R Carletto
email: romain.carletto@univ-tours.fr

H Cardot

N Ragot

Automatic Generation of Web Advertising Layouts: A Synthetic Dataset and a Deep Learning Baseline Model

Keywords: Open source dataset, Automatic document layout, Web Advertising, Deep neural networks, Residual blocks

Automatic generation of advertising layouts shows high economic interest, but as identified with our industrial partner, there is no public document layout dataset that matches this particular application. In this context, we produced two synthetic datasets that allow both the evaluation and training of any learning model on web advertising layout generation, and a small dataset of real cases to demonstrate the contribution of our work. We compared the results obtained by different learning models on the real cases, with and without prior use of our synthetic datasets, and our results show that these datasets allow to build and decisively improve models for the generation of real-world advertising layouts. Our three datasets, as well as useful data processing tools, are available at :

Introduction 1.Automatic advertising layout applications

Laying out advertisements consists essentially of positioning and sizing rectangular bounding boxes containing the graphic elements. It is a long and still very manual job, which may seem simple due to the small number of input parameters, but it is not: the diversity and multiplicity of criteria combinations to be integrated, as well as their subjective nature require a certain level of abstraction, and small variations in a layout parameters can strongly impact the visual rendering.

Restricted access to real data

Due to advertisement ownership policies, bringing together thousands of advertising layouts within a dataset would require the agreement of too many separate entities. Furthermore, the same information (e.g. the position of an element) can be coded in thousands of different ways, due to the specificity of HTML and CSS languages, and it is currently impossible to retrieve the positions and dimensions of the elements by only parsing the code of the advertisement. The acquisition of a public dataset, to develop, evaluate and compare models for online advertising layout is therefore a problem in its own right, to which our work seeks to respond. Figure 1. Real case application of an automatic advertising layout. Types and dimensions of input assets are processed by the model to generate dimensions and positions of these assets within a given rectangular area. Assets contents have been warped for privacy matters.

Related works 2.1 Former works

Early work on automatic layout problems sought to provide basic tools for designers. Some such as Myers [START_REF] Myers | User interface software tools[END_REF] or Lok et al. [START_REF] Lok | Evaluation of visual balance for automated layout[END_REF] focus on the simple use of templates, others such as Feiner et al. [START_REF] Feiner | An Experimental System for Creating and Presenting Interactive Graphical Documents[END_REF], propose to the user to combine low-level rules. But these first tools are either too limited in terms of functionalities or too complicated to use.

More recent studies aim to bring the layout guidelines together in elaborate, multi-dimensional quantitative functions. Merrell et al. [START_REF] Merrell | Interactive Furniture Layout Using Interior Design Guidelines[END_REF] incorporate these guidelines as terms in a density function, where the inputs are the positions and the dimensions of the elements to be arranged within a given space, and then apply a Markov Chain Monte Carlo sampler on this density function to generate layout suggestions. Other authors combine the layout guidelines into an objective function to be optimized. To this nonlinear optimization problem, Lin [START_REF] Lin | Active Layout Engine: Algorithms and Applications in Variable Data Printing[END_REF] adapt a linear two-pass method based on the simplex algorithm. He considers images and texts as inputs to its system, and output the distribution of these elements on a rectangular page, ideally seeking to leave no empty space, contrary to what is sought in online advertising. Lisa Purvis et al. [START_REF] Purvis | Creating Personalized Documents: An Optimization Approach[END_REF] encode various layout criteria, such as the degree of alignment, the balance of empty spaces, the legibility and compactness of the elements on the layout space, and the balance between images and text areas. They generate layouts that optimize the sum of these different encoded criteria by evolving a population of randomly initialized sets of layout parameters via a genetic algorithm. However, the authors put the quality of the examples thus generated into perspective and points out that very close scores at the output of their system can give very different quality renderings.

Therefore, the prioritization of terms within the objective functions defined by these various works remains problematic and these first methods remain on the whole too reductive to integrate the aesthetic canons of online advertising, which are highly complex and constantly evolving.

Learning methods

More recent work on automatic document layout has made use of deep learning models. These models have the advantage of being able to learn implicit rules of arrangement via very simple objective functions, and the depth of these models makes it possible to achieve abstraction capacities which are very well adapted to the complexity of document layout tasks.

Li et al [START_REF] Li | Layoutgan: generating graphic layouts with wiredframe discriminators[END_REF] apply generative networks to different document layout problems. Their architecture is particularly elaborate and integrates several state-of-the-art concepts. They model the elements to be arranged in the form of wireframes, train and apply a Generative Adversarial Network (GAN) model to these inputs, and nevertheless add corrections at the GAN output to ensure, among other things, the alignment of the elements among themselves. Their results are significantly better than those obtained by conventional non-learning methods. However, deep learning models require a lot of training examples, which is not an obstacle in the applications covered by Li et al. article, but which is very problematic in the field of online advertising, where there is no dataset available.

Related datasets, unsuitable for online advertising

Datasets exist on applications that are partially related to advertising layout, including the Layout Analysis dataset, which presents articles containing mostly text, the RICO dataset, which contains mobile application pages, and the Abstract Scenes dataset, which presents layouts of a small number of different elements: characters, accessories, and background elements. But these datasets do not address the critical application issues of web advertising layout. Logos and call to action (CTA), for example, are two predominant categories of elements in the field of online advertising, and they do not present any similarity with the elements of these related datasets.

Furthermore, we find neither the spacing of the elements, nor the balance of spaces, nor the asymmetry that are systematically observed in online advertising. Finally, the reading order of the elements within an online advertisement is always sequential and circumvolutive and the combination of these two criteria is not found in any of these datasets. Therefore, we built two synthetic datasets in order to be able to develop and evaluate truly incisive models for online advertising placement. The first synthetic dataset contains 100,000 samples. For each sample, the input variables correspond to the information given to the designers at the start of any layout job : the categories (which in this first dataset are fictitious), the starting dimensions and the sequential order of the assets to be arranged. The corresponding output layout is then generated via a series of two sets of fictitious rules.

General rules, which apply to a large majority of samples, randomly size and position the elements within given constraints. Among these constraints, elements must not overlap, must not exceed the background frame and a minimum space is set between the elements.

Then, specific rules reposition and resize the elements of certain samples via specific and deterministic functions. An example of a specific rule is as follows: if the three elements are of the fictitious "blue" category, they are positioned against the left border. Yet, their dimensions are not affected by this rule and remain randomly defined within general rules constraints presented previously. These specific rules each relate to a particular order and combination of categories, to which a very small minority of the dataset samples correspond: about 3% of the entire dataset for each of the specific rules.

These artificial rules recreate and even increase tenfold the learning difficulties found in the real cases of advertising layout. The precision and diversity of our layout rules, the degree of contradiction between some of them, the fact that some rules concern only a very small minority of elements when others concern almost all the elements: each of these difficulties is inspired by a real problem in advertising layout and is arbitrarily increased within our first dataset.

