations as in the "Random Sketching" section. We denote by λ 1 (G T G T ), . . . , λ d (G T G T ) the eigenvalues of the matrix G T G T sorted by decreasing order, and by Ω k := d i=k+1 λ i (G T G T ) the sum of the k smallest of them. Then, one has the following bounds for the η slave of the "Frequent Directions Sketched Meta-Grad".

G := T t=1 η 2 2 g t A -1 t,fd gt 1 2 mΩ k ε(m -k) + log(det(I d + 1 ε S T S T )) R D := T t=1 1 2 (u -w η t ) A t,fd (u -w η t ) - 1 2 (u -w η t+1 ) A t,fd (u -w η t+1 )- η 2 (u -w η t ) M t (u -w η t ) ε 2 u -w 1 2 2 (4.2.3) 4.
Improvements on an online convex optimization algorithm: MetaGrad Therefore, the following bound holds for the Slave algorithm:

T t=1 η t (w η t ) T t=1 η t (u) + ε 2 u -w η 1 2 2 + 1 2 mΩ k ε(m -k) + log(det(I d + 1 ε S T S T )) (4.2.4)
for any u ∈ K.

One asset of this "frequent directions" approach is that it is deterministic and does not require the knowledge of rank(G T ). The bound on R G gives several guarantees simultaneously for various values of k.

Proof. For R D , one first sees that A t,fd -A t-1,fd = S t S t -S t-1 S t-1 . Due to the sketch construction, if one denote S t-1 the modified version of S t-1 with √ 2ηg t inserted in the last row (replacing a null row), then S t-1 S t-1 = S t S t + ρ t V t V t and a direct computation gives that S t-1 S t-1 = S t-1 S t-1 + 2η 2 g t g t . So: .2.5) Therefore:

A t,fd -A t-1,fd = S t S t -S t-1 S t-1 = 2η 2 g t g t -ρ t V T t V t 2η 2 g t g t = 2η 2 M t . ( 4
1 2 (u -w η t ) A t,fd (u -w η t ) - 1 2 (u -w η t ) A t-1,fd (u -w η t ) η 2 (u -w η t ) M t (u -w η t )
So one can use the same argument as in previous Sections (with a telescopic inequality sum instead of a normal telescopic sum) to get that:

R D T -1 t=1 1 2 (u -w η t ) A t,fd (u -w η t ) - 1 2 (u -w η t+1 ) A t+1,fd (u -w η t+1 )+ η 2 (u -w η t+1 ) M t+1 (u -w η t+1 ) -η 2 (u -w η t ) M t (u -w η t ) + 1 2 (u -w η T ) A T,fd (u -w η T ) - 1 2 (u -w η T +1 ) A T,fd (u -w η T +1 )- η 2 (u -w η T ) M T (u -w η T ) 1 2 (u -w η 1 ) A 1,fd (u -w η 1 ) -η 2 (u -w η 1 ) M 1 (u -w η 1 ) - 1 2 (u -w η T +1 ) A T,fd (u -w η T +1 ) (u -w η 1 ) 1 2 A 1,fd -η 2 M 1 (u -w η 1 ) = ε 2 u -w 1 2 2
Let us address R G now. As previously, since η ∈ (0, 1/(5DG)], one has that Since g t A -1 t,fd g t is a scalar, one has (with Tr denoting the trace of a matrix):

g t A -1 t,fd g t = Tr g t A -1 t,fd g t = Tr A -1 t,fd g t g t = 1 2 Tr A -1 t,fd (A t,fd -A t-1,fd + ρ t V T t V t ) (4.2.
7) The second equality is an elementary property of the trace (Tr(AB) = Tr(BA)), and the last one comes from (4.2.5).

Then, we split the right-hand side of the previous equality into two parts.

As A -1 t,fd

(1/ε)I d and V t V t = I m , one has that:

Tr(A -1 t,fd V t V t ) 1 ε Tr(V t V t ) = m ε . (4.2.8)
We will now use the following bound (cf. Theorem 1.1 of [START_REF] Ghashami | Frequent directions: Simple and deterministic matrix sketching[END_REF]) on the eigenvalues that are "deleted" by the Frequent Directions sketching:

∀ k < m, T t=1 ρ t Ω k m -k .
Combining this result and (4.2.8) gives:

T t=1 Tr(ρ t A -1 t,fd V t V t ) mΩ k ε(m -k) . ( 4

.2.9)

As for Tr A -1 t,fd (A t,fd -A t-1,fd ) , we will use the concavity on symmetric positive definite matrices of the function log(det(.)), the fact that its gradient at a matrix A is A -1 (cf. [START_REF] Boyd | Convex optimization[END_REF]), and we recall that the corresponding inner product is: A, B = Tr (AB). Thus, the concavity gradient inequality reads:

Tr A -1 t,fd (A t,fd -A t-1,fd ) log (det(A t,fd )) -log (det(A t-1,fd )) .
This leads to:

T t=1 Tr A -1 t,fd (A t,fd -A t-1,fd ) log det(A T,fd ) det(A 0,fd ) log(det(I d + 1 ε S T S T )). ( 4
R G := T t=1 η 2 2 g t A -1 t,fd gt 1 2 mΩ k ε(m -k) + log(det(I d + 1 ε S T S T )) .
4. Improvements on an online convex optimization algorithm: MetaGrad 4.3. Improved "online-to-batch" conversions for Online Newton Step and MetaGrad

In the previous Section, we considered the Online Convex Optimization framework, dealing with individual deterministic sequences and getting uniform deterministic bounds (except for the Random Sketching part, where some randomness was introduced). In this Section, we switch towards a probabilistic "batch" framework, where the learner observes directly a whole learning sample of i.i.d. losses, and is evaluated on only one attempt.

In the introductory Chapter 2, we presented a "standard" way to convert individual sequences algorithm to the batch setting. In this Section we switch to more sophisticated approaches. We investigate which rates can be obtained by combining the techniques of [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF] (applied in their paper to Gradient Descent) with Online Newton Step and MetaGrad. Since these two algorithms obtain optimal (up to multiplicative constants) regret bounds O(d log T ) for exp-concave losses in the online adversarial setting, our hope is that they will lead to optimal rates for exp-concave losses in the batch setting as well. Here we take a first step towards showing such a result by instead applying ONS and MetaGrad to strongly convex losses. This simplifies the analysis, because we stay closer to the setting of [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF], but since the regret bounds for ONS and MetaGrad on strongly convex losses are suboptimal by a factor d, we obviously also lose this factor in the batch rates that we obtain.

Framework

First-order information stochastic batch setting. We consider the batch setting of stochastic convex optimization. In this setting, we face i.i.d. convex loss functions f t , t ∈ 1, . . . , T on a convex domain K ⊂ R d , and the goal is to minimize

F : x ∈ K → E[f 1 (x)].
Therefore, we want to mimic the performance of:

x * = argmin x∈K F (x).
(we will assume that such x * exists).

Before each time step t, we choose a point x t ∈ K. For any t, we do not have access directly to f t (x t ) (nor, a fortiori, to F (x t )) but only to its (sub)gradient g t := ∇f t (x t ) ∈ ∂f t (x t ) -we will assume that such a subgradient exist. This is a "noisy version" of ∇F (x t ).

Assumptions. We will assume that the subgradients of f t are bounded by G, and that the diameter of the domain is not greater than D. Therefore, one has, for all

f t , x 1 ∈ K, x 2 ∈ K, that f t (x 2 ) -f t (x 1 ) ∇f t (x 2 ) (x 2 -x 1 ) and f t (x 1 ) -f t (x 2 ) ∇f t (x 1 ) (x 1 -x 2 ) so that the Cauchy-Schwarz inequality ensures that |f t (x 2 ) -f t (x 1 )| G x 2 -x 1 2 GD. Therefore, by Jensen's inequality, |F (x 1 ) -F (x 2 )| E[|f t (x 2 ) -f t (x 1 )|] GD.
We make an assumption stronger than convexity: we assume that all functions f t are λ-strongly convex, which means that: ∀x, y ∈ K, ∀ ∇f t (y) ∈ ∂f t (y), f t (x) f t (y) + ∇f t (y) (xy) + λ 2

xy 2 4.3. Improved "online-to-batch" conversions for Online Newton Step and MetaGrad

In particular, as ∇f t (y) (xy) 2 G 2 xy 2 2 (via the Cauchy-Schwarz inequality), one has:

∀x, y ∈ K, ∀ ∇f t (y) ∈ ∂f t (y), f t (x) f t (y) + ∇f t (y) (x -y) + λ 2G 2 ∇f t (y) (x -y) 2
Reversing this expression will allow us to bound the regret:

∀x, y ∈ K, ∀ ∇f t (y) ∈ ∂f t (y), f t (y) -f t (x) ∇f t (y) (y -x) - λ 2G 2 ∇f t (y) (x -y) 2 (4.
3.1) We will assume that F is also λ-strongly convex. Some consequences of strong convexity. One can see that if the f t are differentiable everywhere, then the λ-strong convexity of F is a direct consequence of the λ-strong convexity of the f t . Indeed, in this case the boundedness of the gradients guarantees that F is also differentiable and allows to exchange gradient and expectation, by writing that E [∇f t (y)] = ∇E [f t (y)] = ∇F (y) for any y and t. Therefore, one has:

F (x) -F (y) -∇F (y) (x -y) = E f 1 (x) -f 1 (y) -∇f 1 (y) (x -y) E λ 2 x -y 2 2 = λ 2 x -y 2 2
(Notice that in the general case, expectation of subgradients is trickier; in particular it is unclear, when a random function f t can have a subgradient with multiple elements in a point y, what E [∇f t (y)] means).

Another consequence of strong convexity with bounded gradient, is that it limits the domain. Consider two points x and y in K such that f t has a non-empty subgradient in x and in y, and such that xy 2 D/2 (we assume that two such points exist). Then ∀ ∇f t (y) ∈ ∂f t (y), ∇f t (y) (yx) G xy 2 , and similarly for x. So, applying the definition of the λ-strong convexity to x and y gives that λ xy 2 2 /2 is not greater than f t (x)-f t (y)+G x-y 2 and than f t (y)-f t (x)+G x-y 2 . Therefore, λ x-y 2 2 /2 G x-y 2 , so xy 2 2G/λ, and since xy 2 D/2, one has D 4G/λ. Conversely, knowing D and G enables to upper bound λ by 4G/D.

Outline of the section. We will present "online-to-batch" conversions of two algorithms, we call these conversions "Epoch Online Newton Step" and "Epoch MetaGrad". "Epoch Online Newton Step" is more straightforward (and so is its analysis), but it requires to tune the learning rate η correctly, which requires the knowledge of the parameter λ of strong convexity. On the contrary, "Epoch MetaGrad" benefits from one of the main assets of MetaGrad: the learning rate η does not need to be tuned (and thus no knowledge about λ is required). The order of magnitude of the bounds is the same as for "Epoch Online Newton

Step", up to a O(log log(T )) factor and a worsening of the constants.

4. Improvements on an online convex optimization algorithm: MetaGrad

Epoch Online Newton Step

Deterministic analysis

Let us first begin by an analysis of "classical" Online Newton Step (Algorithm 7, introduced in [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]). It follows the same line as the proof of Lemma 4.3, since the slave algorithm in MetaGrad is directly inspired by Online Newton Step.

The values of the parameters η > 0, m > 0 and ε > 0 will be fixed later.

Lemma 4.8. Starting from an arbitrary point x 1 ∈ K, and A 1 = εI d + mg 1 g 1 , apply T iterations of the update:

y t+1 = x t -ηA -1 t g t x t+1 = argmin x∈K (x -y t+1 ) A t (x -y t+1 ) A t+1 = εI d + m t+1
s=1 g s g s Then for any point x * ∈ K, one has:

T t=1 g t (x t -x * ) - m 2η g t (x t -x * ) 2 ηd 2m log(mG 2 T /ε + 1) + ε 2η x 1 -x * 2 2 (4.3.2)
An important corollary is that for η and m such that m/η λ/G 2 , then by λ-strong convexity, as seen in ( 4.3.1), the left-hand side of (4.3. 2) is an upper bound on the regret T t=1 f t (x t )f t (x * ), and therefore the right-hand side of (4.3.2) too:

T t=1 f t (x t ) -f t (x * ) ηd 2m log(mG 2 T /ε + 1) + ε 2η x 1 -x * 2 2 (4.3.3) Proof. y t+1 -x * = x t -x * -η t A -1 t g t so (y t+1 -x * ) A t (y t+1 -x * ) = (x t -x * ) A t (x t -x * ) -2ηg t (x t -x * ) + η 2 g t A -1 t g t
A classical property of projections (cf. [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF]) give that: (y t+1x * ) A t (y t+1x * ) (x t+1x * ) A t (x t+1x * ). As a consequence,

g t (x t -x * ) 1 2 ηg t A -1 t g t + 1 η (x t -x * ) A t (x t -x * ) - 1 η (x t+1 -x * ) A t (x t+1 -x * )
Recalling that A t+1 -A t = mg t+1 g t+1 , one gets by an Abel's transform:

T t=1 g t (x t -x * ) η 2 T t=1 g t A -1 t g t + 1 2η (x 1 -x * ) A 1 (x 1 -x * ) + T t=2 1 2η (x t -x * ) (A t -A t-1 )(x t -x * ) - 1 2η (x T +1 -x * ) A T (x T +1 -x * )) η 2 T t=1 g t A -1 t g t + T t=1 m 2η (x t -x * ) (g t g t )(x t -x * ) + ε 2η x 1 -x *
T t=1 g t (x t -x * ) - m 2η (x t -x * ) (g t g t )(x t -x * ) ηd 2m log(mG 2 T /ε + 1) + ε 2η x 1 -x * 2 2
If one takes ε = T , then the right-hand side of the previous inequality is equal to (ηd/2m) log(mG 2 + 1) + T x 1x * 2 2 /(2η). This will be key in our approach.

The algorithm

Classical "online-to-batch" conversion. The bound (4.3.3) would allow to use a classical "online-to-batch" conversion (cf. Section 2.3.3): starting from an arbitrary point x 1 , run on the whole learning sample ("as if it were online") the version of Online Newton Step described in Lemma 4.8, and then output x := T t=1 x t /T . Then:

E[F (x)] -F (x * ) E 1 T T t=1 F (x t ) -F (x * ) = 1 T E T t=1 f t (x t ) -f t (x * ) ηd log(mG 2 T /ε + 1) 2mT + ε x 1 -x * 2 2 2ηT
the first inequality coming from the convexity of F , and the last one using (4.3.2).

E [F (x t )] = E [f t (x t )
] relies on the fact that f t is independent of x t , that depends only on f 1 , . . . , f t-1 (which are random functions, so x t is random, therefore E [F (x t )] is indeed an expectation, even if F is deterministic).

In terms of d and T , this bound is O(d log(T )/T ), which is suboptimal in the strongly convex set-up. We then detail now an improved algorithm, that we call "Epoch Online Newton

Step", which achieves an accuracy of O(d/T ) as for d and T (unfortunately, it is still not the optimal rate, which is O(1/T ) for strongly convex losses).

Epoch Online Newton

Step algorithm. The algorithm, and its analysis, are strongly inspired by [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF]. They apply their ideas to a "linear first-order" method (Gradient Descent) whereas we adapt them to a "quadratic first-order method": Online Newton Step.