Finally, while layout rules are only available in their implicit form in real document datasets, our artificial rules can both be presented in their implicit form during training (through the synthetic layouts they have created) and in their explicit form during evaluation, which makes it possible to evaluate any learning layout model with absolute precision.

Second synthetic dataset : realistic layouts, matching real advertising layout characteristics

The second synthetic dataset aims to allow a model to be pretrained on it then used on real cases layouts. It contains 100,000 samples and was created through several realistic layout rules, identical to decisive rules in online advertising layout. As in the first dataset, the input elements to be laid out are rectangles of different types and sizes, except that here their categories are real categories used in advertising : "image", "text", "CTA" and "logo" The constraints of dimensions and positions of our explicit layout rules are here specific to these real categories, with for example the "image" elements being often bigger than CTA elements, position of the logo being imperatively in first or last position, etc. Output layouts parameters are then very similar to those of the real advertising layouts.

Real cases dataset for evaluation

To verify the contribution of these two synthetic datasets, we created a small dataset of a hundred real-life layouts, by manually gathering and segmenting advertisements arranged by our industrial partner. The quantitative and visual results obtained on the real cases presented in this article come from this dataset.

Models

Previous research on automatic document layouts jumped from methods without learning to unsupervised deep generative networks, that are particularly heavy to set up and train. Since there is no published evidence that supervised models can't achieve layout generation and since our synthetic datasets aim to enhance the performance of the broadest range of learning models, experiments have been run on our datasets with supervised networks, which are more common and whose training is also more stable.

We conceived and experimented several adaptations of the resnet architecture [START_REF] He | Deep Residual Learning for Image Recognition[END_REF], presented in Figure 3. Model δ, which systematically reached better performances against the other deep residual models, is presented and referenced within our experiment results as our deep residual network. It is composed of 14 residual blocks followed by a fully connected layer of 12 neurons, activated by a sigmoid activation function. Each residual block contains one fully connected layer of 15 neurons, activated by a ReLU activation function.

A simple neural network of three fully connected layers (of respectively 12, 12 and 15 neurons) has also been experimented, aside of our deep residual model, to evaluate the added value of a deeper model on our datasets.

Experimental Setup

In the first dataset, samples where only general rules apply and samples concerned by a specific additional rule are shuffled all together. In each dataset, data has been split as usual into a train set, a validation set (to compare training and validation accuracy and assert the absence of over-fitting during training) and a test set (for post-training performance evaluation). The results presented in this article have been recorded on the test set, which represents 20% of the samples in each dataset.

Since our goal is to integrate layout rules without giving their explicit form to the model during training, a very general learning metric has been used in all experimented models :

L(y, ŷ) = 1 N N n=0 (y i -ŷi ) 2 . ( 1
)
L(y, ŷ) represents the mean squared error between the vectors y and ŷ, that respectively contain the original and predicted parameters of each layout, and where N is the number of layouts to predict. Optimization is run through the Adam algorithm, which adapts the learning rate to each parameter and to the first and second moments of recent gradient magnitudes.

While the only purpose of our fictitious synthetic dataset is to evaluate a model on artificially challenging layouts, our realistic synthetic dataset aims to pre-train any learning model to generate real advertising layouts. Therefore, two training protocols were experimented and compared to assert such contribution. The first protocol is to train and evaluate a model on our small dataset of real cases only. The second protocol consists of training a model on our second synthetic data set, then re-training this model on the real cases train set and finally evaluating the model on the real cases test set.

6 Evaluation and results

Evaluation metrics

While in real cases layout, training and evaluation have to deal with implicit layout rules, the artificial rules that created our synthetic layouts can be shown implicitly to an experimented model during training (through the synthetic layouts themselves) then used in their explicit form during evaluation of the model results.

Once a model has been trained on a synthetic dataset, the layouts generated by this model must thus comply with a set of explicit layout rules, depending on the category of each element and the order of these elements, vertically and horizontally. These rules are defined exhaustively in our git directory, indicated in the abstract. In the real cases dataset, the layouts have been designed by hand by designers who follow intuitive, implicit layout rules, so our evaluation is limited to critical criteria : assets must not overlap and must not exceed screen borders. For each dataset, the respect of the dataset criteria by the generated layouts is evaluated through two different metrics.

A first metric is the Samples Average Maximum Error. For each sample, we measure the difference between the expected value of each feature, defined by the set of rules impacting this sample, and the value of this feature within the layout generated by the model. We then consider the maximum of these differences for each layout and compute the average of theses maximums to get the Samples Average Maximum Error. This value is set in pixels, and has to be put into perspective with the banner frame dimensions, which is 300x600 pixels. A second metric is the percentage of Samples Containing Critical Errors, where, as specified, critical errors are those identified as strongly unacceptable in the industry : when one of the layout elements overlap on another or when it exceeds a screen border. Concerning layouts generated after a training on the first synthetic dataset, only layouts where general rules apply are evaluated by this metric. The details and code of our evaluation methods are also available in our git directory. Samples Containing Critical Errors 8% 0%

Results on our synthetic datasets

Table 2. Models performances on our realistic synthetic dataset

In Figure 4, generated layouts are compared to original synthetic layouts. As indicated in the dataset section, some parameters have been set randomly in each layout, so while some layout parameters are expected to be the same between a generated layout and an original layout, some other parameters are not concerned by this criteria and can be different without impacting the validity of a generated layout, at the evaluation step.

On the left column layout, due to the sequence of asset categories (which here is red-green-red), only general rules apply : a large range of positions and dimensions are valid but no over-lap is tolerated. On other columns, on the contrary, additional virtual rules apply, which are specific to the related sequence of asset categories and can even override some of the general rules. Therefore on each of these other columns, exact values are expected for some particular parameters, e.g. lefts must all be equal to zero on the second column, precise overlap patterns are expected on the third column (which is deliberately contradictory with the general no-overlap criteria), tops must all be equal to zero on the fourth column.

While the three-layers network results show important issues (prohibited overlaps in the first column and unexpected values in other columns), the visual accuracy of our deep residual model is in line with industrial requirements, in terms of distances between generated parameters and closest valid asset parameters. These visual results are confirmed quantitatively in Table 1, where concerning our deep residual model, the Samples Average Maximum Error is only a few pixels long and the percentage of Samples in Critical Error is very close to zero. Same performance shifts are observed on our realistic synthetic dataset, as quantitatively displayed in In Figure 5, we compare real advertising layouts generated by our deep residual network after different training protocols. For now our experiment focus on the layout of assets on a plain background. More sophisticated backgrounds and their graphical content are not considered yet, therefore they are replaced by a plain background in the real layouts generated by our models. Also, assets graphical content have been warped for privacy matters but their categories are indicated by a color frame around each asset : logos, cta, texts and images respectively correspond to red, yellow, green and blue frame colors.

The first column shows poor results, generated by our deep residual model after training on our small real cases dataset only. The second column shows much better results, reached by the same model being pre-trained on our synthetic realistic dataset, then trained again on on our small real cases dataset. These results are confirmed by quantitative results displayed in Table 4.

In Figure 6, we visually compare the results obtained by a simple three-layer model with those achieved by our deep residual model. These visual differences, in addition to the quantitative performance shift seen in Finally, we see through both visual and quantitative evaluations that the performances shifts (between two different models) are positively correlated between our synthetic and our real-cases datasets, showing that our synthetic datasets match similar problematic with the real cases dataset.

Conclusion

Identified contributions

We created two synthetic datasets that allow for strong quantitative evaluation in document layout generation, which is sorely lacking in current state of the art, where only user studies are available. These datasets also allow for the training of deep learning models on advertising layout generation, in a context where real web advertising layouts are highly restricted and where related document layout datasets do not match decisive characteristics of advertising layout.

Our fictitious synthetic dataset allows to evaluate the capacity of a given model to integrate layout rules that match and even surpasses specific challenges of advertising layouts generation. Our results also show that the complexity of these virtual layout rules is not accessible to trivial neural network models, and can be addressed only by deeper models. Our realistic synthetic dataset makes it possible to pre-train a model to generate real advertising layouts that match industry critical criteria, even when only a few real training samples are available.

Finally, the tools we built to create our synthetic datasets can easily be handled by designers to teach learning models any creative bias, in the form of new explicit rules of their choice, in addition to implicit rules that those models directly learn on real data. This combines the advantage of both learning and non-learning approaches that were opposed so far, as described in our related works section.

Future works

Since our work aimed to enhance the evaluation and the training of main deep learning models, we focused our experimentation on supervised models which are widespread, stable to train and allow, as our results show, to generate real advertising layouts that match industry decisive requirements. However, generative unsupervised models are often experimented in related works on layout generation and they are never compared with supervised models. Therefore, it could be very interesting to compare unsupervised generative models with our supervised model. This comparison can be run on our synthetic datasets but also on related datasets, which do not match advertising layout requirements but address other valuable layout problems.

Another interesting approach would be to replace dimensional inputs by actual images content and analyze the graphical and semantic content of input elements in the process of generating layouts. In addition, adapting saliency maps and segmentation networks to integrate the background graphical content and then position elements between important areas of the background could be of great interest.

Figure 2 .

 2 Figure 2. Layouts of our first dataset, created by fictitious layout rules. Layouts (a) have been created from general rules while layouts (b) and (c) have been generated through special rules, activated on layouts where the colors of the elements match specific combinations.

Figure 4 .

 4 Figure 4. layouts generated by a three-layers fully connected network (first row) and by our deep residual model (middle row), after training on our synthetic dataset. Original synthetic layouts are presented in the lower row.

Figure 5 .

 5 Figure 5. Visual comparison between real advertising layouts generated with and without pre-training (resp. middle and left columns) on our second synthetic dataset. Ground-truth is presented in the right column. Several results in the first column show critical overlapping issues while results of the middle column match all decisive criteria.

  Figure 3. Experimented deep residual models. Best performance is reached by model δ, referenced in this article as our deep residual model.

	Residual block	Residual block	Residual block	Residual block
	of type A1	of type A2	of type B	of type C
	Input X	Input X	Input X	Input X
	×2		×3	×1
	Fully connected layer	×2	Fully connected layer	Fully connected layer
	+ ReLU activation	Residual block A1	+ ReLU activation	+ ReLU activation
	function		function	function
	Concatenation	Concatenation	Concatenation	Concatenation
	Output y	Output y	Output y	Output y
	model α	model β	model γ	model δ
	Input X	Input X	Input X	Input X
	×7	×3	×7	×14
	Residual block A1	Residual block A2	Residual block B	Residual block C
		Concatenation
	Fully connected layer	Fully connected layer	Fully connected layer	Fully connected layer
	+ sigmoid activation	+ sigmoid activation	+ sigmoid activation	+ sigmoid activation
	function	function	function	function
	Output y	Output y	Output y	Output y

Table 1 .

 1 Models performances on our fictitious synthetic dataset

		Three Layers	Our Deep
	Evaluation Metric	Fully Connected	Residual
		Network	Network
	Samples Average Maximum Error	17	3
	Samples Containing Critical Errors	13%	0.7%
		Three Layers	Our Deep
	Evaluation Metric	Fully Connected	Residual
		Network	Network
	Samples Average Maximum Error	9	2

Table 2 6

 2 .3 Results on real cases

		Training	Training on Synthetic
	Evaluation Metric	on Real	Samples Then Training
		Cases	on Real Cases
	Samples Average Maximum Error	25	11
	Samples Containing Critical Errors	60%	4%

Table 3 .

 3 Performance gains of our deep residual model on real cases due to pre-training on our realistic synthetic dataset

Table 4 ,

 4 show that trivial models are not suitable for real advertising layout tasks, while

		Three Layers	Our Deep
	Evaluation Metric	Fully Connected	Residual
		Network	Network
	Samples Average Maximum Error	38	11
	Samples Containing Critical Errors	16%	4%

Table 4 .

 4 Models performances on real cases. Each model has been pre-trained on our realistic synthetic dataset our own deep residual model matches reliability expectations on critical layout criteria.

Acknowledgements

This work is financed by Centre Val de Loire Region (France) and by Madmix Digital, a creative studio based in Paris and New-York, which has enabled us to identify and scientifically match the major challenges of layout generation in the online advertising industry.

https: //github.com/romain-rsr/synth_datasets_ for_web_advertising_layout/


Effet du croisement sur la qualité des agneaux de lait des brebis Manech tête rousse

  Effet du croisement sur la qualité des agneaux de lait des brebis Manech tête rousse Keranflech L., Papillon S., Carpentier D., Arranz J.M., Ellies-Oury M.P. de la future IGP « Agneau des Pyrénées », incluant déjà les agneaux en race pure, les producteurs des races laitières expérimentent l'intérêt des agneaux croisés races laitières avec la race « Montagne noire » présentant de meilleures qualités bouchères, en substitution ou complément du croisement traditionnel avec des races bouchères non locales. En dépit de l'intérêt technique et économique (croissance plus rapide, conformation plus adaptée aux marchés), les éleveurs ne pratiquent que trop peu les croisements. Dans ce contexte, cette étude préliminaire a pour objectif de quantifier l'intérêt du croisement par rapport à la race pure de Manech tête rousse, en termes de gain de performances et d'adaptation au marché.Matériel etMéthode 21 agneaux de lait abattus à 49 ± 8 jours (11,6 ± 2,0 kg). Composition des carcasses (dissection totale) Propriétés sensorielles (couleur et fermeté) de 3 morceaux xMN > MTR > xB (***) a* : xB = MTR > xMN (***) b* : ns Interaction : ns Conclusion Les agneaux croisés ont de meilleures performances, sans altération significative des propriétés des viandes. Les agneaux xMN ont des propriétés sensorielles (fermeté et couleur des gras) légèrement plus conformes aux attentes des consommateurs.