The details are given in Algorithm 12. We will define epochs of size T j (for the j-th epoch), and we will use a double indexation in t and j: f j t , x j t , with t the "local" time inside epoch j, corresponding to a global time ( j-1 i=1 T i ) + t. In other words, we will write f j t and x j t , instead respectively of f ( j-1 i=1 T i )+t and

x ( j-1 i=1 T i )+t
The approach relies on the following key idea. The theoretical bounds on the average regret that one gets on running classical Online Newton Steps (derived from Lemma 4.8) decrease 4.3. Improved "online-to-batch" conversions for Online Newton Step and MetaGrad Algorithm 12 Epoch Online Newton Step Input: parameters η, m, ε 0 , T 1 and total time T , domain K.

Initialization:

1. Initialize x 1 1 arbitrarily 2. Set j = 1 while j i=1 T i T do: ### Epoch j 1. ε j = 2ε j-1 2. A j 0 = ε j I d 3. for t = 1 to T j do: 4. obtain g j t ∈ ∂f j t (x j t ) with f j t := f ( j-1 i=1 T i )+t
### Update:

5.

A j t = A j t-1 + mg j t g j t 6. y j t+1 = x j tη A j t -1 g j t 7.

x j t+1 = argmin of the algorithm satisfies:

E[F (x k+1 1 )] -F (x * ) 4G 2 d log(4G 2 + 1) λT + 2GD T This is a O(d/T ) bound.
One can notice that there exists also a trivial bound, independent of T and d, for E[F (x k 1 )]-F (x * ):

E[F (x k+1 1 )] -F (x * ) GD 4G 2 /λ. (4.3.4)
The first inequality comes from:

E[F (x k+1 1 )] -F (x * ) E[|F (x k+1 1 ) -F (x * )|] GD,
and the second one is a direct consequence of strong convexity, already stated in Section 4.3.1.

Proof. By the hypothesis of λ-strong convexity of f , since m/η = λ/G 2 , one has, similarly to (4.3.1):

T j t=1 f j t (x j t ) -f j t (x * ) T j t=1 g j t (x j t -x * ) - m 2η (x j t -x * ) (g j t g j t )(x j t -x * )
and the right-hand-side can itself be upper bounded using Lemma 4.8. Since F is λ-strongly convex, the null vector is in ∂F (x * ) (because x * ∈ argmin F ), and the definition of λ-strong convexity leads to:

x 1 -x * 2 2 2(F (x 1 ) -F (x * ))/λ. (4.3.5)
Moreover, the independence of x j t and f j t gives that E[f j t (x j t )] = E[F (x j t )] (which remains an expectation since x j t depends on the previous f s ). The same result holds for the expectation conditioned on all randomness until the end of the epoch j -1 (that we will write E j-1 ):

E j-1 [f j t (x j t )] = E j-1 [F (x j t )
]. The convexity of the f t , and therefore of F , gives that:

F   1 T j T j t=1 x j t   1 T j T j t=1 F (x j t )
and this also holds in expectation.

The initialization of T 1 and ε 0 leads to: ε j = T j for any j 1.

One can now bring all these pieces together. Since x j+1 1 = ( T j t=1 x j t )/T j for any j 1, one has by Lemma 4.8, conditioning on all randomness until the end of the (j -1)-th epoch:

E j-1 [F (x j+1 1 )] -F (x * )] 1 T j E j-1   T j t=1 F (x j t ) -F (x * )   = 1 T j E j-1   T j t=1 f j t (x j t ) -f j t (x * )   ηd log(mG 2 T j /ε j + 1) 2mT j + ε j F (x j 1 ) -F (x * ) ηλT j = ηd log(mG 2 + 1) 2mT j + F (x j 1 ) -F (x * ) ηλ .
4.3. Improved "online-to-batch" conversions for Online Newton Step and MetaGrad Denote ∆ j := F (x j 1 )-F (x * ). Taking the unconditional expectation of the previous expression, and using the fact that ηλ = 4, gives:

E[∆ j+1 ] U η T j + E[∆ j ] 4
with U η = ηd log(mG 2 +1)/2m = dG 2 log(4G 2 +1)/2λ. Note that we picked T 1 = 4U η /(GD) , which yields U η /T 1 GD/4. Then, define V j = 2 1-j GD.

Lemma 4.10. For all j 1, E[∆ j ] V j .

Proof. Let us show it by induction. It is true for j = 1, because GD is a bound on F (x 1 1 ) -F (x * ) so E[∆ 1 ] GD. Let us assume that the property is true for some j 1, and show that it is also true for j + 1. One has:

E[∆ j+1 ] U η T j + E[∆ j ] 4 = U η 2 j-1 T 1 + E[∆ j ] 4 2 1-j GD 4 + V j 4 = V j+1 .
The first inequality has been seen before, and the second inequality relies on the fact that T 1 = 4U η /(GD) and on the induction hypothesis. This finishes to prove the lemma.

One can then complete the proof of Theorem 4.9.

A straightforward computation gives that the number k of complete epochs run by the algorithm is: k = log 2 (1+T /T 1 ) . Moreover, we recall that T 1 (4U η /B)+1. Consequently, by Lemma 4.10, one has:

E[F (x k+1 1 )] -F (x * ) = E[∆ k+1 ] V k+1 = 2 -k GD 2T 1 GD T 8U η + 2GD T .
Since U η = d G 2 log(4G 2 + 1)/2λ, this gives the desired bound.

Epoch MetaGrad

Deterministic analysis

We recall that Rx * T := T t=1 g t (x tx * ) and that

V x * T := T t=1 g t (x t -x * ) 2 .
We also recall the grid of learning rates in the MetaGrad master algorithm (Step 1 of Algorithm 8): η i = 2 -i /(5DG) for i = 0, 1, . . . , log 2 (T )/2 with prior weights

π η i 1 = 1+1/(1+ log 2 (T )/2 ) (i+1)(i+2)
. We re-write for convenience an adapted version of Lemma 4.5, denoting η i instead of η to emphasize that several values of η i , within the grid, are used by MetaGrad. Lemma 4.11. Starting from an arbitrary point x 1 ∈ K, apply T iterations of MetaGrad, using as covariance matrix initialization A η i 0 = ε i I d . Then, for any value η i ∈ (0, 1 5DG ]:

Rx * T η i V x * T + 1 η i ε 2 x 1 -x * 2 2 -log(π η i 1 ) + 1 2 log det I d + 2 η 2 i ε T t=1
f t (x t ) -f t (x * ) Rx * T -η i V x *
T for all η i 1/(5DG), it is only guaranteed for η i smaller than λ/(2G 2 ). To overcome this problem, we introduce the following quantity, that mixes the bounds required by (4.3.1) and by Lemma 4.11:

ζ := min λ/(2G 2 ), 1/(5DG) . Therefore, any η i ζ satisfies T t=1 f t (x t ) -f t (x * ) Rx * T -η i V x *
T and Lemma 4.11. This will allow us to prove the following result. Lemma 4.12. Defining:

ξ T := min (2 log(4 + log 2 (T )/2), 2 log (-log 2 (ζ/2) -log 2 (5DG) + 2))
one has:

T t=1 f t (x t ) -f t (x * ) Rx * T -ζV x * T 2 ζ ε 2 x 1 -x * 2 2 + ξ T + d 2 log(1 + T /(εD 2 )) . (4.3.7)
We will see that this quantity ξ T is actually an upper bound onlog(π η i 1 ) for an η i carefully chosen (either close to ζ, or the smallest value in the grid).

Remark: we could have proved a similar lemma focused on the grid point closest to η -that will be introduced in the proof in (4.3.10)-, instead of ζ; but we chose to introduce ζ to work with a fixed quantity in the following sections.

Proof. We will separate two cases, depending whether ζ is larger or smaller than the smallest point of the grid, that we will denote by η ( standing for low).

First case. First, assume that ζ is larger than η (and by definition ζ η 0 = 1/(5DG)). Then there exists η i in the grid such that ζ ∈ [η i , 2η i ]. Then, using π η i 1 1/(i + 2) 2 and η i = 2 -i /(5DG), one has:

-log(π η i 1 ) 2 log(i + 2) = 2 log (-log 2 (η i ) -log 2 (5DG) + 2) 2 log (-log 2 (ζ/2) -log 2 (5DG) + 2)
We denote by ξ this last quantity: T t=1 g t g t can not be greater than 1 + (T /(εD 2 )). As η i ζ and 1/η i 2/ζ, one has from (4.3.6) (and using the λ-strong convexity for the first inequality):

ξ := 2 log (-log 2 (ζ/2) -log 2 (5DG) + 2) . ( 4
T t=1 f t (x t ) -f t (x * ) Rx * T -ζV x * T 2 ζ ε 2 x 1 -x * 2 2 + ξ + d 2 log(1 + T /(εD 2 ))
4.3. Improved "online-to-batch" conversions for Online Newton Step and MetaGrad

Notice that since ζ is larger than η , one has (due to the construction of the grid):

ζ 2 η 2 1 20DG √ T
and thus, using the definition (4.3.8) of ξ: ξ 2 log(4 + log 2 (T )/2). Therefore, ξ T = ξ and the bound (4.3.7) is satisfied for this first case.

Second case. We will use similar arguments for the case in which ζ is smaller than the smallest grid point η . Due to the construction of the grid, 2η where:

M = ε 2 x 1 -x * 2 2 + 2 log(3 + log 2 (T )/2) + 1 2 log det I d + 1 εG 2 D 2 T t=1 g t g t
The minimizer η of η → ηV x * T + M/η is: T t=1 g t g t )) d log(1 + T /(εD 2 )) gives finally:

η = M/V x * T 2 log(3 + log 2 (T )/2)/(T G 2 D 2 ) 1/(5GD √ T ) η ζ ( 4
T t=1 f t (x t )-f t (x * ) Rx * T -ζV x * T 1 ζ ε 2 x 1 -x * 2 2 + 2 log(3 + log 2 (T )/2) + d 2 log(1 + T /(εD 2 ))
One can notice that in that case, since ζ is smaller than the smallest grid point η , one has ζ/2 1/(10GD √ T ), so (due to the definition of ξ) 2 log(3 + log 2 (T )/2) ξ. Therefore, 2 log(3 + log 2 (T )/2) ξ T and the bound (4.3.7) is satisfied in this second case.

The algorithm

A classical "online-to-batch" conversion (running the original MetaGrad algorithm on the whole learning sample, "as if it were online", and then outputting the average of the forecasts, cf. Section 2.3.3) would lead to a O(d log(T )/T ) upper bound on the expected regret (i.e., the bound (4.3.6) divided by T : cf. Theorem 2.9). To improve on this, we modify the conversion into an algorithm that we call Epoch MetaGrad (Algorithm 13).

We keep the same idea as in the Epoch Online Newton Step algorithm: dividing the learning sample into epochs, and running MetaGrad within each epoch, with an adequate choice of parameters and starting from the average of the outputs in the previous epoch. Thus, we can guarantee that at each epoch we get closer to the objective x * , both in terms of distance x j 1x * and in terms of expected regret (more precisely, we get upper bounds on these two quantities that decrease quickly at each epoch). Initialization:

1. Initialize x 1 1 arbitrarily. 2. Set j = 1, T 1 = 2 and ε 1 = T 1 /(2 log(T 1 )). while j i=1 T i T do: ### Epoch j 1. Run T j rounds of MetaGrad, initialized with ε j I d and x j 1 , get the outputs

x j t . 2. Set x j+1 1 = 1 T j T j t=1 x j t 3. Set T j+1 = 2T j 4. Set ε j = T j /(2 log(T j )) 5. Set j ← j + 1 end while Output: x j 1 ### If the number of completed epochs is k, this is x k+1 1

Theoretical bounds

Our main result shows that we get with Epoch MetaGrad nearly the same order of magnitude of expected error than for the Epoch Online Newton Step algorithm: O(d log log(T )/T ) (as far as d and T are concerned), but without having to tune η using an a priori knowledge on λ: we gain adaptivity in λ.

We focus on theses dependencies in d and T and might be a bit loose towards the other parameters.

We have previously introduced the following quantities, independent of d and (for T large enough) of T : ζ := min λ/(2G 2 ), 1/(5DG) and

ξ T := min (2 log(4 + log 2 (T )/2), 2 log (-log 2 (ζ/2) -log 2 (5DG) + 2)) .
Define now: ρ T := 16ξ T /(GDζ) .

One can easily see that ρ T is constant for T large enough. For T large enough, 2 log 2 (4 + log 2 (T )/2) 2 log (-log 2 (ζ/2)log 2 (5DG) + 2) so ξ T = 2 log (-log 2 (ζ/2)log 2 (5DG) + 2) and ρ T = 16 log (-log 2 (ζ/2)log 2 (5DG) + 2) /(ζGD) .

Theorem 4.13. The output x k+1 1 of the "Epoch MetaGrad" algorithm satisfies:

E[F (x k+1 1 )] -F (x * ) 8GD T max ρ T , 8d GDζ , exp 4 ζλ max 1, log 1 + 2 log(T ) D 2 .
4. Improvements on an online convex optimization algorithm: MetaGrad

• 2ε j /(T j ζλ) = 1/4 (since 2 log(T j ) 8/(ζλ)),

• 2ξ T j /(ζT j ) 2 1-j GD/8 (since T j 1 = ρ T := 16ξ T /(GDζ) ),

• d log(1 + T j /(ε j D 2 ))/(ζT j ) 2 1-j GD log(1 + 2 log(T j )/D 2 )/8 (since T j 1 8d/(GDζ)

So:

E[∆ j+1 ] V j 4 + 2 1-j GD 8 + 2 1-j GD log(1 + 2 log(T j )/D 2 ) 8 V j 2 V j+1
This finishes to prove the lemma.

We can now finish the theorem in the case where there exists an epoch satisfying (4.3.12). After renumbering, the number k of epoch run by the algorithm is: k = log 2 (1 + T /T 1 ) , with T T /2, so 2 -k 4T 1 /T , with T 1 2 max (ρ T , 8d/(GDζ), exp (4/(ζλ))). Another useful fact will be that T k T /2.

Consequently, by Lemma 4.14, one has:

E[F (x k+1 1 )] -F (x * ) = E[∆ k+1 ] V k+1 = 2 -k GD max 1, log(1 + 2 log(T k+1 )/D 2 ) 4T 1 GD max 1, log(1 + 2 log(2T k )/D 2 ) T 8GD T max ρ T , 8d GDζ , exp 4 ζλ max 1, log 1 + 2 log(T ) D 2 .
This finishes the proof for the case where at least one completed epoch satisfies (4.3.12). If it is not the case, that means that T 4 max (ρ T , 8d/(GDζ), exp (4/(ζλ)))

And then one can transform the trivial E[F (x k+1 1 )] -F (x * ) GD into:

E[F (x k+1
1 )] -F (x * ) 4GD max (ρ T , 8d/(GDζ), exp (4/(ζλ))) /T.

Putting together the two possible cases about the existence, or not, of a completed epoch satisfying (4.3.12), gives the result for all cases:

E[F (x k+1 1 )] -F (x * ) 8GD T max ρ T , 8d GDζ , exp 4 ζλ max 1, log 1 + 2 log(T ) D 2 .
For T large enough, ρ T = 16 log (-log 2 (ζ/2)log 2 (5DG) + 2) /(ζGD) so this bound is actually a O(d log log(T )/T ) as far as d and T are concerned.

Remark: we chose the value of ε j to guarantee 2ε j /(T j ζλ) 1/4 in the proof of Lemma 4.14. As the parameter λ (on which depends also ζ) is unknown, we have used ε j = T j /(2 log(T j )), but actually, for any real function ψ that tends to +∞ in +∞, taking ε j = T j /(ψ(T j )) would work, because it would satisfy for j big enough 2ε j /(T j ζλ) 1/4; and if ψ log, it would lead to a better O(d log(ψ(T ))/T ) rate.