Olivier Marchal

Julyan Arbel

On the sub-Gaussianity of the Beta and Dirichlet distributions

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta momentgenerating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.

Introduction

The sub-Gaussian property (Buldygin andKozachenko, 1980, 2000;[START_REF] Pisier | Subgaussian sequences in probability and Fourier analysis[END_REF] and related concentration inequalities [START_REF] Boucheron | Concentration inequalities: A nonasymptotic theory of independence[END_REF][START_REF] Raginsky | Concentration of measure inequalities in information theory, communications, and coding[END_REF] have attracted a lot of attention in the last couple of decades due to their applications in various areas such as pure mathematics, physics, information theory and computer sciences. Recent interest focused on deriving the optimal proxy variance for discrete random variables like the Bernoulli distribution [START_REF] Buldygin | The sub-Gaussian norm of a binary random variable[END_REF][START_REF] Kearns | Large deviation methods for approximate probabilistic inference[END_REF][START_REF] Berend | On the concentration of the missing mass[END_REF] and the missing mass [START_REF] Mcallester | On the convergence rate of Good-Turing estimators[END_REF][START_REF] Mcallester | Concentration inequalities for the missing mass and for histogram rule error[END_REF][START_REF] Berend | On the concentration of the missing mass[END_REF][START_REF] Ben-Hamou | Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications[END_REF]. Our focus is instead on two continuous random variables, the Beta and Dirichlet distributions, for which the optimal proxy variance was not known to the best of our knowledge. Some upper bounds were recently conjectured by [START_REF] Elder | Bayesian adaptive data analysis guarantees from subgaussianity[END_REF] that we prove in the present article by providing the optimal proxy variance for both Beta and Dirichlet distributions. Similar concentration properties of the Beta distribution have been recently used in many contexts including Bayesian adaptive data analysis [START_REF] Elder | Bayesian adaptive data analysis guarantees from subgaussianity[END_REF], Bayesian nonparametrics [START_REF] Castillo | Pólya tree posterior distributions on densities[END_REF] and spectral properties of random matrices [START_REF] Perry | Statistical limits of spiked tensor models[END_REF].

We start by reminding the definition of sub-Gaussian property for random variables:

Definition 1 (Sub-Gaussian variables). A random variable X with finite mean µ = E[X] is sub-Gaussian if there is a positive number σ such that:

E[exp(λ(X -µ))] ≤ exp λ 2 σ 2 2 for all λ ∈ R. ( 1
)
Such a constant σ 2 is called a proxy variance (or sub-Gaussian norm), and we say that X is σ 2 -sub-Gaussian. If X is sub-Gaussian, one is usually interested in the optimal proxy variance:

σ 2 opt (X) = min{σ 2 ≥ 0 such that X is σ 2 -sub-Gaussian}.
Note that the variance always gives a lower bound on the optimal proxy variance: Var[X] ≤ σ 2 opt (X). In particular, when σ 2 opt (X) = Var[X], X is said to be strictly sub-Gaussian.

Every compactly supported distribution, as is the Beta(α, β) distribution, is sub-Gaussian. This can be seen by Hoeffding's classic inequality: any random variable X supported on [0, 1] with mean µ satisfies ∀λ ∈ R, E e λ(X-µ) ≤ e λ 2 8 , thus exhibiting1 4 as an upper bound to the proxy variance. This bound can be improved by taking into account the location of the mean µ within the interval [0, 1]. An early step in this direction is the second inequality in Hoeffding (1963) paper, indexed (2.2). It states that if µ < 1/2, then for any positive , P(X -µ > ) ≤ e -2 g (µ) , where

g(µ) = 1 1 -2µ ln 1 -µ µ (2)
thus indicating that X has a right tail lighter than a Gaussian tail of variance 1 2g(µ) . Hoeffding's result was strengthened by [START_REF] Kearns | Large deviation methods for approximate probabilistic inference[END_REF] to comply with Definition 1 of sub-Gaussianity 1 as follows

E[exp(λ(X -µ))] ≤ exp λ 2 4g(µ) for all λ ∈ R, (3)
thus indicating that 1 2g(µ) is a distribution-sensitive proxy variance for any [0, 1]-supported random variable with mean µ (see also [START_REF] Berend | On the concentration of the missing mass[END_REF], for a detailed proof of this result). If this is the optimal proxy variance for the Bernoulli distribution (see Theorem 2.1 and Theorem 3.1 of [START_REF] Buldygin | The sub-Gaussian norm of a binary random variable[END_REF], it is clear from our result that it does not hold true for the Beta distribution. However, fixing α α+β = µ and letting α → 0, β → 0, the Beta(α, β) distribution concentrates to the Bern(µ) distribution, and we show that we recover the optimal proxy variance for the Bernoulli distribution (Theorem 2).

An interesting common feature between optimal proxy variances for the Bernoulli distribution:

1 2g(µ)
, and that of the Beta distribution derived later on, is that they deteriorate in a similar fashion as the mean µ goes to 0 or 1, see for instance the left panel of Figure 1. We briefly present here classical proof techniques for sub-Gaussianity hinging on certain tools from functional analysis. We show how they apply in the Bernoulli setting, and let as an interesting open problem how our proof in the Beta distribution setting could be supplemented by these same functional analysis tools.

Essentially two (related) functional inequalities allow one to derive a sub-Gaussian property: log-Sobolev inequalities, which date back to [START_REF] Gross | Logarithmic sobolev inequalities[END_REF], and transport inequalities. The relation with the former inequalities is called Herbst's argument. It states that if a probability measure satisfies a log-Sobolev inequality with some constant, then it is sub-Gaussian with the same constant as a proxy variance 2 (see for instance Ledoux, 1999, Section 2.3 and Proposition 2.3). The optimal constant in the log-Sobolev inequality satisfied by the Bernoulli distribution also produces its optimal proxy variance (Ledoux, 1999, Corollary 5.9).