Moreover, if one knows the strong convexity parameter λ, then it is possible to modify the "Epoch MetaGrad" algorithm so that it leads to O(d/T ) bounds (as does the "Epoch Online Newton Step" presented in Section 4.3.2), without the extra O(log log(T )) factor. It suffices to use ε j = ζλT j /8 instead of ε j = T j /(2 log(T j )); then one can modify the proof, in particular by using V k = 2 1-k GD instead of the V k presented above, to get the result with a O(d/T ) bound.

Introduction

The sequential aggregation provides a robust framework to make day-after-day forecasts, on the basis of the predictions of external experts (see [START_REF] Cesa | Prediction, Learning, and Games[END_REF] for a deep presentation). There are many algorithms available, e.g., the exponentially weighted average forecaster (EWA, first described in [START_REF] Littlestone | The weighted majority algorithm[END_REF]), Fixed-Share [START_REF] Herbster | Tracking the best expert[END_REF]), or more recently Squint (Koolen and [START_REF] Wouter | Second-order quantile methods for experts and combinatorial games[END_REF]) and ML-Poly [START_REF] Gaillard | A second-order bound with excess losses[END_REF]). Some classical linear regression algorithms such as LASSO (introduced in [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]) or Ridge [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]) can also be used in this framework.

A short-term objective. The arbitrary sequences setup is very robust, relying on uniform deterministic bounds. Its aim is generally to make single-point (one-step-ahead) forecasts, rather than managing the uncertainty, even if [Gaillard, 2015, Chapter 7], offers some ideas to deal with prediction intervals or distribution functions. The methods in this field rely crucially on the knowledge of the whole set of past observations before the instant of forecast. As a consequence, they are suited for a short term and an online (i.e., "day-after-day") prediction. But they cannot be directly used for a series of long-term forecasts, because of the lack of feedback for the intermediate observations.

Existing methods for long-term forecasts. This issue of long-term forecasts has been tackled in the field of time-series analysis, with methods such as the Box-Jenkins method (cf Box et al. [2015]) or more recent tools [START_REF] Rinke | Information criteria for nonlinear time series models[END_REF] and [START_REF] Lanne | Noncausal vector autoregression[END_REF]). In particular, the issue of managing the uncertainty by forecasting intervals is central in some papers. One can see [START_REF] Chatfield | Prediction intervals for time-series forecasting[END_REF] for a detailed discussion about it, Christoffersen [1998] for an example of a criterion, and [START_REF] Snyder | Prediction intervals for ARIMA models[END_REF] for a study on an ARIMA model. To compute the probability of simultaneous validity for multiple intervals, Ravishanker et al.

[1991] use adapted Bonferroni inequalities.

Nonetheless, the approaches presented in these time-series papers rely on a modeling of the data, rather than on external forecasts of experts.

Aim and outline. In this chapter, we tackle the question of adapting the algorithms of sequential aggregation to long-term forecasts of intervals. In Section 5.2, we provide a framework and a new methodology which benefits from the robustness and the safety of the sequential algorithms. This methodology relies on an optimization upon a large set of "possible sequences of observations". This optimization is a priori computationally costly, but in Sections 5.3 to 5.5, we provide computationally efficient ways of carrying it out for several algorithms. In Section 5.3 we use a closed-form for the Ridge algorithm to compute directly Ridge forecast intervals. In Section 5.4 we tackle the EWA algorithm with a two-step approach: getting "possible sets of weights" from the "possible sequences of observations" (Section 5.4.1) and deriving forecast intervals from these "possible sets of weights" (Section 5.4.2). In Section 5.5, we adapt the approach to get forecast intervals for the Fixed-Share algorithm. The data We divide time into two periods: learning and prediction.

The first set of rounds s = 1, . . . , T 0 -1 ("learning period") corresponds to the classical setup of online single-point forecasts with expert advice (cf. the beginning of Algorithm 1), see Chapter 2.

The actual goal is to forecast at rounds t = T 0 , T 0 + 1, . . . , T ("prediction period") a quantity y t (the observation) with the help of K experts, but without any stochastic assumption on the data. One has access to the forecasts of the experts (f k,t ) 1 k K,1 t<T 0 and to the observations (y t ) s<T 0 until T 0 -1 (i.e., of the learning period). At each instant t from T 0 to T (the prediction period) the learner has also access to the forecasts f k,t of the experts, and has to predict an interval [ y min t , y max t ] aiming at containing y t . The observation y t is simultaneously chosen by the environment, but not revealed immediately: the set of observations (y t ) t=T 0 ,...,T is available only after T , i.e., after all the forecast intervals have been made by the learner. In particular, no "feedback" about the quality of the forecasts after T 0 can be used to build the forecast intervals. It is only at the end, after T , that the observations from T 0 to T are revealed and that the accuracy of the forecast intervals can be evaluated.

A new methodology. The sequential aggregation algorithms are well-adapted to tackle online learning without stochastic assumptions, but they can not be directly used here. Indeed, they require the knowledge of the observations up to the previous instant, which are not all available in our framework. The methodology we present here overcomes this issue. It comes from the idea that these algorithms are robust and able to deal with all sets of observations, even worst-case ones. Therefore, forecast intervals that contain all the possible future forecasts of the algorithm are likely to contain the observation. That is exactly how we build them.

Methodology (formalized in

Ŝ

The set Ŝ of the forecast intervals

Computational issue to solve

A difficulty of this approach is that, as soon as the subset S is large (infinite, possibly uncountable), one just cannot compute the forecasts scenario by scenario (S contains infinitely many of them!). But we actually aim only at the higher and lower forecasts y max t and y min t , which is an optimization problem over the set of the possible scenarios up to round t. This optimization depends heavily on the algorithm, and each algorithm requires a different approach.

We give in Section 5.3 an example of an algorithm (Ridge) for which a (linear in some sense) closed-form expression allows to compute efficiently and quickly the intervals. We then present in Section 5.4 a way to deal with the EWA algorithm, using a split of the weight computations and a separation from the past (resulting in a slightly wider range of possible weights). We then adapt this approach in Section 5.5 for the Fixed-Share algorithm.

5. Providing long-term forecast intervals using sequential aggregation 5.4.1. Weight intervals for the EWA (with fixed learning rate) algorithm Our trick. The method we present here consists in splitting the computations of the weights for each expert, and obtaining them by successive updates.

We provide weight intervals updates that only take into account the next forecasts, the weight to be updated and bounds on observations (but no past data, and no other weight). That is, we restrict our attention to weight updates for which, for any j and t, the next weight p j,t+1 must only be a function of p j,t , of f t+1 = (f 1,t+1 , . . . , f K,t+1 ) and of known bounds on the observations. This scheme leads to a significant decrease in the computation time, as there are only around K variables taken into account. It leads to a slight broadening of the weight intervals.

We focus here on the EWA algorithm, because its definition relies on a multiplicative update of the weights, so it is well adapted to our approach. Recall that in the EWA algorithm with fixed learning rate η, the weights are convex and satisfy:

p j,t+1 = p j,t i p i,t α i,j,t where α i,j,t = exp -η y t -f i,t 2 + η y t -f j,t 2 = exp η f 2 j,t -f 2 i,t + 2y t (f i,t -f j,t )
. This quantity α i,j,t , which contains the experts interactions in the evolution of the weights of EWA, will be key in our arguments.

Warm-up: straightforward but loose bound. The quantity max (y tf j,t ) 2 : t ∈ {T 0 , .., T } and j ∈ {1, .., K} is unknown beforehand, but if one has access to an upper bound Q on it, then one gets directly the following bounds on the weights (proved in the supplementary material).

Lemma 5.3. For any j, the weights of the j-th expert satisfy:

p min j,t = p j,T 0 exp -(t -T 0 )ηQ p j,t 1 -(1 -p j,T 0 ) exp -(t -T 0 )ηQ = p max j,t .
This bound is not sharp, in particular it does not take into account the real positions of the experts. Also, if tηQ is much larger than one (which happens in many practical applications), then the lower bound in the lemma is nearly null and the upper bound is close to 1. This leads to a lowest and a highest forecast that correspond respectively to the highest and the lowest forecast of the experts. Such an interval, too wide and relying only on two extreme experts, is not very relevant in practice.

So we show hereunder how to use more accurately the information at our disposal, to get sharper bounds on the weights.

More accurate bounds. So far, we have not used any precise information on the set S of the possible observations, and we will do so now. We make the same assumption (H t ) as for Ridge: we assume that at each instant τ , there are known bounds B τ and B τ for the 5.4. Forecast intervals with the EWA algorithm observation: B τ y τ B τ . For given j and t, define:

M j,t = max yt∈[B t ,Bt] max i∈{1...K} α i,j,t = max yt∈[B t ,Bt] max i∈{1...K} exp -η y t -f i,t 2 + η y t -f j,t 2 and m j,t = min yt∈[B t ,Bt] min i∈{1...K} α i,j,t = min yt∈[B t ,Bt] min i∈{1...K} exp -η y t -f i,t 2 + η y t -f j,t 2 .
These two quantities will be studied in deeper details, with a more explicit computation, in Lemma 5.5. For the moment we only use their definition.

The following forecast interval [p min j,t , p max j,t ] only makes the weight intervals larger: 5.4.1) Contrary to the bounds in Lemma 5.3, here the real forecast of the expert is taken into account via M j,u and m j,u , which prevents one from an overcautious uniform evolution of the weights.

p min j,t = p min j,T 0 p min j,T 0 + (1 -p min j,T 0 ) t-1 u=T 0 M j,u and p max j,t = p max j,T 0 p max j,T 0 + (1 -p max j,T 0 ) t-1 u=T 0 m j,u (
Proof for these bounds. The facts that α j,j,t = 1 and that i p i,t = 1 lead to: p j,t+1 p j,t p j,t + i =j p i,t M j,t = p j,t p j,t + (1p j,t )M j,t and p j,t+1 p j,t p j,t + (1p j,t )m j,t • Thus, a reasonable update for the weight intervals, that can make them (and the forecast intervals) only larger, is:

p min j,t+1 = p min j,t p min j,t + (1 -p min j,t )M j,t and p max j,t+1 = p max j,t p max j,t + (1 -p max j,t )m j,t •
One can then deduce (by an induction given in the supplementary material) the direct complete form of the weight intervals.

Lemma 5.4. The previous updates lead to the weight intervals [p min j,t , p max j,t ] given in (5.4.1).

Final piece: efficient computation. It remains to be able to compute efficiently M j,t and m j,t , by identifying which observation and which expert forecast maximizes or minimizes α i,j,t . That is what the next lemma does.

Lines of future works

Fixed-Share forecast intervals. Minimizing (or maximizing) p j,t+1 is equivalent to minimizing (or maximizing) p j,t / ( i p i,t α i,j,t ), so the same approach as for EWA can be applied. Define again M j,t = max (yt,i) α i,j,t , and m j,t = min (yt,i) α i,j,t . They can be computed by Lemma 5.5. One has then upper and lower bound on the weights: p j,t+1 α p j,t p j,t + (1p j,t )M j,t + α K and p j,t+1 α p j,t p j,t + (1p j,t )m j,t + α K •

Theses inequalities provide reasonable updates (which, as in the EWA case, will make the weight and forecast intervals only larger):

p min j,t+1 = α p min j,t p min j,t + (1 -p min j,t )M j,t + α K and p max j,t+1 = α p max j,t p max j,t + (1 -p max j,t )m j,t + α K •
Then, one can apply the methodology of Subsection 5.4.2 to get forecast intervals from the weight intervals.

Lines of future works

The methods in this chapter for the computation of the forecast intervals rely heavily on known closed-forms for the algorithms weights. It would be interesting to try to apply the methodology on other algorithms, especially some for which no closed-form formula is known, e.g., the LASSO algorithm.

A line of thought would be to try and get some theoretical guarantees. This requires to define new and adapted benchmarks and criteria of performance. Indeed, the lack of modeling on the observations process prevents any guarantee on the proportion of observations falling inside the forecast intervals (unless of course the forecast intervals contain the totality of the possible observations, which is a trivial case). And many other criteria such as the quadratic loss do not make obvious sense in the forecast intervals setup. 145 5. Providing long-term forecast intervals using sequential aggregation

Supplementary material

Proof of Lemma 5.3 For any i, j, t, the coefficient α i,j,t satisfy 0 < α i,j,t exp (ηQ). So for any j, one has:

p j,t+1
p j,t i p i,t exp (ηQ)

= exp (-ηQ) p j,t i p i,t = p j,t exp (-ηQ) (recall that i p i,t = 1). By an immediate induction, p j,t p j,0 exp (-tηQ).

As a consequence, applying this bound to all the other experts, and using twice the fact that the weights sum up to 1, leads to:

p j,t = 1 - i =j p i,t 1 - i =j p i,0 exp (-tηQ) = 1 -(1 -p j,0 ) exp (-tηQ) . Proof of Lemma 5.4 p min j,t+1 = p min j,t p min j,t + (1 -p min j,t )M j,t
can be re-written as:

1 p min j,t+1 -1 = M j,t 1 p min j,t -1 . A direct induction (starting at T 0 ) gives: 1 p min j,t -1 = 1 p min j,T 0 -1 t-1 u=T 0 M j,u so that p min j,t = p min j,T 0 p min j,T 0 + (1 -p min j,T 0 ) t-1 u=T 0 M j,u •
The same computation (replacing M j,t by m j,t ) holds for p max j,t and leads to the desired result.

Proof of Lemma 5.5

Let us consider the expert j at time t (whose prediction is f j,t ), and tackle the maximization and minimization of α i,j,t with respect to y t and to f i,t (with f i,t ∈ {f j,t } j=1..K ). Split the experts into three groups: the ones that forecast higher than f j,t (group 1), the ones that forecast lower than f j,t (group 2), and the ones that forecast f j,t (group 3).

One has:

∂α i,j,t ∂y t = 2(f i,tf j,t )α i,j,t and α i,j,t > 0; so ∂α i,j,t

∂y t > 0 ⇔ (f i,t -f j,t ) > 0.
For the experts of the group 1, α i,j,t is increasing with y t . So its maximum is attained for the maximal value of y t : y t = B t ; and its minimum is attained for the minimum value of y t : y t = B t . On the contrary, for the experts of the group 2, α i,j,t is maximal for y t = B t and minimal for y t = B t . As for the experts of the group 3, the value of y t has no influence, for them α i,j,t = 1 whatever y t is.

As a consequence, there are only two values of y t to consider in order to maximize or minimize (in y t ) α i,j,t : B t and B t . Hence:

Introduction

To optimize the development of a reservoir, engineers need to forecast its production in response to potential development plans. To that purpose, numerical representations -or models -of the reservoir can be considered. They consist of a grid reproducing the structure of the reservoir and populated with facies and petrophysical properties. A fluid-flow simulator is then used to assess the dynamic evolution of the fluids in response to the production scheme. The main difficulty consists in identifying models that are representative of the reservoir. Indeed, many properties of the reservoir cannot be directly related to measurements, and strong uncertainties exist. They can be related to the structure of the reservoir, such as fault location and throw. The reservoir grid can be composed of millions of grid blocks, in which petrophysical properties are mostly unknown. To populate this grid, facies and petrophysical properties are generally considered as random functions characterized by statistical properties inferred from static data such as logs and seismic. Geostatistical approaches are then applied to generate distributions of these properties conditioned to the static data.

Time-dependent measurements, referred to as dynamic data, are also considered to build the reservoir models. They are acquired during the production period, and include measurements at wells such as pressure, oil and water rates, or 4D-seismic related attributes. However, these data are not linearly related to the reservoir properties, so that constraining models to dynamic data is generally a challenging task. Assessing the validity of a model requires to simulate its dynamic behavior. However, fluid-flow simulations can be very long, up to several hours, and the number of potential uncertain parameters very large. Constraining reservoir models to dynamic data, the history-matching process, is thus generally particularly time-consuming. Several methods have been investigated to solve this inverse problem. The variational approach consists in applying minimization algorithms to reduce the objective function that quantifies the error between the production data and the corresponding simulated properties (see [START_REF] Tarantola | Inverse problem theory: Method for data fitting and model parameter estimation[END_REF]). Ensemble methods can also be considered, such as the Ensemble Kalman Filter: cf. Aanonsen et al. [2009]. Interested readers can refer to Oliver and Chen [2011], for instance, for a review of history-matching approaches. One of the main challenges consists in properly parameterizing the problem. Indeed, considering all potential uncertain parameters, such as petrophysical distributions in all grid blocks, would lead to over-parameterization. In addition, the geological consistency of the model needs to be preserved. Several parameterization techniques have thus been proposed in the literature, which aim at mitigating the ill-posedness of the problem while preserving the geological realism. A review of parameterization techniques for petrophysical properties are proposed in Oliver and Chen [2011], Vo and Durlofsky [2016], for instance. However, the history-matching problem remains challenging, especially when discrete properties are considered.

The identification of models constrained to dynamic data is not necessarily the end-goal of reservoir modelling. Indeed, testing potential development scenarios and predicting the corresponding uncertainty on future production generally appears as a crucial step, especially to help decision-making [START_REF] Scheidt | Prediction-focused subsurface modeling: investigating the need for accuracy in flow-based inverse modeling[END_REF]). Based on this observation, some authors recently focused on the generation of production forecasts constrained to dynamic data without explicitly generating the corresponding updated reservoir models: see [START_REF] Satija | Direct forecasting of subsurface flow response from non-linear dynamic data by linear least-squares in canonical functional principal component space[END_REF], [START_REF] Satija | Direct forecasting of reservoir performance using production data without history matching[END_REF], [START_REF] Scheidt | Prediction-focused subsurface modeling: investigating the need for accuracy in flow-based inverse modeling[END_REF], [START_REF] Sun | A new data-space inversion procedure for efficient uncertainty quantification in subsurface flow problems[END_REF]. These approaches rely on 6. Sequential model aggregation for production forecasting a Bayesian framework, and use a set of reservoir models to represent prior uncertainty. Fluidflow simulations performed for this ensemble provide a sampling of the data variables and prediction variables, corresponding to the values simulated for the measured dynamic properties during the history-matching and prediction periods, respectively. The Prediction-Focused Approach (PFA) introduced in [START_REF] Scheidt | Prediction-focused subsurface modeling: investigating the need for accuracy in flow-based inverse modeling[END_REF] consists in applying a dimensionality reduction technique, namely the non-linear principal component analysis (NLPCA), to the two ensembles of variables (data and prediction). The statistical relationship estimated between the two sets of reduced-order variables is then used to estimate the posterior distribution of the prediction variables constrained to the observations using a Metropolis sampling algorithm. This approach is extended in [START_REF] Satija | Direct forecasting of subsurface flow response from non-linear dynamic data by linear least-squares in canonical functional principal component space[END_REF] to Functional Data Analysis. A Canonical Correlation Analysis is also considered to linearize the relationship between the data and prediction variables in the low-dimensional space. This additional step makes it possible to sample the posterior distribution of prediction variables using simple regression techniques. The resulting approach was demonstrated on a real field case in [START_REF] Satija | Direct forecasting of reservoir performance using production data without history matching[END_REF]. In [START_REF] Sun | A new data-space inversion procedure for efficient uncertainty quantification in subsurface flow problems[END_REF], the data and prediction variables are considered jointly in the Bayesian framework. They are first parameterized using Principal Component Analysis (PCA) combined to some mapping operation that aims at reducing the non-Gaussianity of the reduced-order variables. A randomized maximum likelihood algorithm is then used to sample the distribution of the variables given the observed data.

Our approach

The approach we follow also aims at the generation of production forecasts constrained to dynamic data without explicitly generating the corresponding updated reservoir models. Its distinguishing feature with respect to the approaches mentioned above is that it does not rely on a Bayesian framework: it actually relies on no stochastic modeling at all, which actually is in strong contrast with any forecasting method for reservoir production.

This approach uses as building blocks an ensemble of base geological models, from which production forecasts are generated. These models are not updated over time via historymatching, they are simulated once for all. The ensembles of forecasts issued by these models quantify in some sense some uncertainty (the larger the convex hull of forecasts, the more uncertain); see Figure 6.2. Now, a machine-learning algorithm combines these base forecasts at each prediction step, by using convex or linear weights set based on past performance of each base model. This dependency on past performance is where something with a flavor of history-matching is performed. These machine-learning algorithms are called aggregation algorithms. We use them "from the book", with no tweak or adjustment that would be specific to the case of reservoir production, at least as far as one-step-ahead predictions are concerned. This is similar to the studies performed for the forecasting of air quality [START_REF] Mauricette | Ozone ensemble forecast with machine learning algorithms[END_REF], electricity consumption [START_REF] Devaine | Forecasting the electricity consumption by aggregation of specialized experts; application to Slovakian and French country-wide (half-)hourly predictions[END_REF], [START_REF] Gaillard | Forecasting the electricity consumption by aggregating experts; how to design a good set of experts[END_REF], and exchange rates [START_REF] Amat | Fundamentals and exchange rate forecastability with machine learning methods[END_REF]. The book [START_REF] Cesa | Prediction, Learning, and Games[END_REF] is an excellent introduction to this sub-field of machine learning called, among other names, prediction with expert advice.

The only methodological modification of this well-established methodology actually consists in an addition: we extend it to not only provide one-step-ahead predictions but also 6.2. Brugge case

Outline of this article

Section 6.2 discusses the (artificial) data set used. Section 6.3 contains a high-level exposition of the machine-learning approach followed, with some additional technical details on the implementations of the algorithms being provided in appendix. Section 6.4 discusses our onestep-ahead predictions while Section 6.5 shows our longer-term predictions. These two sections use two specific machine-learning algorithms, called the ridge regression and the exponentially weighted average forecaster. A final section-Section 6.6-studies a third algorithm, called the LASSO, that not only aggregates the model forecasts but also first selects a subset of the models on which aggregation is then to be performed.

Brugge case

To assess the potential of the proposed approach for reservoir engineering, we consider the Brugge case, defined by TNO for benchmark purposes: see [START_REF] Peters | Results of the Brugge benchmark study for flooding optimization and history matching[END_REF]. This field, inspired by North Sea Brent-type reservoirs, has an elongated half-dome structure with a modest throw internal fault as shown in Figure 6.1. Its dimensions are about 10km × 3km. It consists of four main reservoir zones, namely Schelde, Waal, Maas and Schie. The formations with good reservoir properties, Schelde and Maas, alternate with less permeable regions. The average values of the petrophysical properties in each formation are given in Table 6.1. The reservoir is produced by 20 wells located in the top part of the structure. They are indicated in black in Figure 6.1. Ten water injectors are also considered. They are distributed around the producers, near the water-oil contact (blue wells in Figure 6.1).

Reminders on the units

For the mathematical audience intended, we remind the quantities typically measured or monitored when producing oil and gas, as well as their units. We first have the bottomhole pressure (BHP) at the wells, which is measured in pounds per square inch-psi in short.

Wells can be of two types, injectors (I) or producers (P). For the latter only, the oil and water production are measured in terms of flow rates, assessed in bbl/day, where bbl is an 6.3. How to combine the forecasts of the 104 models considered ered, using the production history of the reference case. However, our objective here was to assess the robustness of the proposed approach.

6.2.3. Summary of the 70 properties to be predicted

We respectively index injector wells by i ∈ {1, . . . , 10} and producer wells by j ∈ {1, . . . , 20}. Table 6.2 summarizes the 70 time-series to be predicted. They will be referred to by codes of the form QO P19 or BHP I2 in the following. Table 6.3 also provides some descriptive statistics pertaining to their orders of magnitude. They should be put in perspective with the root mean-square errors calculated later in Section 6.4.2. In this table, we report both descriptive statistics for the original (nominal) time-series, as well as for the time-series of unit changes1 (variations between two prediction steps). The latter are the most interesting ones in our view, as far as one-step-ahead forecasting is concerned.

How to combine the forecasts of the 104 models considered

Several paradigms and theories exist in the field of machine learning to combine (aggregate) the forecasts output by a set of models. Some are designed for the batch case, when all data are available and when only one aggregation is to be performed, while others are sequential in nature: aggregation is to be performed on a regular (e.g., monthly) basis. Some of these aggregation techniques deal with stochastic data: the observations to be forecast may be modeled by some stochastic process, with stationarity often required; on the contrary, other techniques work on deterministic data and come with theoretical guarantees of performance even when the observations cannot be modeled by a stochastic process. Most often, batch methods require stochasticity of the data while sequential methods may get rid of this assumption. Examples of popular aggregation methods include Bayesian model averaging (batch, stochastic) and random forests (batch, stochastic), as well as robust online aggregation (sequential, deterministic), also known as prediction of individual sequences or prediction with expert advice.

6. Sequential model aggregation for production forecasting More precisely, given a set W of reference weights (e.g., the set of all convex weights, or of all linear weights in some compact ball), good algorithms ensure that, no matter what the observations y t and the forecasts m j,t of the models were, where ε T is a small term, typically of order 1/ √ T . More details are given in Section 6.3.3, for each specific algorithm.

The reference set W will always include the weights (v 1 , . . . , v 104 ) of the form (0, . . . , 0, 1, 0, . . . , 0) that only put non-zero mass equal to 1 on one model. Thus, the infimum over elements in W will always be smaller than the cumulative square loss of the best of the 104 models. For some algorithms, this reference set W will be much larger and will contain all weights of some Euclidean ball of R 104 with radius larger than 1, thus in particular, all convex weights. Note also that no stochastic modeling of the observations or of the forecasts of the models is required; all possible (bounded) sequences can be considered.

The algorithms we will consider (and the way we will refer to them in the sequel) are: the exponentially weighted average (EWA) forecaster; the ridge regression (Ridge); the LASSO regression (LASSO). Their statement and theoretical guarantees-in terms of the quantities W and ε T in (6.3.2)-are detailed in Section 6.3.3.

Before we describe them in details, we provide a high-level view on the second aspect of our methodology, pertaining to longer-term predictions.

High-level methodology: longer-term predictions

The point aggregation indicated above is for one-step-ahead predictions: it may be performed as long as properties are measured on a regular basis (in technical words: as long as y t is measured after outputting y t and before y t+1 is to be output). Denote by T the last step of measurement of the property considered. Note that the geological models still provide forecasts for rounds t T + 1. Now, we may also be interested in providing interval forecasts for longer-term forecasts, i.e., for rounds t = T + k, where k 1 and k can be possibly large. We determine our interval for round T + k by first determining a set W T +k where the desirable weights (w 1,T +k , . . . , w 104,T +K ) lie in; this set of weights depends on the algorithm considered and on its performance and behavior on the T rounds of sequential prediction. The interval forecast for round t = T + k is then (6.3.3) where conv denotes a convex hull, possibly with some enlargement to take into account the noise level.

S T +k = conv
The question is how to determine these sets W T +k of desirable weights.

Statement of the aggregation algorithms considered

We provided hereabove the general methodological framework for sequentially aggregating forecasts in a robust manner, not relying on any stochastic modeling of the observations or of the forecasts of the models. We now provide the statements and the theoretical guarantees of the considered algorithms; the theoretical guarantees refer to (6.3.2) and consist in providing the values of W and ε T for the considered algorithm. We only deal with point aggregation in this subsection. The extension to interval aggregation is briefly discussed in Section 6.7.

For the sake of generality, we write K to denote the number of underlying models when stating our algorithms (K = 104 with the Brugge data set).

The ridge regression (Ridge)

The ridge regression (which we will refer to as Ridge when reporting the experimental results) was introduced by [START_REF] Hoerl | Ridge regression: Biased estimation for nonorthogonal problems[END_REF] in a stochastic and non-sequential setting. What follows relies on recent new analyses of the ridge regression in the machine learning community; see the original papers by [START_REF] Katy | Relative loss bounds for on-line density estimation with the exponential family of distributions[END_REF], [START_REF] Vovk | Competitive on-line statistics[END_REF] and the survey in the monograph by [START_REF] Cesa | Prediction, Learning, and Games[END_REF], as well as the discussion and the optimization of the bounds found in these references proposed by [START_REF] Gerchinovitz | Prediction of individual sequences and prediction in the statistical framework: some links around sparse regression and aggregation techniques[END_REF].

Ridge relies on a parameter λ > 0, called a regularization factor. At round t = 1, it picks arbitrary weights, e.g., uniform (1/K, . . . , 1/K) weights. At rounds t 2, it picks (w 1,t , . . . , w K,t ) ∈ argmin 6.3.4) i.e., it picks the best constant weights to reconstruct past observations based on the model forecasts subject to an 2 -regularization constraint v 2 j , which is useful to avoid overfitting to the past. The performance bound relies on two bounds V and B and is over

W =    (v 1 , . . . , v K ) : K j=1 v 2 j V 2    ,
the Euclidean ball of R K with center (0, . . . , 0) and radius V 1. This ball contains in particular all convex combinations in R K . The bound (6.3.2) with the above W reads: for all bounded sequences of observations y t ∈ [-B, B] and model forecasts m j,t ∈ [-B, B],

ε T 1 T λV 2 + 4KB 2 1 + KB 2 T λ ln 1 + B 2 T λ + 5B 2 .
In particular, for a well-chosen λ of order √ T , we have ε T = O (ln T )/ √ T . The latter choice on λ depends however on the quantities T and B, which are not always known in advance. This is why in practice we set the λ to be used at round t based on past data. More explanations and details are provided below.

The LASSO regression (LASSO)

The LASSO regression was introduced by [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF], see also the efficient implementation proposed in [START_REF] Efron | Least angle regression[END_REF]. Its definition is similar to the definition (6.3.4) of Ridge, except that the 2 -regularization is replaced by an 1 -regularization: at rounds t 2, (w 1,t , . . . , w K,t ) ∈ argmin As can be seen from this definition, LASSO also relies on a regularization parameter λ > 0.

One of the key features of LASSO is that the weights (w 1,t , . . . , w K,t ) it picks are often sparse: many of its components are null. Unfortunately, we are not aware of any performance guarantee of the form (6.3.2): all analyses of LASSO we know rely on (heavy) stochastic assumptions and are tailored to non-sequential data. We nonetheless implemented it and tabulated its performance, as well as its selection power (since the weights are sparse, many models are discarded). See Section 6.6 for details.

The exponentially weighted average (EWA) forecaster

The previous two forecasters were using linear weights: weights that lie in R K but are not constrained to be nonnegative or to sum up to 1. In contrast, the exponentially weighted average (EWA) forecaster picks convex weights: weights that are nonnegative and sum up to 1. The aggregated forecasts y t lie therefore in the convex hull of the forecasts m j,t of the models, which may be considered a safer way to predict.

EWA (sometimes called Hedge) was introduced by [START_REF] Littlestone | The weighted majority algorithm[END_REF], [START_REF] Vovk | Aggregating strategies[END_REF] and further understood and studied by, among others, [START_REF] Auer | Adaptive and self-confident on-line learning algorithms[END_REF], [START_REF] Cesa-Bianchi | How to use expert advice[END_REF], [START_REF] Cesa-Bianchi | Analysis of two gradient-based algorithms for on-line regression[END_REF]; see also the monograph by [START_REF] Cesa | Prediction, Learning, and Games[END_REF].

6.4. Results of point aggregation for one-step-ahead forecasts tuning of η t and λ t on past data, which somehow adapts to the data without overfitting; it corresponds to a grid search of the best parameters on available past data.