The relation with transport inequalities is usually referred to as Marton's argument (see for instance Raginsky and Sason, 2013, Section 3.4). Define the Wasserstein distance between two probability measures P and Q on a space X by

W (P, Q) = inf π∈Π(P,Q) X ×X d(x, y)π(dx, dy),
where Π(P, Q) is the set of probability measures on X × X with fixed marginal distributions respectively P and Q. The Wasserstein distance depends on some choice of a distance d on X . A probability measure P is said to satisfy a transport inequality with constant c, if for any probability measure Q dominated by P ,

W (P, Q) ≤ 2cD(Q||P ), (4)
where D(Q||P ) is the entropy, or Kullback-Leibler divergence, between P and Q. The transport inequality (4) is denoted by T(c). [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic sobolev inequalities[END_REF] proved that T(c) implies c-sub-Gaussianity. See also Proposition 3.6 and Theorem 3.4.4 of [START_REF] Raginsky | Concentration of measure inequalities in information theory, communications, and coding[END_REF] for general results. Further developments in the discrete X setting are interesting for our purposes. Equip a discrete space X with the Hamming metric, d(x, y) = {x =y} . The induced Wasserstein distance then reduces to the total variation distance, W (P, Q) = P -Q TV . In that setting, [START_REF] Ordentlich | A distribution dependent refinement of pinsker's inequality[END_REF] proved the distribution-sensitive transport inequality:

P -Q TV ≤ 1 g(µ P ) D(Q||P ), (5)
where the function g is defined in Equation (2) and the coefficient µ P is called the balance coefficient of P , and is defined by µ P = max A⊂X min{P (A), 1 -P (A)}. In particular, the Bernoulli balance coefficient is easily shown to coincide with its mean. Hence, applying the result of [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic sobolev inequalities[END_REF] to the T 1 2g(µ P ) transport inequality (5) yields a distribution-sensitive proxy variance of 1 2g(µ) for the Bernoulli with mean µ. It is optimal, see for instance Theorem 3.4.6 of [START_REF] Raginsky | Concentration of measure inequalities in information theory, communications, and coding[END_REF]. This viewpoint highlights the key role played by the balance coefficient in the non-uniformity of the optimal proxy variance for discrete distributions such as the Bernoulli. However, it is not clear how this argument would carry over to non discrete distributions such as the Beta distribution for explaining similar sensitivity to the mean. However, to quote [START_REF] Raginsky | Concentration of measure inequalities in information theory, communications, and coding[END_REF], the general approach may not produce optimal concentration estimates, that often require case-by-case treatments. This is the route followed in this note for the Beta distribution.

The outline of the note is as follows. We introduce the Beta distribution and state the main result (Theorem 1) in Section 2.1. We then prove our result depending on whether α = β (Section 2.2) or α = β (Section 2.3). In the first case, the proof is elementary and based on comparing the coefficients of the entire series representations of the functions of both sides of inequality (1). However, it does not directly carry over to the second case, whose proof requires some finer analysis tool: the study of the ordinary differential equation (ODE) satisfied by the confluent hypergeometric function 1 F 1 . Although the second proof also covers the case α = β upon slight modifications, the independent proof for the symmetric case is kept owing to its simplicity. As a by-product, we derive the optimal proxy variance for the Bernoulli and the Dirichlet distributions in Section 3. The R code for the plots presented in this note and for a function deriving the optimal proxy variance in terms of α and β is available at http://www.julyanarbel.com/software.

2 Optimal proxy variance for the Beta distribution

Notations and main result

The Beta(α, β) distribution, with α, β > 0, is characterized by a density on the segment [0, 1] given by:

f (x) = 1 B(α, β) x α-1 (1 -x) β-1 , where B(α, β) = ∞ 0 x α-1 (1 -x) β-1 dx = Γ(α)Γ(β) Γ(α+β)
is the Beta function. The moment-generating function of a Beta(α, β) distribution is given by a confluent hypergeometric function (also known as Kummer's function):

E[exp(λX)] = 1 F 1 (α; α + β; λ) = ∞ j=0 Γ(α + j)Γ(α + β) (j!)Γ(α)Γ(α + β + j) λ j . ( 6
)
This is equivalent to say that the j th raw moment of a Beta(α, β) random variable X is given by:

E[X j ] = (α) j (α + β) j , (7)
where (x

) j = x(x + 1) • • • (x + j -1) = Γ(x+j) Γ(x)
is the Pochhammer symbol, also known in the literature as a rising factorial. In particular, the mean and variance are given by:

E[X] = α α + β , Var[X] = αβ (α + β) 2 (α + β + 1)
.

The Beta distribution is ubiquitous in statistics. It plays a central role in the binomial model in Bayesian statistics where it is a conjugate prior distribution (the associated posterior distribution is also Beta): if X ∼ Binomial(θ, N ) and θ ∼ Beta(α, β), then θ|X ∼ Beta(α + X, β + N -X). It is also key to Bayesian nonparametrics where it embodies, among others, the distribution of the breaks in the stick-breaking representation of the Dirichlet process and the Pitman-Yor process; marginal distributions of Polya trees [START_REF] Castillo | Pólya tree posterior distributions on densities[END_REF]; the posterior distribution of discovery probabilities under a Bayesian nonparametrics model [START_REF] Arbel | Bayesian nonparametric inference for discovery probabilities: credible intervals and large sample asymptotics[END_REF]. Our main result opens new research avenues for instance about asymptotic (frequentist) assessments of these procedures.

Our main result regarding the Beta distribution is the following:

Theorem 1 (Optimal proxy variance for the Beta distribution). For any α, β > 0, the Beta distribution Beta(α, β) is σ 2 opt (α, β)-sub-Gaussian with optimal proxy variance σ 2 opt (α, β) given by:

       σ 2 opt (α, β) = α (α+β)x0 1F1(α+1;α+β+1;x0) 1F1(α;α+β;x0)
-1 where x 0 is the unique solution of the equation

ln( 1 F 1 (α; α + β; x 0 )) = αx0 2(α+β) 1 + 1F1(α+1;α+β+1;x0) 1F1(α;α+β;x0) . (8)
A simple and explicit upper bound to

σ 2 opt (α, β) is given by σ 2 0 (α, β) = 1 4(α+β+1) : -for α = β we have Var[Beta(α, β)] < σ 2 opt (α, β) < 1 4(α+β+1) -for α = β we have Var[Beta(α, α)] = σ 2 opt (α, α) = 1 4(2α+1) .
Equation ( 8) defining x 0 is a transcendental equation, the solution of which is not available in closed form. However, it is simple to evaluate numerically. The values of the variance, optimal proxy variance and its simple upper bound are illustrated on Figure 1. Note that for a fixed value of the sum of the parameters, α + β = S, the optimal proxy variance deteriorates when α, or equivalently β, gets close to 0 or to S. This is reminiscent of the Bernoulli optimal proxy variance behavior which deteriorates when the success probability moves away from 1 2 [START_REF] Buldygin | The sub-Gaussian norm of a binary random variable[END_REF]. The intuition of the proof can be seen from Figure 2 (Section 2.3.3) where we represent the difference

λ → exp E[X]λ + σ 2 2 λ 2 -E[exp(λX)]
for various values of σ 2 . The main argument is that the optimal proxy variance is obtained for the curve (in magenta) whose positive local minimum equals zero, thus leading to the system of equations of Theorem 1.