More precisely, we respectively denote by R λ and E η the algorithms Ridge run with constant regularization factor λ > 0 and EWA run with constant learning rate η > 0. We further denote by L t-1 the cumulative loss they suffered on prediction steps 1, 2, . . . , t -1:

L t-1 ( • ) = t-1 s=1 y s -y s ) 2 ,
where the y s denote the predictions output by the algorithm considered, R λ or E η . Now, given a finite grid G ⊂ (0, +∞) of possible values for the parameters λ or η, we pick, at round t 2, λ t ∈ argmin λ∈G L t-1 (R λ ) and η t ∈ argmin η∈G L t-1 (E η ) , and then form our aggregated prediction y t for step t by using either the aggregated forecast output by R λt or E λt . We resorted to wide grids in our implementations, as the various properties to be forecast have extremely different orders of magnitude:

• for EWA, 300 equally spaced points in logarithmic scale between 10 -20 and 10 10 ;

• for LASSO, 100 such points between 10 -20 and 10 10 ;

• for Ridge, 100 such between 10 -30 and 10 30 . However, how fine the grids are has not a significant impact on the performance; what matter most is that the correct orders of magnitude for the hyperparameters be covered by the considered grids.

Results of point aggregation for one-step-ahead forecasts

We discuss here the application of the Ridge and EWA algorithms on the Brugge data set. The latter covers 10 years of time but we only consider 127 evenly spaced prediction steps, which are thus roughly separated by a month. The one-step-ahead predictions discussed in this section thus (roughly) correspond to one-month-ahead predictions. The main comment would be that the aggregated forecasts look close enough to the true observations, even though most of the model forecasts they build on may err. The aggregated forecasts evolve typically in a smoother way than the observations. The right-most part of the BHP I1 picture reveals that Ridge is typically less constrained than EWA by the ensemble of forecasts: while EWA cannot provide aggregated forecasts that are out of the convex hull 6. Sequential model aggregation for production forecasting of the ensemble forecasts, Ridge resorts to linear combinations of the latter and may thus output predictions out of this range.

Qualitative study on a few representative properties

Also, the middle part of the QO P19 picture reveals that Ridge typically adjusts better and faster to regime changes, while EWA takes some more time to depart from a simulation it had sticked to. The disadvantage of this for Ridge is that it may at times react too fast: see the bumps in the initial time steps on pictures BHP P12 and QW P18.

Quantitative study on all properties

A more objective assessment of the performance of Ridge and EWA can be obtained through an accuracy criterion. As is classical in the literature, we resort to the root mean-square error (RMSE), which we define by taking into account a training period: by taking out of the evaluation the first 31 predictions (that is, roughly 1/4 of the observations). Hence, the algorithms are only evaluated on time steps 32 to 127, as follows: where C denotes the set of all convex weights, i.e., all vectors of R 104 with non-negative coefficients summing up to 1. The best model and the best convex combination of the models vary by properties; this is why we will sometimes write the "best local model" or the "best local convex combination".

Note that the orders of magnitude of the properties are extremely different, depending on what is measured (they tend to be similar within a given category). We did not correct for that and did not try to normalize the RMSEs. (Considering other criteria like the mean absolute percentage of error-MAPE-would help to get such a normalization.)

The various RMSEs introduced above are represented in Figure 6.6. A synthetical summary of performance would be that Ridge typically gets an overall accuracy close to that of the best local convex combination while EWA rather performs like the best local model. This is perfectly in line with the theoretical guarantees described in Section 6.3.3. But Ridge has a drawback: the instabilities (the reactions that might come too fast) already underlined in our qualitative assessment result in a few absolutely disastrous performance, in particular for BHP P5, BPH P10, QW P16, QW P12. The EWA algorithm seems a safer option, though not being as effective as Ridge. The deep reason why EWA is safer comes from its definition: it only resorts to convex weights of the model forecasts, and never predicts a value larger (smaller) than the largest (smallest) forecast of the models. Ridge (top graphs) and EWA (bottom graphs). The RMSEs of the latter are depicted in blue whenever they are smaller than that of the best model for the considered property, in red otherwise. 166 6.5. Results for interval aggregation

Results for interval aggregation

In this section we again discuss only the application of the Ridge and EWA algorithms on the Brugge data set. We use the first two thirds of the observations (times steps 1 to 84) as a training sample. Based on these observations, the methods considered output longer-term forecasts in the form of prediction intervals as explained in Section 6.3.2. They do so for the prediction steps 85 to 127. (We note that Ridge selects first a sub-sample of the simulations to build these interval forecasts, see Section 6.7.2 for details.)

We now provide a qualitative assessment of the interval forecasts obtained.

Some good results

Figures 6.7 and 6.8 report interval forecasts that look good: they are significantly narrower than the sets of scenarios while containing most of the observations. They were obtained, though, by using some hand-picked parameters λ or η: we manually performed some trade-off between the widths of the interval forecasts (which is expected to be much smaller than the set S of all scenarios) and the accuracy of the predictions (a large proportion of future observations should lie in the interval forecasts).

We were unable so far to get any totally satisfactory automatic tuning of these parameters (however, two attempts are presented in Section 6.8.3, with interesting results: Section 6.8.4). Hence, the good results achieved on Figures 6.7 and 6.8 merely hint at the potential benefits of our methods once they will come with proper parameter-tuning rules.

Sequential model aggregation for production forecasting

As far as the selection is concerned, we actually study how often coefficients w j,t in the aggregation equations (6.3.1) are non-zero. It is well documented that LASSO outputs sparse vectors of weights w j,t , i.e., that most of the coefficients picked are zero. Indeed, Figure 6.13 illustrates this fact while Figure 6.14 quantifies it. We read a high selection for the prediction of bottomhole pressure at injectors: only 4 models are active more than 20% of the time (when the averages are computed over time steps of the evaluation period 32 to 127, and over all properties of this family). The selection is even more extreme for oil production rate: no model is active more than 15% of the time. On the contrary, all models are active more than 40% of the time to predict the water production rate. .13: Illustration of the selection power of LASSO: same pictures and legend as in Figure 6.11, except that only forecasts of those models that are used in the aggregation are depicted; the forecasts associated with zero weights are omitted.

6. Sequential model aggregation for production forecasting 6.7. Appendix: Technical details for interval forecasts These technical details are best described in Chapter 5 of this manuscript. They are of two sorts: a computational issue and some further specific descriptions on the general methodology explained in Section 6.5.

A computational issue to solve

First, a computational issue is to be discussed: getting a numerical value of the convex hull (6.3.3) is computationally challenging as S typically contains infinitely many scenarios. We could however provide a solution for two of the considered algorithms, namely Ridge and EWA, see Chapter 5.

For Ridge, we were able therein to determine a closed-form expression of the upper and lower bounds of the sets S T +k in (6.3.3).

As for EWA, we offer an efficient and recursive computation of a series of sets W T +k containing the target sets W T +k , from which it is then easy to compute intervals containing the target prediction intervals S T +k . Indeed, it suffices to compute the maximum and the minimum of each S T +k .

Some further specific descriptions on the methodology

Determining the set S of scenarios. We compute the maximal variations upwards M or downwards m of the observations on the learning part of the data set and of any single trajectory of model forecasts on the prediction part of the data set. We do so by considering variations averaged out over 10 consecutive steps. The maximal variation downwards m can be negative or positive, depending on the property considered; the same remark holds for the maximal variation upwards. The set S T +k where the putative observations z T +k lie is then equal to the interval [y T + k m, y T + k M ].

Correcting the interval forecasts for the noise level. We first study the learning part of the data set and estimate some upper bound σ max on the noise level of the observations, as detailed below. Then, denoting by c T +k the center of each interval forecast S T +k , we replace

S T +k by max S T +k , [c T +k -σ max , c T +k + σ max ] ,
where the maximum in the right-hand side has to be understood in terms of the inclusion ⊆ operator.

Our estimate σ max is formed as follows. We first determine, among the observations available, time steps corresponding to some local stability of the property studied; those steps t are the ones when the observation y t is within 150 psi or 150 bbl/day (depending of the property) of all y t-r , where r varies between -15 and +15. We denote by S the set of those time steps with local stability. Then, our estimate is .15: Efficiencies of the benchmark (yellow bars) for each property, as well as the efficiencies of the Ridge forecast intervals, using the "objective-driven" tuning (top graphs) and the " hybrid" tuning (bottom graphs). The Ridge efficiencies are depicted in blue whenever they are larger than that of the benchmark for the considered property, in red otherwise. Abstract: This thesis tackles the topic of linear regression, within several frameworks, mainly linked to statistical learning. The first and second chapters present the context, the results and the mathematical tools of the manuscript. In the third chapter, we provide a way of building an optimal regularization function, improving for instance, in a theoretical way, the LASSO estimator. The fourth chapter presents, in the field of online convex optimization, speed-ups for a recent and promising algorithm, MetaGrad, and shows how to transfer its guarantees from a so-called "online deterministic setting" to a "stochastic batch setting". In the fifth chapter, we introduce a new method to forecast successive intervals by aggregating predictors, without intermediate feedback nor stochastic modeling. The sixth chapter applies several aggregation methods to an oil production dataset, forecasting short-term precise values and long-term intervals.

σ max = max

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

  générale de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Cadre mathématique de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Chapitre 3 : Obtention d'une régularisation optimale dans un cadre batch stochastique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Chapitre 4 : Amélioration d'un algorithme adaptatif : MetaGrad . . . . . . . . générale de la thèse . . . . . . . . . . . . . . . . . . . . 1.2 Cadre mathématique de la thèse . . . . . . . . . . . . . . . . . . . . 1.2.1 Différents cadres pour la régression linéaire, adaptés à différentes situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Un point commun : la mesure des performances est souvent relative 1.3 Chapitre 3 : Obtention d'une régularisation optimale dans un cadre batch stochastique . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Cadre mathématique . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Régularisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Apports de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Définition d'un critère d'optimalité pour la fonction de régularisation 1.3.5 Construction d'une fonction de régularisation optimale . . . . . . . . 1.3.6 Pistes de recherche futures . . . . . . . . . . . . . . . . . . . . . . . 1.4 Chapitre 4 : Amélioration d'un algorithme adaptatif : MetaGrad 1.4.1 Optimisation convexe séquentielle . . . . . . . . . . . . . . . . . . . 1.4.2 Algorithme MetaGrad, apports de la thèse . . . . . . . . . . . . . . . 1.4.3 Pistes de recherches futures . . . . . . . . . . . . . . . . . . . . . . . 1.5 Chapitre 5 : Faisceaux de prévision par agrégation séquentielle . 1.5.1 Problématique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Apports des travaux de cette thèse . . . . . . . . . . . . . . . . . . . 1.5.3 Pistes de recherche futures . . . . . . . . . . . . . . . . . . . . . . . 1.6 Chapitre 6 : Application de méthodes d'agrégation à la prévision pétrolière . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Différentes approches pour la prévision de production pétrolière . . . 1.6.2 Jeu de données étudié . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Apports de la thèse . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Pistes de recherche futures . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 2 :

 12 Figure 1.2: Zones de R d dans lesquelles sont choisis les coefficients de régression selon les chapitres (la pastille numérotée du chapitre est située dans la/les zone(s) où sont choisis les coefficients)

Figure 1

 1 Figure 1.3: (A) : Régularisation non nécessaire. (B) : La régularisation peut être nécessaire.

Figure 1 . 5 :

 15 Figure 1.5: Plan d'étude du jeu de données Brugge

2. 1 .

 1 Aggregation for individual sequences and in the batch setting 2.1.1. Aggregation for individual sequences

2. 1 .

 1 Aggregation for individual sequences and in the batch setting Setting 3 Online forecasting with expert advice

2 .

 2 The statistician chooses a forecast y 3. The environment reveals a value y N +1 4. The statistician suffers the loss ( y, y N +1 )

2 Figure 2

 22 Figure 2.1: Several p balls: " 0 ∩ ∞ ball", 1 ball, 2 ball

Figure 2 . 2 :

 22 Figure 2.2: Sparsity induced by the 1 regularization for a two-dimensional estimator

  3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.1.1 Regularization and Model Selection . . . . . . . . . . . . . . . . . . 66 3.1.2 General approach provided in this chapter . . . . . . . . . . . . . . . 68 3.1.3 Overview of the chapter; main results . . . . . . . . . . . . . . . . . 69 3.2 Proof of Theorem 3.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2.1 Probabilistic control of the processes . . . . . . . . . . . . . . . . . . 78 3.2.2 Deterministic part of the proof . . . . . . . . . . . . . . . . . . . . . 83 3.3 Technical material and proof of Proposition 3.1.6 . . . . . . . . . . 88 3.3.1 Localization with balls and spheres . . . . . . . . . . . . . . . . . . . 88 3.3.2 Control of the probability estimate . . . . . . . . . . . . . . . . . . . 89 3.3.3 Proof of Proposition 3.1.6 . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4 Minimax regularization function in the fixed design setup . . . . . 92 3.4.1 Proof of Theorem 3.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . 96

  d 1 norm. Let us recall the statistical model we used in both Definition 3.1.2 and Definition 3.1.1: the Gaussian linear regression model with a Gaussian design Y = X, t * + ξ (3.1.7)

  Theorem 3.1.3. Consider the Gaussian linear model with Gaussian design introduced in (3.1.7). Let ρ > 0. The minimax rate of convergence for constant confidence parameter δ N = 1/4 over ρB d 1 is achieved by the ERM and is given (up to absolute constants) by min r 2 (ρ), ρ 2 where r(ρ) = max r Q (ρ), r M (ρ) . (3.1.11) 3.1. Introduction Up to multiplicative absolute constants, this rate is given for some ζ < 1 < ζ , 1. when N log d, by ρ 2 , 2. when log d N ζd, by

2 min r 2

 22 (ρ), ρ 2 . Moreover, there are no estimator that can do uniformly better than the ERM t ERM ρ over ρB d 1 when N / ∈ (ζd, ζ d).

4 .

 4 There are absolute constants η, Q, ζ, ζ , ∆ 0 , c 0 such that the following holds. When ζ d N or ζd N , a minimax regularization function for the d 1 -norm over R d \(∆ 0 σ log(ed)/N B d 1 ) for the confidence parameter δ N = 1/4 is given by the following function: for all ρ > 0, Ψ(ρ) = c 0 r 2 (ρ)

Figure 3 . 1 :

 31 Figure 3.1: Shape of the graph of the minimax regularization function ρ → r 2 (ρ) of the d 1 -norm for the cases N ζd (left) and N ζ d (right)

  Remark 3.1.5 (Regularization function for the LASSO). The LASSO is the RERM procedure obtained for a linear regularization function Ψ(ρ) = σρ log d/N which is obtained by using a trivial upper bound on the complexity of the model ρB d 1 in (3.1.8): comp ρB d 1 = r 2 M (ρ) σ * ρB d

  Technical material and proof of Proposition 3.1.6 3.3.3. Proof of Proposition 3.1.6

4 .

 4 Improvements on an online convex optimization algorithm: MetaGrad Algorithm 7 Online Newton Step Input: Learning rate η. Parameters ε (initialization) and m (update). Starting point w 1 . Domain K.

  Number of time steps T . Diameter of the domain D: D = sup u,v∈K vu 2 .Uniform bound G on the (sub)gradients: G sup

Theorem 4 . 7 .

 47 Let A t,fd = εI d + S T t S t and η ∈ (0, 1 5GD ].Then the following holds for any k < m.

R

123 4 .

 4 If the number of completed epochs is k, this is x k+1 1 Improvements on an online convex optimization algorithm: MetaGrad Then the output x k+1 1

  .3.10) By convexity of η → ηV x * T + M/η, one gets that η V x * T + M/η ζV x * T + M/ζ, so (4.3.9) gives: Rx * T -ζV x * T M/ζ. The previously seen inequality log(det(I d + 1 εG 2 D 2

4 .

 4 Improvements on an online convex optimization algorithm: MetaGradAlgorithm 13 Epoch MetaGradInput: total time T , domain K, diameter of the domain D, uniform bound on the (sub-)gradients G.

5. 2 .

 2 The forecast intervals framework and methodology5.2. The forecast intervals framework and methodologylearning

Figure 5 . 1 :

 51 Figure 5.1: Framework of the forecast intervals

Figure 5 . 2 :

 52 Figure 5.2: The set S of the possible scenarios (left, green), combined with the expert forecasts (right, pink), leads to the set S = ( S t ) t T 0 of forecast intervals (right, orange).

  T +k m j,T +k : (w 1,T +k , . . . , w 104,T +k ) ∈ W T +k    ,

Figure 6 Figure 6 . 4 :

 664 Figure 6.4: An example of the set of scenarios S calculated for QO P19 on the last third of the sample.

6. 3 .

 3 How to combine the forecasts of the 104 models considered

Figure 6 .

 6 Figure 6.5 reports the forecasts of Ridge and EWA for 8 selected properties (which are representative of the 70 properties to be predicted).The main comment would be that the aggregated forecasts look close enough to the true observations, even though most of the model forecasts they build on may err. The aggregated forecasts evolve typically in a smoother way than the observations. The right-most part of the BHP I1 picture reveals that Ridge is typically less constrained than EWA by the ensemble of forecasts: while EWA cannot provide aggregated forecasts that are out of the convex hull

2 .

 2 Similar formulae determine on this time interval 32-127 the performance of the best model and of the best convex combination of the models

Figure 1 Figure 6 . 6 :

 166 Figure 1.1 -RMSE of EWA (blue/red, above) and Ridge (blue/red, below) vs RMSE of the best simulation (yellow) and of the best constant convex combination (pink) 3 Performance summary for EWA

Figure 6

 6 Figure 6.13: Illustration of the selection power of LASSO: same pictures and legend as in Figure6.11, except that only forecasts of those models that are used in the aggregation are depicted; the forecasts associated with zero weights are omitted.

Figure 6

 6 Figure 6.15: Efficiencies of the benchmark (yellow bars) for each property, as well as the efficiencies of the Ridge forecast intervals, using the "objective-driven" tuning (top graphs) and the " hybrid" tuning (bottom graphs). The Ridge efficiencies are depicted in blue whenever they are larger than that of the benchmark for the considered property, in red otherwise.

Titre:

  R égression lin éaire et apprentissage : contributions aux m éthodes de r égularisation et d'agr égation Mots Clefs : Apprentissage, r égression lin éaire, r égularisation, agr égation, processus empiriques, optimisation convexe s équentielle R ésum é : Cette th èse aborde le sujet de la r égression lin éaire dans diff érents cadres, li és notamment à l'apprentissage. Les deux premiers chapitres pr ésentent le contexte des travaux, leurs apports et les outils math ématiques utilis és. Le troisi ème chapitre est consacr é à la construction d'une fonction de r égularisation optimale, permettant par exemple d'am éliorer sur le plan th éorique la r égularisation de l'estimateur LASSO. Le quatri ème chapitre pr ésente, dans le domaine de l'optimisation convexe s équentielle, des acc él érations d'un algorithme r écent et prometteur, MetaGrad, et une conversion d'un cadre dit "s équentiel d éterministe" vers un cadre dit "batch stochastique" pour cet algorithme. Le cinqui ème chapitre s'int éresse à des pr évisions successives par intervalles, fond ées sur l'agr égation de pr édicteurs, sans retour d'exp érience interm édiaire ni mod élisation stochastique. Enfin, le sixi ème chapitre applique à un jeu de donn ées p étroli ères plusieurs m éthodes d'agr égation, aboutissant à des pr évisions ponctuelles court-terme et des intervalles de pr évision long-terme.Title: Linear regression and learning: contributions to regularization and aggregation methodsKeys words: Learning, linear regression, regularization, aggregation, empirical processes, online convex optimization

  t * où P N L t est un processus aléatoire (la différence des risques empiriques de t et t un événement de grande probabilité défini dans la preuve) pas sûr que P N L t > 0. Or, on verra que cette zone est une boule 2 .On est ainsi amené à considérer des intersections des modèles (des boules 1 ) avec cette boule 2 "où la régularisation peut être nécessaire" : cette approche est nommée "localisation". Elle est représentée en Figure1.3 (où l'on n'a pas représenté une boule entière t * + ρB d 1 , avec B d 1 désignant la boule unité fermée de R d pour la norme 1 , mais plutôt la couronne t * + ρB d 1 \(ρ/2)B d 1 vu que ce sont les vecteurs de grande norme 1 que l'on souhaite écarter). La fonction de régularisation f présentée dans l'article est précisément choisie de façon à contrôler P N L t (et plus précisément le seul terme de P N L t potentiellement négatif) sur cette intersection.

* ) et R t,t * = ψ(t)ψ(t * ) une quantité déterministe (la différence des régularisations en t et t * ), proche de Ψ(t) si t 1 t * 1 . Si t 1 t * 1 , R t,t * est positif et donc dans ce cas, P N L t > 0 suffit à garantir que P N L Ψ t > 0. La régularisation n'est par conséquent "utile" que dans la zone où l'on n'est (sur

  • S t et (εI t + S t S t ) -1 sont calculables facilement (par mise à jour par exemple).L'approche consiste alors à remplacer dans les calculs G t par S t et (εI d + G t G t ) -1 par :

  4. Chapitre 4 : Amélioration d'un algorithme adaptatif : MetaGrad pour une fonction g 2 liée à g 1 . Cette récurrence permet de garantir, pour une constante C liée aux données du problème : Elle aboutit (moyennant notamment le calcul de la taille minimale nécessaire pour la première période, ce qui introduit en particulier un facteur d de dimension) au théorème suivant, tiré du théorème 4.13. Theorem 1.3. Soient f 1 , . . . , f T des fonctions aléatoires tirées de manière i.i.d., fortement convexes, et définies sur un sous-ensemble convexe K de R

	E F (x j 1 ) -F x *	C	log log(T j ) 2 j	•

d . On suppose que l'on connaît une borne uniforme G sur les (sous-)gradients des f t , et une borne D sur le diamètre de K ; mais le paramètre λ de convexité forte des f t n'est pas supposé connu. Soit

  One has for all j ∈ {0, 1, . . . , 2d} that, if φ = j then tt *

												separated set in ρB d 1 w.r.t. d 2 .
	Now, let us define the test statistics
							φ ∈ argmin j∈{0,••• ,2d}	t * j -t 2 .
												j 2	ρ/4. Indeed, if φ = j then
	there exists k ∈ {0, 1, . . . , 2d}\{j} such that t * k -t 2 the result holds otherwise t * k -t 2 < ρ/4 and so t * j -t 2 t * j -t 2 . If t * k -t 2 t * j -t * k 2 -t * k -t 2 ρ/4 then ρ/4. Therefore, we have, for all τ > 0
	P t * 0	t -t * 0 2	ρ/4	P t * 0 φ = 0 =	2d j=1	P t * 0 φ = j	2d j=1	τ P t * j	φ = j and	dP t * 0 j dP t *	τ
	τ	2d j=1	P t * j φ = j -P t * j	dP t * 0 dP t * j	< τ	τ	2d j=1	P t * j	t -t * j 2 < ρ/4 -P t * j	dP t * 0 j dP t *	< τ .
												(3.3.1)
	{1, • • • , 2d},	P t * j	t -t * j 2 < ρ/4	P t * j	t -t * j	2 2 < χ 1 r 2 (ρ)	3/4	∈ (3.3.2)
	when χ 1 r 2 (ρ) tion 2.3 in Tsybakov [2009] (based on second Pinsker inequality), we obtain ρ 2 /16. Let j ∈ {1, • • • , 2d}. Following the same argument as in Proposi-

It follows from the adaptation property of t over {±ρe 1 , • • • , ±ρe d } that for every j

  Algorithm 15): T 0 , . . . , z T ) ∈ S: 1. Feed any classical learning algorithm with (y 0 , . . . , y T 0 -1 , z T 0 , . . . , z t-1 ) and(f k,τ ) 1 k K,1 τ t 2. Predict z t 3. Update S t ← S t ∪ { z t }

		5.2. The forecast intervals framework and methodology
	Algorithm 15 Methodology
	Preliminaries:
	Observe (y 0 , . . . , y T 0 -1 )
	for t = T 0 , . . . , T :
	I. Building S t :
	Initialize S t = ∅ for each (z II. Output:
	Output the forecast interval [ y min t , y max t	] defined as the smallest interval containing S t
	The set S of the possible scenarios
		scenarios some
		S
	learning	T 0 prediction	learning	T 0 prediction

Table 6 .

 6 1: Petrophysical properties of the Brugge field

	Formation Depositional	Average	Average Average	Average
		environment	thickness (m) Porosity Permeability (mD) Net to Gross
	Schelde	Fluvial	10	0.207	1105	60
	Waal	Lower shoreface 26	0.19	90	88
	Maas	Upper shoreface 20	0.241	814	97
	Schie	Sandy shelf	5	0.194	36	77
	longer-term predictions, as is typically done when forecasting reservoir production. See Sec-
	tion 6.3.2 for details.

Table 6 .

 6 2: Summary of the 70 times-series to be predicted.

	Code	Well type Property measured	Units	Number
	BHP Ii	injector	bottomhole pressure psi	10
	BHP Pj producer bottomhole pressure psi	20
	QO Pj	producer flow rate of oil	bbl/day	20
	QW Pj producer flow rate of water	bbl/day	20

log(K) .

We suppressed the extreme changes caused by an opening or a closing of the well when computing the mean, the median, and the standard deviation of the absolute values of these changes.

Remerciements

(1) M ) 2 η) + 1 t * 1 , that ρ 0 is the smallest ρ such that r M (ρ) = r M (ρ 0 ) for all ρ ρ 0 and that K 0 = min{k ∈ N : 2 k ρ * 2ρ 0 }.

Hereunder, K 0 is the number of shelves used to partition the intermediate "peeling zone" defined below. We use ρ * and K 0 to construct a partition of R d into three main zones:

• the "central zone" t * + ρ * B d 1 ,

• the intermediate "peeling zone" {t ∈ R d : ρ * < tt * 1 2 K 0 ρ * } (recall that 2 K 0 ρ * 2ρ 0 ). This zone is considered only when K 0 1. We use a "peeling", i.e. a partition of this zone into K 0 sub-areas called the "shelves":

• the "exterior zone" {t ∈ R d : tt * 1 > 2 K 0 ρ * }, on which r M (•) is constant.

Our main objective is now to show that, on the event Ω 0 , t belongs to the central zone.

.2: P N M t-t * is dominated by P N Q t-t * in (A) and by R t,t * in (B). The regularization function Ψ(•) is designed in order to dominate P N M t-t * in (B).

3.4. Minimax regularization function in the fixed design setup Define r 0 as the non-zero solution to the equation σ * (rB N 2 ∩ Im(X)) = η r 2 √ N , where Im(X) is the image of X in R d , i.e.

)/(η √ N ) = (σ/η ) Rank(X)/N Let ρ 0 be the smallest ρ such that (ρ/ √ N )XB d 1 contains r 0 B N 2 ∩ Im(X). An argument similar to the one used in the random design case shows that ρ 0 is the smallest radius such that for all ρ ρ 0 , r X (ρ) = r X (ρ 0 ) = r 0 . Finally, we consider K 0 = min{k ∈ N : 2 k ρ * 2ρ 0 }.

The fixed point function r X (•) depends on the Gaussian mean width of XB d 1 intersected with rB N 2 for various radii r. This quantity has been recently controlled in Proposition 2 in [START_REF] Bellec | Localized Gaussian width of M -convex hulls with applications to Lasso and convex aggregation[END_REF].

Proposition 3.4.3 (Proposition 2 in [START_REF] Bellec | Localized Gaussian width of M -convex hulls with applications to Lasso and convex aggregation[END_REF]). Let X ∈ R N ×d . Assume that the column vectors of X are in B N 2 . Then, for all r 0, * (XB d 1 ∩ rB N

2 ) min 4 log(8ed), 4 log(8edr 2 ), r Rank(X) .

It follows from some calculations (similar to the one used to obtain the closed form of r M (•) in (3.1.9)) that there exists an absolute constant C X such that for all ρ, r X (ρ) r X (ρ) with

(3. 4.3) We see that r X (ρ) in (3. 4.3) and r M (ρ) in (3.1.9) are very close. The only difference comes from the rank of X and when N ζ d and Rank(X) ∼ d, the two fixed points r M and r X are equal up to absolute constants. Furthermore, one can check that there are two absolute constants 0 < C 1 < C 2 and C X = C X (ρ, d, σ, N ) such that C 1 C X C 2 and

(3.4.4)

We will use ρ → r X 2 (ρ) as a regularization function (up to an absolute constant).

Main result

In this section, we obtain bounds on the rates provided by the regularization function Ψ(ρ) = c 0 r X 2 (ρ) for some well-chosen absolute constant c 0 when the design matrix X satisfies RIP. We first present the minimax rate over d 1 -balls in the fixed design setup. Such a result was obtained in [START_REF] Rigollet | Exponential screening and optimal rates of sparse estimation[END_REF]. Let us now recall this result in our context (see (5.25) in Section 5.2.2 in [START_REF] Rigollet | Exponential screening and optimal rates of sparse estimation[END_REF]).

Probabilistic control of the multiplier process

The following lemma shows how the fixed point r X allows to control the multiplier process.

Lemma 3.4.6. Let ρ > 0 and take η = 1/8. Then, for all r r X (ρ), with probability greater than 1exp -N r 2 /(128σ 2 ) , for all t ∈ t * + ρB d 1 ,

Proof. Let r r X (ρ). We denote by B 2 X the unit ball associated with the pseudo-metric

r its unit ball of radius r. We have

Then, it follows from Borell's concentration inequality (cf. [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]) that for all x > 0, with probability at least 1exp(-x 2 /2),

where σ(V ) := sup v∈V E v, ξ 2 = sup v∈V σ v 2 σr. Moreover, given that ξ ∼ N (0, σ 2 I N ),

we have

where the last inequality follows from the definition of r X (ρ) and because r r X (ρ). Gathering all the pieces together, it follows for x = η r √ N /σ, that, with probability at least 1exp(-r 2 N/(128σ 2 )), sup

(3. 4.6) Now, the proof follows from an homogeneity argument. Indeed, let us assume that (3.4.6) holds. Let t ∈ t * + ρB d 1 be such that

Deterministic part of the proof

We start with two lemmas on the growth behavior of r X (•). Their proofs are almost identical to the one of Lemmas 3.2.3 and 3.2.4 and are therefore omitted.

4. Improvements on an online convex optimization algorithm: MetaGrad with T , but increase with the initial distance to the objective: x 1x * 2 . Therefore, if the algorithm tends to converge fast enough to the objective (and the analysis will show that it is the case), then it is reasonable to make a trade-off between length of run T and accuracy of the starting point. Roughly, we "sacrifice" the first half of the learning sample to find a point close to the objective x * , and we run Online Newton Step on the second half of the learning sample, initializing on this point "close to the objective".

As distance to the objective and performance are linked, the search for a point close to the objective (first half of the learning sample) leads us to seek a point with a good performance -that is why we iterate the previous idea in this first half of the learning sample. This leads us to some kind of dichotomy and to divide the learning sample into "epochs" whose size is exponentially increasing.