Corollary 1. The Beta distribution Beta(α, β) is strictly sub-Gaussian if and only if α = β. As a direct consequence, we obtain the strict sub-Gaussianity of the uniform, the arc-sine and the Wigner semicircle distributions, as special cases up to a trivial rescaling of the Beta(α, α) distribution respectively with α equal to 1, 1 2 and 3 2 .

2.2 The Beta(α, α) distribution is strictly sub-Gaussian

Let σ 2 0 (α) = Var[Beta(α, α)] = 1 4(2α+1) . Since a random variable X ∼ Beta(α, α) is symmetric around 1 2
, only its even centered moments are non-zero. The reason why

E[exp(λ(X -E[X]))] ≤ exp λ 2 σ 2 0 (α) 2
is because the coefficients of the series expansions at λ = 0 of each side:

E[exp(λ(X -E[X]))] = ∞ j=0 E (X -1/2) 2j λ 2j (2j)! , (9) exp λ 2 σ 2 0 (α) 2 = ∞ j=0 σ 2j 2 j λ 2j j! = ∞ j=0 λ 2j 2 2j 2 j (2α + 1) j (j!) , (10)
satisfy the inequalities:

E X - 1 2 2j 1 (2j)! ≤ σ 2j 0 (α) 2 j 1 j! . ( 11
)
Indeed, algebra yields:

E X - 1 2 2j = (2j)! 2 2j j! Γ(2α)Γ(α + j) Γ(α)Γ(2(α + j)) = (2j)! 2 2j j! (α) j (2α) 2j . ( 12
)
Combining the expression of the raw moments (7) with the following inequality:

(α) j (2α) 2j = 1 2 j j l=1 (2α + 2l -1) ≤ 1 (2(2α + 1)) j , (13)
in ( 9) concludes the proof.

Remark 1. The non-symmetrical distribution with α = β has even centered moments whose expressions are not as simple as (12). Moreover, it has obviously non-zero odd centered moments. For this last reason, the present proof does not carry over to the case α = β.

2.3 Optimal proxy variance for the Beta(α, β) distribution

Connection with ordinary differential equations

In this section, we assume that X ∼ Beta(α, β) with β = α. We denote σ 2 0 = 1 4(α+β+1) (we omit the dependence on α and β for compactness) and define for all t ∈ R:

σ 2 t = (1 -t) 4(1 + α + β) + t αβ (α + β) 2 (α + β + 1) = 1 4(1 + α + β) + (β -α) 2 4(α + β) 2 (1 + α + β) t.
In other words, the decreasing function

t → σ 2 t maps the interval [0, 1] to the interval [σ 2 1 , σ 2 0 ] with σ 2 1 = Var[X]
. Then, we introduce the function u t defined by:

u t (x) def = exp α α + β x + σ 2 t 2 x 2 -E[exp(xX)] , ∀ x ∈ R,
where σ 2 t -sub-Gaussianity amounts to non negativity of u t on R. Since the confluent hypergeometric function y : x → y(x) = 1 F 1 (α, α+β; x) satisfies the linear second order ordinary differential equation xy (x) + (α + β -x)y (x) -αy(x) = 0, we obtain together with equation ( 6) that u t is the unique solution of the Cauchy problem:

     xu t (x) + (α + β -x)u t (x) -αu t (x) = x 16(α+β) 4 (1+α+β) 2 exp α α+β x + σ 2 t 2 x 2 P 2 (x; t), u t (0) = 0 and u t (0) = 0, ( 14
)
where P 2 is a polynomial of degree 2 in x:

P 2 (x; t) = 4(1 -t)(α 2 -β 2 ) 2 (1 + α + β) 2 -4(β 2 -α 2 )(1 + α + β) (α + β) 2 -t(β -α) 2 x + (α + β) 2 -t(β -α) 2 2 x 2 .
For normalization purposes, we also define:

v t (x) = 16(α + β) 4 (1 + α + β) 2 u t (x) exp - α α + β x - σ 2 t 2 x 2 , = 16(α + β) 4 (1 + α + β) 2 1 -E[exp(xX)] exp - α α + β x - σ 2 t 2 x 2 . ( 15
)
The function v t is the unique solution of the Cauchy problem:

xv t (x) + Q 1 (x; t)v t (x) + Q 0 (x; t)v t (x) = xP 2 (x; t), v t (0) = 0 and v t (0) = 0, (16)
with:

Q 1 (x; t) = α + β - β -α α + β x + (α + β) 2 -t(β -α) 2 2(α + β) 2 (1 + α + β) x 2 , Q 0 (x; t) = 1 16(α + β) 4 (1 + α + β) 2 xP 2 (x; t).
Note that u t and v t have the same sign hence proving that u t is positive (resp. negative) is equivalent to proving that v t is positive (resp. negative). From standard theory on ODEs [START_REF] Birkhoff | Ordinary Differential Equations[END_REF][START_REF] Robinson | An introduction to Ordinary Differential Equations[END_REF], we get that the functions u t and v t are C ∞ (R). Indeed, the only possible singularity is at x = 0 but the initial conditions imply that the function is regular at this point. In particular, a Taylor expansion at x = 0 shows that:

v t (0) = P 2 (0; t) 1 + Q 1 (0; t) = 4(β 2 -α 2 ) 2 (1 + α + β)(1 -t). (17)
We also observe that the discriminant of the polynomial x → P 2 (x; t) is given by:

∆ t = 16t(1 + α + β) 2 (β 2 -α 2 ) 2 (α + β) 2 -t(β -α) 2 2 .
Hence we conclude that for t > 0, P 2 admits two distinct real zeros that are positive, while for t < 0 it remains strictly positive on R. For t = 0, P 2 admits a double zero and thus remains positive on R appart from its zero.