We present now the algorithm (Algorithm 12 below) with further details. We divide the time t = 1 . . . , T in epochs in the following way. Choose T 1 , and for any j ∈ N, T j = 2 j-1 T 1 . Let k be the number of completed epoch, i.e., such that k i=1 T i T < k+1 i=1 T i . We define the first epoch as t = 1, . . . , T 1 and the j-th epoch as t = T j-1 + 1, . . . , T j and we write x j t := x ( j-1 i=1 T i )+t . We start in epoch 1, from an arbitrary point x 1 1 . In epoch j, we run a version of the Online Newton Step algorithm with parameter η constant over the epochs, and parameter ε j = 2 j ε 0 , but we initialise it at the average of the outputs of the previous epoch:

. This allows to decrease the upper bound on x j 1x * 2 at each epoch. Contrary to [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF], we keep a constant learning rate η. This comes from the presence of the quadratic term m g t (x tx * ) 2 /η in Lemma 4.8 and therefore in our subsequent analysis. To handle this term with the strong convexity property, we need to keep m/η small. Not too small though, because its inverse η/m appears in the right-hand side bound of Lemma 4.8. So we decide to keep η/m constant, and modify instead another parameter (which does not exist in the Epoch Gradient Descent of [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF]): the initialization ε j I d of A j t . The output x k+1 1 of the algorithm is the average of the outputs of the last epoch:

Theoretical bounds

The main result of this section shows that the algorithm "Epoch Online Newton step" achieves (as far as the dependency in the dimension d and the learning sample size T are concerned) a regret O(d/T ). This achieves the optimal rate in T . But we were not able to suppress the d factor, that can be seen in the regret bound of Theorem 2 of [START_REF] Hazan | Logarithmic regret algorithms for online convex optimization[END_REF] (in the weaker hypothesis of exp-concavity), but which is suboptimal in the context of strong convexity (cf. [START_REF] Hazan | Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization[END_REF]). Whether this d term is an artifact of our analysis or is intrinsic to our algorithm is an open question.

Theorem 4.9. Initialise the algorithm "Epoch Online Newton Step" with η

λD

, and ε 0 = T 1 /2.

4.3.

Improved "online-to-batch" conversions for Online Newton Step and MetaGrad

In particular, as far as the dependencies in d and T are concerned,

Similarly to (4.3.4), one has also the following trivial bound, independent of T and d:

Proof. The proof follows the line of the proof for Epoch Online Newton Step (Theorem 4.9). The same arguments (independence of x j t and f j t , convexity) give:

Using the fact, already seen in (4.3.5), that

))/λ (by strong convexity and since 0 ∈ ∂F (x * ), because x * minimizes F ), one then has, taking the expectation of (4.3.7):

We will for the moment assume that there exists a completed epoch j 1 satisfying:

T j 1 max (ρ T , 8d/(GDζ), exp (4/(ζλ))) (4.3.12)

We will start our analysis at the beginning of the first epoch satisfying the previous inequality, and re-number everything by defining this epoch as epoch 1, without loss of generality. The only impact is that the remaining learning sample is of size T T /2 (and, of course, T 1 and ε 1 have no longer the value given in the description of the algorithm).

After this re-numbering, denote ∆ j = F (x j 1 ) -F (x * ) and

The same lemma as in section 4.3.2 holds (although the value of V j is slightly different).

Lemma 4.14. For all j 1, E[∆ j ] V j .

Proof. We show it by induction. It is clear for j = 1: GD is a bound on f t (x)f t (y) for any x, y ∈ K, so

Let us assume the property holds for some j 1, and show that it is also true for j + 1. From (4.3.11), one has:

Therefore, given the value of ε j and T j (after renumbering, cf. (4.3.12)), one has:

Chapter 5

Providing long-term forecast intervals using sequential aggregation

This chapter is a link between the batch and the individual sequences settings. We aim at providing forecast intervals, relying only on external short-term and long-term expert forecasts, without stochastic modeling of the data. However, contrary to the classical individual sequences process, here we do not get any intermediate feedback, and we have to provide at once a whole sequence of forecast intervals for the short and the long term. Thus, we aim at adapting the individual sequences algorithms, which generally require this feedback in their very definition, to this framework.

We introduce a new methodology to do so, which relies on a set of "possible future scenarios", and on an optimization of the algorithms outputs with respect to this set.

We explain how to solve this optimization for three algorithms: exactly for the Ridge regression algorithm, and with some approximations which only widen a bit the resulting intervals for the EWA and Fixed-Share EWA algorithms.

Algorithm 14 The framework of forecast intervals I. Learning period for s = 1, . . . , T 0 -1:

1. The observation y s is chosen by the environment 2. Get the expert forecasts (f k,s ) 1 k K 3. Provide a single-point forecast y s = K k=1 u k,s f k,s 4. Observe y s II. Prediction period for t = T 0 , T 0 + 1, . . . , T :

1. The observation y t is chosen by the environment but not revealed

for t = T : Observe y T 0 , . . . , y T

The starting point is to assume that the sequence of observations to be predicted (y T 0 , . . . , y T ) can be any element of a large subset S ⊂ R T -T 0 +1 , the set of the "possible scenarios for the future". At round t, for a given aggregation algorithm, the t-T 0 first components (z T 0 , . . . , z t-1 ) of any "possible scenario" (z T 0 , . . . , z T ) ∈ S, in combination with the expert forecasts, lead to a prediction z t : the prediction that would be made at round t in the classical sequential setting, if the observations sequentially revealed had been y 1 , . . . , y T 0 -1 , z T 0 , . . . , z t-1 . So the set S of possible scenarios leads to a set of "possible predictions at round t", which we denote S t := z t : (z T 0 , . . . , z t-1 ) is the prefix of a sequence in S . The forecast interval at round t is then defined as the smallest interval containing S t .

The set S of the "possible scenarios". The choice of this subset S is important. It is the main "modeling" choice of the statistician, most of the "data modeling" being supposedly included into the expert forecasts. It can be for example a product of intervals:

The set S is typically constructed based on y T 0 and on what we know about the evolution of the observations. E.g., if one assumes that the absolute variation of the observations between two time steps is bounded by some value ∆, then the observation y t will lie between:

The set S is in this case a cone, that will contain all the observations. It is this approach that is shown in Figure 5.2 where the red cone contains all the considered sequences of observations (three of them are drawn in green).

Forecast intervals with the Ridge regression forecaster

In this section, we address the Ridge algorithm, for which a closed-form expression of the forecasts (Subsection 5.3.1) allows to compute directly the forecast intervals (Subsection 5.3.2).

The Ridge regression forecaster

The Ridge regression forecaster is a regularized "least squares" estimator, introduced by [START_REF] Hoerl | Application of ridge analysis to regression problems[END_REF]. It relies on a regularization parameter λ, and on a Euclidean regularization u → u 2 2 . Consider some instant t, and denote:

-by f t the vector of all the experts' forecasts at time t:

the t × K matrix of the experts' forecasts up to time t (notice that we inverse the order of indices of the experts' forecasts), -by Y t = (y 1 , . . . , y t ) the column-vector of the observations up to time t (. meaning "transpose").

The weight vector output by Ridge u R t+1 :=

(5.3.1)

We recall now Lemma 2.6 (proved in Chapter 2), which shows that the weight vector generated by Ridge, and therefore the Ridge regression forecasts, are linear mappings of the observations. Lemma 5.1. The weights and forecasts provided by the Ridge algorithm (with regularization parameter λ) are linear with respect to the observations vector Y t :

The forecast intervals for the Ridge regression

Recall that the observations y s are known for all instants s < T 0 of the training period. We consider the following known bounds: ] output by Ridge following the methodology given in Section 5.2 and under the bounds: B s y s B s for any instant T 0 s t, satisfy:

and

Proof. Writing y t+1 = t u=1 V t+1 u y u emphasizes that the highest possible value for y t+1 corresponds to a scenario in which the value of the observation at any time u (such that

On the contrary, y t+1 reaches its lowest possible value when the value of the observation at any time u (such that

Some remarks on the Ridge forecast intervals. The width of the forecast interval at time t is a sum of t non-negative terms (which change at each t). It scales linearly with the set of bounds {B u , B u } u t , in particular, it does not necessarily belong to the interval generated by the expert forecasts.

Forecast intervals with the EWA algorithm

In this section, we address the exponentially weighted average (EWA) algorithm. Contrary to the Ridge algorithm, it does not provide an easy (e.g., linear) link between the observations and the forecasts. So optimizing directly upon the whole set of possible observations scenarios (e.g., T

t=T 0 [B t , B t ]) seems to be computationally intractable. In this section, we therefore present a compromise: using computationally cheap methods, at the cost of being less sharp and having slightly larger weight intervals.

Our approach. We adopt a two-step approach. First, for any round t, on the basis of the set S of the possible scenarios and on the expert forecasts, we compute separately for each expert an interval of possible weights [p min j,t , p max j,t ]. We provide in Subsection 5.4.1 a computationally efficient way to do so. The second step consists in passing from weight intervals to forecast intervals. This amounts to choosing a weight vector compatible with each weight interval, and see which highest and lowest possible forecasts can be obtained this way. We implicitly accept thus to aggregate weights that do not come from the same observations scenario. We provide in Subsection 5.4.2 fast and easy ways to deal with two aggregation problems in this setup: linear and convex aggregations. It is the convex aggregation case that corresponds to the EWA algorithms; the linear case is presented to allow future use of linear algorithms.

5. Providing long-term forecast intervals using sequential aggregation Lemma 5.5. For given j and t, the maximum M j,t of α i,j,t is obtained for an observation equal to one of the bounds B t or B t , and for the expert that is the closest to this bound:

As for the minimum m j,t of α i,j,t , it is obtained for an observation equal to one of the bounds B t or B t , and for the expert that is the farthest to this bound:

The proof, detailed in the supplementary material, relies on the study (for a given instant t and an expert j) of the function (y t , f i,t ) → α i,j,t .

High-level remarks. An important remark is that, since we allow in this section to separate the weight intervals computations, we get a wider range of weights than what a direct optimization upon the possible scenarios would have provided, and so this leads to larger forecast intervals.

As EWA is a convex algorithm, passing from weight intervals to forecast intervals requires normalization (cf. Subsection 5.4.2), so the forecast interval is always inside the expert forecasts interval. As a consequence, the choice of the possible scenarios, though still important, is less crucial than for non-convex algorithms such as Ridge.

From weight intervals to forecast intervals

Section 5.4.1 shows how to get weight intervals. It then remains to form weight vectors from these intervals. Two aggregation frameworks are studied hereunder: linear aggregation and convex aggregation.

A preliminary example: linear aggregation. The case of linear aggregation (i.e., with weights in R and without normalization), is straightforward. The maximum M and the

are given as follows:

Lemma 5.6.

Proof. If f j > 0, then for all u ∈ [u min j , u max j ], one has u min j f j uf j u max j f j ; and if f j < 0, then for all u ∈ [u min j , u max j ], one has u max j f j uf j u min j f j .

Convex aggregation

In this case, one is given f

, but this time the weights have to be normalized. One aims at computing: max

and the corresponding argmax ( * ) and argmin ( * * ).

We assume that at least one weight p min j is positive, so all expressions are well-defined. We focus on the first problem ( * ), the second one ( * * ) is addressed by symmetry below.

Discretization of the problem. The next lemma shows that what seems a continuous optimization problem is actually a discrete optimization problem.

Lemma 5.7. The argmax ( * ) contains at least one element for which all the weights are either maximal or minimal:

The proof, given in the supplementary material relies on the idea that if a weight p j is different from p min j and p max j , then (in most cases) either increasing or decreasing p j will increase

Efficient computation via a ranking. The previous lemma shows that there exists a weight vector leading to a maximal forecast, and whose components are (before normalization) p min j or p max j . So one only has to test the 2 K vectors of {p min 1 , p max 1 } × . . . × {p min K , p max K } and get the one which leads to the highest value. But 2 K is quite a lot, and actually, one can show that a "ranking" of the coordinates allows one to test only K + 1 weight vectors. These K + 1 vectors are the ones that "give their maximum possible weights to the experts that forecast the highest values and the minimum possible weight to the others", which is quite intuitive.

Lemma 5.8. Denote by n 1 , . . . , n K the indexes of the coordinates of f sorted by decreasing order. There exists R K and a weight vector v in the argmax of Lemma 5.7 such that:

The proof, based on similar ideas as the previous one, can be found in the supplementary material.

Providing long-term forecast intervals using sequential aggregation

From maximization to minimization. Let us move now to the second problem: the minimum (and argmin) problem. It can be solved using the argmax result, with a symmetry argument: it suffices to replace f j by -f j . argmin

A decreasing order on (-f 1 , . . . , -f K ) corresponds to an increasing order on (f 1 , . . . , f K ). This directly leads to the following lemma (the difference with the previous one is that an increasing order is used, instead of a decreasing one). Lemma 5.9. Denote by n 1 , . . . , n K the indexes of the coordinates of f sorted by increasing order. There exists R K and a weight vector v in the argmin such that:

Similarly to the maximization problem, only K + 1 vectors need to be tested.

Remark.

One can note that even in the cases where the weight intervals get wider over time, it is not sufficient to guarantee that the forecast intervals will also get wider, since they depend on the expert forecasts. Even if the set of expert forecasts get broader over time (i.e., the extremal expert forecasts are moving away from each other), it is not enough to guarantee that the size of the forecast intervals will increase, since it is mostly driven by the experts with the highest weights, which are not necessarily the extremal experts. To sum up, there is no standard evolution of the forecast intervals, it is really the behaviour of the experts (along with the chosen algorithm and the possible scenarios) that will determine the forecast intervals.

Extension to the Fixed-Share algorithm

The Fixed-Share algorithm. The approach developed hereabove can be applied to another important algorithm, derived from EWA: Fixed-Share EWA. This algorithm involves two parameters: α ∈ [0, 1] and η > 0, and its weights satisfy:

where, as in EWA, α i,j,t , we proceed in a symmetric way, since min

Proof of Lemma 5.7 Let f be the vector of the expert forecasts: f = (f 1 , . . . , f K ), and recall that the set of the possible weights before normalization is j [p min j , p max j ], with all p j being non-negative, and at least one of them being positive.

Denote by W the set of these possible weights after normalization:

It is clear that W is a compact set (because it is the image of a compact set by the mapping x → (1/ x 1 )x which is continuous outside the null vector, thus on the set at hand). Problem ( * ) is equivalent to getting the maximum of the function w ∈ W → w • f , which is continuous (because it is linear in finite dimension). Therefore it reaches its maximum on the compact set W , so the argmax ( * ) is not empty. We will show that the argmax ( * ) contains at least one element located (before normalization) at a corner of j [p min j , p max j ]. To do so, let u be a weight vector belonging to the argmax ( * ), before normalization, and let u be obtained by replacing the j-th component u j of u by another value u j . Denote by S u the "normalized dot product":

5. Providing long-term forecast intervals using sequential aggregation

The first equality comes from the fact that u and u only differ on their j-th component.

The second one introduces S u by writing f j = f j -S u + S u . The third one is based on u 1u 1 = u ju j and on S u u 1 = u • f . Three cases appear for the coordinates of u.

First case: f j = S u . Then the second term of the sum (5.7.2) is null, the value of u j has no impact and can be replaced at will by p min j or p max j . Second case: f j > S u . If u j = p max j , consider u j > u j , then the second term of the sum (5.7.2) is positive, and one has S u < u • f / u 1 ; this is not possible since u belongs to the argmax. So u j = p max j . Third case: f j < S u . If u j = p min j , consider u j < u j , then the second term of the sum (5.7.2) is positive, and one has S u < u • f / u 1 ; this is not possible since u belongs to the argmax. So u j = p min j . To sum up, the only coordinates of u that are not extremal are the ones of the "first case" and can be replaced at will by an extremal value without changing the result S u . Doing so leads to an element of the argmax that has only extremal coordinates.