By definition (15), we want to study the sign of v t on R. Indeed, showing that v t is positive on R then X is equivalent to showing σ 2 t -sub-Gaussianity. We first observe that we may restrict the sign study on R + . Indeed, if we prove that:

∀ λ ≥ 0 , E[exp(λX)] = 1 F 1 (α; α + β; λ) ≤ exp λα α + β + λ 2 σ 2 t 2 . ( 18
)
then, the case λ < 0 is automatically obtained by noting that 1 -X ∼ Beta(β, α), whose mean is

β α+β = 1 -α α+β .
Therefore, applying (18) to 1 -X gives that for all λ < 0:

E[exp(λX)] = exp(λ)E[exp(-λ(1 -X))] ≤ exp(λ) exp -λ(1 - α α + β ) + σ 2 t λ 2 2 = exp λ α α + β + σ 2 t λ 2 2 .
Eventually, in agreement with the general theory, we observe that for t > 1 (i.e. σ 2 t < Var[X]), X is not σ 2 t -sub-Gaussian. Indeed, the series expansion at x = 0 (17), shows that for t > 1, v t is strictly negative in a neighborhood of 0. On the contrary, for t < 1, the function v t is strictly positive in a neighborhood of 0 so that we may not directly conclude. Note also that for any value of t, we always have lim

x→∞ v t (x) = +∞.
2.3.2 Proof that the Beta(α, β) distribution is σ 2 0 -sub-Gaussian In this section, we take t = 0. As explained above, this corresponds to a case where P 2 is positive on R (apart from its double zero). We prove that u 0 (x) > 0 for x > 0 by proceeding by contradiction. Let us assume that there exists x 1 > 0 such that u 0 (x 1 ) = 0. Since the non-empty set {x > 0 / u 0 (x) = 0} is compact (because u 0 ∈ C ∞ (R)) and excludes a neighborhood of 0, we may define x 0 = min{x > 0 such that u 0 (x) = 0} > 0. Let us now define the set: M = {0 < x < x 0 such that u 0 (x) = 0 and u 0 changes sign at x}. Since u 0 (0) = u 0 (x 0 ) = 0 and the facts that u 0 is strictly positive in a neighborhood of 0 and u 0 is continuous on R, Rolle's theorem shows that M is not empty and that: m = min{x ∈ M } exists and 0 < m < x 0 .

Evaluating the ODE ( 14) at x = m and using the fact that the polynomial P 2 is positive on R (appart from its double zero) leads to:

mu 0 (m) + 0 -αu 0 (m) ≥ 0 ⇒ u 0 (m) ≥ α m u 0 (m) > 0. ( 19
)
However, combined with u 0 (m) = 0, this contradicts the fact that u 0 changes sign at x = m.

Thus, we conclude that there cannot exist x 1 > 0 such that u 0 (x 1 ) = 0. Since u 0 is strictly positive in a neighborhood of 0 and continuous on R, we conclude that it must remain strictly positive on R * + .

Remark 2. In this proof, the case β = α requires an adaptation since u 0 (0) = 0. Thus, we must determine u

(4) 0 (0) > 0 (u (3)
0 (0) = 0 by symmetry) to ensure that the function u 0 is locally convex and remains strictly positive in a neighborhood of 0. Apart from this minor verification, the rest of the proof applies also to this case.

Proof of the optimal proxy variance for the Beta(α, β) distribution

In this section we assume that β = α. From general theorems regarding ODEs, we have that the application:

g : [0, +∞) × [0, +∞) → R (x, t) → g(x, t) = v t (x), ( 20
) is smooth (g ∈ C ∞ ([0, +∞) × [0, +∞)))
. Indeed, the t-dependence of the coefficients of the ODE ( 16) is polynomial and thus smooth. The x-dependence of the coefficients of the ODE ( 16) is polynomial and as explained above, the only possible singularity in x is at x = 0 but initial conditions ensure that the solutions x → v t (x) are always regular there. Since for all t ≥ 0 we have lim x→∞ v t (x) = +∞, we also have that the function:

h : [0, +∞) → R t → min{v t (x), x ∈ R * + }, is continuous.
We now observe that for any 0 ≤ t < 1, the functions v t are strictly positive in a neighborhood of 0. More precisely, if we choose a segment [0, t 0 ] with t 0 < 1, then for all 0 ≤ t ≤ t 0 we have that

v t (0) ≥ v t0 (0) = 4(β 2 -α 2 ) 2 (1 + α + β)(1 -t 0 ) > 0.
Hence, we may choose η > 0 such that for all 0 ≤ t ≤ t 0 , we have v t is strictly positive on ]0, η]. Moreover, since lim x→∞ v 0 (x) = +∞, v 0 is bounded from below on [η, +∞[ by a constant A > 0. Thus, since g is continuous, there exists a neighborhood of t = 0 in which all solutions v t remain greater than A 2 on [η, +∞[ and thus strictly positive on R * + . This shows that for t > 0, σ 2 0 is not optimal. Let us now introduce the set:

T + = {t ≥ 0 such that v t is positive on R * + }.
Then, from the results presented above, we know that T + is non-empty, that it contains a neighborhood of 0 and that it is bounded from above by 1. Moreover, by connection with the initial problem (15), T + is an interval and thus is of the form [0, t opt ] with 0 < t opt < 1. Indeed t ∈ T + implies by construction that for all s ≤ t, s ∈ T + . Note also that t < 1 since v 1 is strictly negative in a neighborhood of 0 and thus min{v 1 (x), x ∈ R * + } < 0. Hence, the continuity of h shows that there exists a neighborhood of t = 1 in which the solutions v t are non-positive on R * + . For t = t opt the function v topt must have a zero on R * + otherwise by continuity of h we may find a neighborhood of t opt for which min{v t (x), x ∈ R * + } remains strictly positive thus contradicting the maximality of t opt . Since v topt must remain positive, the zero is at least a double zero and therefore we find that there exists x 0 > 0 such that v topt (x 0 ) = 0, v topt (x 0 ) = 0 and v topt (x 0 ) ≥ 0. From ( 6) and ( 15), the conditions v topt (x 0 ) = 0, v topt (x 0 ) = 0 are equivalent to the following system of equations (we use here the contiguous relations for the confluent hypergeometric function:

1 F 1 (a; b; x) = a b 1 F 1 (a; b; x)):      1 F 1 (α; α + β; x 0 ) = exp α α+β x 0 + σ 2 t opt 2 x 2 0 , α α+β 1 F 1 (α + 1; α + β + 1; x 0 ) = α α+β + σ 2 topt x 0 1 F 1 (α; α + β; x 0 ). (21)
This is equivalent to say that x 0 ≡ x 0 (α, β) is the solution of the transcendental equation:

1 F 1 (α; α + β; x 0 ) = exp αx 0 2(α + β) 1 + 1 F 1 (α + 1; α + β + 1; x 0 ) 1 F 1 (α; α + β; x 0 )
, and that σ 2 topt is given by:

σ 2 topt = α (α + β)x 0 1 F 1 (α + 1; α + β + 1; x 0 ) 1 F 1 (α; α + β; x 0 ) -1 .
Note that by symmetry, we have

x 0 (β, α) = -x 0 (α, β) hence, σ 2 topt (β, α) = σ 2 topt (α, β). Moreover, if β > α then x 0 (α, β) > 0 while α > β implies x 0 (α, β) < 0.
We may illustrate the situation with Figure 2 which displays the difference function x → u t (x).