Proof of Lemma 5.8 Let us show that a vector u which does not fit the form stated in the lemma cannot be in the argmax. More precisely, assume that there exists j 1 and j 2 such that f j 1 < f j 2 but u j 1 = p max j 1 and u j 2 = p min j 2 . Then, using the notations of the previous proof, either f j 1 < S u or f j 2 > S u , and in each case the previous proof shows that S u is not maximal, so u does not belong to the argmax.

Chapter 6

Sequential model aggregation for production forecasting

This chapter is a joint work with Gilles Stoltz, Charles-Pierre Astolfi, Véronique Gervais-Couplet and Sébastien Da Veiga. It has been submitted for publication.

In this chapter, we apply aggregation methods to an oil production dataset, consisting in monthly simulations and observations of several petrophysical properties over ten years. Classical procedures for oil production forecasting ("history matching") require time-consuming re-computations to take into account intermediate observations along time; our goal is to decrease substantially this computation time by aggregating the simulations results and changing only the weights each month.

We first apply three one-step-ahead individual sequences algorithms: the EWA algorithm, the Ridge regression and the LASSO regression. They give good results, being usually competitive (and even better for LASSO) with the best simulation at hand.

We then apply and adapt the methodology of Chapter 5 to provide forecast intervals for this dataset. At the end, we define a performance measure and a benchmark for the forecast intervals, which we compare with our results. abbreviation for oilfield barrel, a volume of 42 US gallons, that is, 0.16 m 3 . The flow rates of oil, respectively, water, are denoted by q o (or QO on our pictures), respectively, q w (or QW).

Sequential model aggregation for production forecasting

Description of the construction of the 104 models considered

A fine-scale reference geological model of 20 million grid blocks was initially generated and populated with properties. It was then upscaled to a 450 000 grid block model used to perform the fluid-flow simulation considered in the following as the reference one. The reservoir is initially produced under primary depletion. The producing wells are opened successively during the first 20 months of production. They are imposed a target production rate of 2 000 bbl/day, with a minimum bottomhole pressure of 725 psi. Injection starts in a second step, once all producers are opened. A target water injection rate of 4 000 bbl/day is imposed to the injectors, with a maximal bottomhole pressure of 2 611 psi. A water-cut constraint of 90% is also considered at the producers.

Static data extracted from the reference case were used to generate 104 geological models of 139 × 48 × 9 grid blocks (∼ 60 000) provided to the project participants. These models were built considering various approaches for the simulation of facies, fluvial reservoir zones, porosity and permeability. They differ from the distribution of facies, porosity, net-to-gross, water saturation and permeability. More details can be found in [START_REF] Peters | Results of the Brugge benchmark study for flooding optimization and history matching[END_REF], together with examples of permeability realizations.

The dynamic data provided for the benchmark are oil and water rates at the producers plus bottomhole pressure at all wells during a period of 10 years. They were obtained by adding some noise to the results of the reference fluid-flow simulation.

In what follows, the 104 models are used to represent the prior geological uncertainty, see Figure 6.2. (The letter codes explaining which outputs are shown are detailed below.) Two-phase flow simulations are performed for each of them, considering the same production constraints as for the reference case. Additional production constraints could also be consid- In this article, we will focus on the latter theory, which was developed in the 1990s and is summarized in the monograph [START_REF] Cesa | Prediction, Learning, and Games[END_REF].

High-level methodology: point aggregation for one-step-ahead forecasts

Unless mentioned otherwise, the aggregation of the forecasts will take place well by well, property by property. A first setting is the case of one-step-ahead forecasts. For a given well and a given property, we denote by y 1 , y 2 , . . . , y T the sequence of the observed values of the property over time and by m j,t , where j ∈ {1, . . . , 104} and t ∈ {1, . . . , T }, the sequence of the sets of forecasts output by the 104 models considered. At each step t, we linearly combine the forecasts m j,t to form our aggregated forecast y t : (6.3.1) where the weights w j,t are determined based on the past observations y s and past forecasts m j,s , where s t -1. The precise formulae to set these weights (some specific algorithms designed by the literature) are detailed in Section 6.3.3 below. The basic idea is to put higher weights on models that performed better in the past.

The main interest of this methodology is given by its performance guarantee: the weights can be set to mimic the performance of some good constant combination of the forecasts. and on what we know about the evolution of the observations. E.g., if one assumes that the absolute variation of the observations between two time steps is bounded by some value ∆, then the observation y t will lie between:

The set S is in this case a cone, that will contain all the observations. It is this approach that is shown in Figure 2.2 where the red cone contains all the considered sequences of observations (three of them are drawn in green).

Computational issue to solve

A difficulty of this approach is that, as soon as the subset S is large (infinite, possibly uncountable), one just cannot compute the forecasts scenario by scenario (S contains infinitely many of them!). But we actually aim only at the higher and lower forecasts ŷmax 1. On the first part of the data set (called the learning or training part), we use the algorithms as explained above for point aggregation.

2. On the second part of the data set (called the prediction part),

• we consider the set S = S T +1 × . . . × S T +k × . . . of all plausible continuations z T +1 , . . . , z T +k , . . . of the observations y 1 , . . . , y T ; this set S will be referred to as the set of scenarios;

• for a given scenario y 1 , . . . , y T , z T +1 , z T +2 , . . ., we compute the weights (w 1,T +k , . . . , w 104,T +k ) to use at round t + k by running the considered algorithm on the putative past observations y 1 , . . . , y T , z T +1 , . . . , z T +k-1 and past model forecasts;

• we form the aggregated forecast z T +k = j w j,T +k m j,T +k .

3. The interval forecasts S T +k are the convex hulls of all possible aggregated forecasts z T +k obtained by running all scenarios in S (with possibly some enlargement to take into account the noise level).

The main constructions remaining to be explained is (i) how the set S of plausible continuations is determined; (ii) how we may efficiently compute the interval forecasts S T +k , as there are infinitely many scenarios; (iii) what we mean by an enlargement to account for the noise level. We provide all needed explanations in Section 6.7 (which actually mostly refers to Chapter 5 of this thesis).

The high-level idea for (i) is however that we look on available data how large the typical variations were, which yields an interval [m, M ] of typical 1-step average variations. The set of scenarios is then the cone formed by the product of the intervals [y T + km, y T + kM ], where k = 0, 1, 2, . . .. See Figure 6.4 for an illustration. As for (ii), we should note that only the upper and lower bounds of S T +k need to be computed (or bounded), which is not too difficult a task.

Sequential model aggregation for production forecasting

This algorithm picks uniform (1/K, . . . , 1/K) weights at round t = 1, while at subsequent rounds t 2, it picks weights (w 1,t , . . . , w K,t ) such that

.

The weight put on model j at round t depends on the cumulative accuracy error suffered by j on rounds 1 to t -1; however, the weight is not directly proportional to this cumulative error: a rescaling via the exponential function is operated, with a parameter η > 0. We will call this parameter the learning rate of EWA: when η is smaller, the weights get closer to the uniform weights; when η is larger, the weights of the suboptimal models get closer to 0 while the (sum of the) weight(s) of the best-performing model(s) on the past get closer to 1.

To provide the performance bound we first denote by δ j the convex weight vector (0, . . . , 0, 1, 0, . . . , 0), where the unique non-zero coordinate is the j-th one. The set W of reference weights is given by W = δ j : j ∈ {1, . . . , K} .

The performance bound (6.3.2) with the above W relies on a boundedness parameter B and reads: for all bounded sequences of observations y t ∈ [0, B] and model forecasts m j,t ∈ [0, B],

In particular, ε T = O(1/T ) if η is well-calibrated, which requires the knowledge of a plausible bound B. Here again, we may prefer to set the η to be used at round t based on past data; see the next section.

How to implement these algorithms (i.e., pick their parameter λ or η)

First, note that the algorithms described above rely each on a single parameter λ > 0 or η > 0, which is in strong contrast with the geophysical models constructed. (These parameters λ and η are actually rather called hyperparameters to distinguish them from the model parameters.)

In addition, the literature provides theoretical or practical guidelines on how to choose these parameters. The key idea was introduced by [START_REF] Auer | Adaptive and self-confident on-line learning algorithms[END_REF]. It consist in letting the parameters η or λ vary over time: we denote by η t and λ t the parameters used to pick the weights (w 1,t , . . . , w K,t ) at round t. Theoretical studies offer some formulas for η t and λ t (see, e.g., [START_REF] Auer | Adaptive and self-confident on-line learning algorithms[END_REF], [START_REF] Cesa-Bianchi | Improved second-order bounds for prediction with expert advice[END_REF]) but the associated practical performance are usually poor, or at least, improvable, as noted first by [START_REF] Devaine | Forecasting the electricity consumption by aggregation of specialized experts; application to Slovakian and French country-wide (half-)hourly predictions[END_REF] and later by [START_REF] Amat | Fundamentals and exchange rate forecastability with machine learning methods[END_REF]. This is why [START_REF] Devaine | Forecasting the electricity consumption by aggregation of specialized experts; application to Slovakian and French country-wide (half-)hourly predictions[END_REF] suggested and implemented the following .7: Simulations of the models (green lines -or grey lines -, depending on whether the simulations were selected for the interval forecasts), observations (red solid line -), set S of scenarios (upper and lower bounds given by black dotted lines ---), and interval forecasts output by Ridge (upper and lower bounds given by blue solid lines -). Values of λ used are written on the graphs.

Some disappointing results

Figures 6.9 and 6.10 show, on the other hand, that for some properties, neither Ridge nor EWA may provide useful interval forecasts: the latter either completely fail to accurately predict the observations or they are so large that they cover (almost) the set of all scenarios -hence, they do not provide any useful information.

We illustrate this by letting λ increase (Figure 6.9) and η decrease (Figure 6.10): the interval forecasts become larger as the parameters vary in this way. They first provide inaccurate interval forecasts and finally resort to intervals (almost) covering all scenarios. .9: Simulations of the models (green lines -or grey lines -, depending on whether the simulations were selected for the interval forecasts), observations (red solid line -), set S of scenarios (upper and lower bounds given by black dotted lines ---), and interval forecasts output by Ridge (upper and lower bounds given by blue solid lines -). Values of λ used are written on the graphs. 6. Sequential model aggregation for production forecasting

LASSO

We conclude our experiments with a discussion of the performance of LASSO. Two points are to be discussed: first, the accuracy of the one-step-ahead forecasts, as we did for EWA and Ridge, and second, the selection power of LASSO.

As far as the forecasting is concerned, we recall that we were only able to produce onestep-ahead forecasts with LASSO, not longer-term interval forecasts. 12 reveal, compared to the similar graphs for Ridge (in particular, Figure 6.6,top), that the accuracy achieved by LASSO is slightly better than that of Ridge, with only one exception, well number 9 in terms of oil production rate. Otherwise, LASSO basically gets the best out of the accuracy of EWA (which predicted well all properties for producers, namely, bottomhole pressure, oil production rate and water production rate) and that of Ridge (which predicted well the bottomhole pressure for injectors). 6. Sequential model aggregation for production forecasting

Supplementary material

In this section, we tackle the question of quantifying and evaluating the performance of forecast intervals algorithms. We therefore define a performance measure: the efficiency. We then introduce a benchmark.

In a third part, we present two ways of calibrating the regularization parameter for the Ridge forecast intervals, and compare their efficiency with the benchmark.

A performance measure for the forecast intervals

The aim of a forecast interval is to contain the future observation, so it seems natural to take into account the number (or the proportion) of observations lying inside the forecast intervals. But this count is not a satisfying measure on its own, since it can be trivially maximized, at least when there are known lower bound m and an upper bound M on the observations, by outputting the interval [m, M ]. This interval is too wide: a good forecast interval should be narrow.

Therefore we introduce a measure that is based on this trade-off between width and accuracy of the interval forecasts: the efficiency. It is defined as the ratio of the number of observations lying within the forecast intervals, over the total sum of the width of the forecast intervals:

where S T +k = S T +k , S T +k is the forecast interval at round T + k.

Obviously, the higher the efficiency, the better. Let us emphasize the fact that this measure aims at comparing algorithms on the same data set, but its value for one precise algorithm, which depends for instance on the units, does not tell much about the quality of this algorithm. This measure should therefore be used in a relative way rather than an absolute way.

A possible benchmark

Classical benchmarks of sequential point aggregation (cf. (6.3.2)) can not be applied in the forecast intervals framework, since they forecast points and not intervals.

We suggest to use as benchmarks, the convex hulls of the forecasts of fixed subsets of experts. In order to keep computation time reasonable, we restrict ourselves to the subsets that contain only the best simulations on the learning sample. That is, we focus on the subsets of the K simulations with the smallest RMSE on the learning sample, for any K between 2 and K. Our benchmark eff will be the best efficiency among the efficiencies obtained using these K -1 subsets. It is formalized in Algorithm 16. This benchmark will be used in the next section. We present, in this section, two ways of choosing the regularization parameter λ in the case of the Ridge forecast intervals.

An "objective-driven" tuning of the parameters

This tuning aims at getting a precise forecast interval with a chosen width W desired (this will often make other forecast intervals have "reasonable" widths). We decided to work on the last forecast interval (t = T f ), for which we chose the width:

with σ the estimated noise level. The tuning consists in three steps:

• 1. Choosing the desired width W desired for a precise forecast interval.

• 2. Computing the forecast intervals corresponding to each regularization parameter belonging to a set of parameters (we used a grid of logarithmic-spaced parameters).

• 3. Keeping the parameter (and the corresponding forecast intervals) that leads to the width that is the closest to the desired width for the chosen forecast interval.

Actually, the widths of the forecast intervals tend to increase when the regularization parameter λ decreases, which can help speed up the computations.

A hybrid tuning

The tuning we present now is hybrid, in the sense that it mixes theoretical and empirical elements. It chooses the regularization parameter λ as the geometric mean of a fully empirical parameter λ emp and a theory-based parameter λ the : λ = λ emp λ the .

λ emp is the geometric mean of the parameters used by the "classical" version of the Ridge algorithm, for the last 10 instants of the learning sample (i.e., from T -9 to T ).

λ the = 2F Y T f K/B is an approximative minimizer of the bound of Theorem 2.7. F and Y are upper bounds respectively on the simulations and the observations; T f is the total number of instants in the study; B is an upper bound on the Euclidean norm of the weight vectors. All these parameters are estimated using empirical data available at the instant of prediction (including simulations forecasts for instants posterior to T ).

6.8.4. Results obtained with the two parameter tunings Figure 6.15 shows the efficiencies (multiplied by 1000 for readability) of the Ridge forecast intervals, tuned with the two approaches of the previous section, compared to the efficiencies of the benchmark presented above. We recall that the higher the efficiency, the better. The efficiencies of the algorithms are drawn in blue when they are larger than the benchmark, in red when they are smaller. One can see improvable performance on the hardest properties (QW P), but quite good performance for the other properties, at the level of the benchmark (QO P) or even better (BHP P, QO P). .16: Simulations of the models (green lines -or grey lines -, depending on whether the simulations were selected for the interval forecasts), observations (red solid line -), set S of scenarios (upper and lower bounds given by black dotted lines ---), and interval forecasts output by Ridge (upper and lower bounds given by blue solid lines -). Values of λ used are written on the graphs. The top graphs correspond to the "objective-driven" tuning of the regularization parameter, the bottom graphs to the "hybrid" tuning.