Remark 3. The system of equations ( 21) admits only one solution on R * + . Indeed, let us transpose the problem from v topt to u topt using (15) and assume that there exist two points 0 < x 0 < x 1 such that u topt (x 0 ) = 0, u topt (x 0 ) = 0 and u topt (x 1 ) = 0, u topt (x 1 ) = 0 with u topt strictly positive on (x 0 , x 1 ) (hence u topt (x 0 ) ≥ 0 and u topt (x 1 ) ≥ 0). Using (14), this implies that P 2 (x 0 ; t opt ) ≥ 0 and P 2 (x 1 ; t opt ) ≥ 0. If we denote x -< x + the potential distinct positive zeros of x → P 2 (x; t opt ) we may exclude that x 0 ≤ x -. Indeed, if x 0 ≤ x -then we may apply the same argument to u topt on the interval [0, x 0 ] as the one developed for u 0 in Section 2.3.2 and obtain a contradiction. Thus, the only remaining case is to assume x 1 > x 0 > x + . In that case, since u topt (x 0 ) = 0, u topt (x 0 ) = 0, u topt (x 1 ) = 0 and x → P 2 (x; t opt ) is positive on [x 0 , x 1 ], we may apply the same argument to u topt on the interval [x 0 , x 1 ] as the one developed for u 0 in Section 2.3.2 and obtain a contradiction.

3 Relations to other distributions

Optimal proxy variance for the Bernoulli distribution

We show that our proof technique can be used to recover the optimal proxy variance for the Bernoulli distribution, known since [START_REF] Kearns | Large deviation methods for approximate probabilistic inference[END_REF]. This is illustrated by the center panel of Figure 1.

Theorem 2 (Optimal proxy variance for the Bernoulli distribution). For any µ ∈ (0, 1), the Bernoulli distribution with mean µ is sub-Gaussian with optimal proxy variance σ 2 opt (µ) given by:

σ 2 opt (µ) = (1 -2µ) 2 ln 1-µ µ . ( 22
)
Proof. In the limit α → 0 with α α+β fixed equal to µ, the differential equation ( 14) simplifies into:

u t,µ (x) -u t,µ (x) = exp µx + x 2 8 - x 2 (2µ -1) 2 t 8 (2µ -1) 2 (1 -t) 4 + (2µ -1)(1 -t + 4tµ(1 -µ)) 4 x - (1 -t + 4tµ(1 -µ)) 2 16 x 2 0 1 2 3 -1 0 1 2 x10 -3 q x 0 u 0 u topt u tnon opt u 1
Figure 2: Difference function x → u t (x). For t = 0 (simple upper bound σ 2 0 ), the curve [dotted black] remains strictly positive. For t = t opt (optimal proxy variance σ 2 opt ), the curve [magenta] has zero minimum (at x 0 ). For t = 1 (leading to the variance), the curve [dashed green] has negative second derivative at x = 0, hence is directly negative around 0. The intermediate case with t non opt in the interval (t opt , 1) produces a curve [orange, dash and dots] which is first positive, then negative, and positive again.

with the Cauchy initial conditions u t,µ (0) = 0 and u t,µ (0) = 0. The solution of this Cauchy problem is explicit and given by:

u t,µ (x) = exp µx + x 2 8 - x 2 (2µ -1) 2 t 8 -µe x + µ -1 (23)
Therefore the optimal proxy variance is given by σ 2 opt (µ) = 1 4 -1 4 x(2µ -1) 2 t 0 where t 0 is determined by the system of equations: u t0,µ (x 0 ) = 0 and u t0,µ (x 0 ), thus defining implicitly t 0 and x 0 as functions of µ. In order to solve explicitly the last system of equations, we perform the change of variables: (µ, t) = s s+1 , 2 t s+1 + 1 so that the solution ( 23) is now given by:

u t,s (x) = exp s s + 1 x - t 4(s + 1) x 2 - s s + 1 e x - 1 s + 1
Consequently, we have to solve the system:

u t0,s (x 0 ) = 0 u t0,x (x 0 ) = 0 ⇔    (s + 1) exp s s+1 x 0 -t0 4(s+1) x 2 0 = se x0 + 1 s -x0 t0 2 exp s s+1 x 0 -t0 4(s+1) x 2 0 = se x0
Introducing another change of variable (x 0 , y 0 ) = (x 0 , x 0 t0 ), the last system is equivalent to:

s -y 0 = s(1 + y 0 )e x0 1 + y 0 = exp -x0 s+1 s -y0 2 ⇔ s -y 0 = s exp y0+2 y0-2s ln(1 + y 0 ) x 0 = s+1 y 0 2 -s ln(1 + y 0 )
We now observe that y 0 = s -1 and x 0 = -2 ln s is a solution of the former system. Performing back the various changes of variables, this is equivalent to say that t0 = y0 x0 = 2(1-s) ln s so that t 0 = 4(1-s) (s+1) ln s +1 or equivalently t 0 = 1 (2µ-1) 2 + 2 (2µ-1) ln 1-µ µ . Consequently, the optimal proxy variance is given by:

σ 2 opt (µ) = 1 4 - 1 4 (2µ -1) 2 t 0 = (1 -2µ) 2 ln 1-µ µ
which is precisely the optimal proxy variance of a Bernoulli random variable with mean µ.

Optimal proxy variance for the Dirichlet distribution

We start by reminding the definition of sub-Gaussian property for random vectors:

Definition 2 (Sub-Gaussian vectors). A random d-dimensional vector X with finite mean µ

= E[X] is σ 2 -sub-Gaussian if the random variable u X is σ 2 -sub-Gaussian for any unit vector u in the simplex S d-1 = {u ∈ [0, 1] d / d i=1 u i = 1}
. This is equivalent to say that:

E[exp(λ (X -µ))] ≤ exp λ 2 σ 2 2 for all λ ∈ R d .
where λ 2 = d i=1 λ 2 i . Eventually, a random vector X is said to be strictly sub-Gaussian, if the random variables u X are strictly sub-Gaussian for any unit vector u ∈ S d-1 .

Let d ≥ 2. The Dirichlet distribution Dir(α), with positive parameters α = (α 1 , . . . , α d ) , is characterized by a density on the simplex S d-1 given by:

f (x 1 , . . . , x d ; α 1 , . . . , α d ) = 1 B(α) d i=1 x αi-1 i , where B(α) = 1 Γ( ᾱ) d i=1 Γ(α i ) and ᾱ = d i=1 α i .
It generalizes the Beta distribution in the sense that the components are Beta distributed. More precisely, for any non-empty and strict subset I of {1, . . . , d}:

X = (X 1 , . . . , X d ) ∼ Dir(α) =⇒ i∈I X i ∼ Beta   i∈I α i , j / ∈I α j   .
However, we remind the reader that the components (X i ) 1≤i≤d are not independent and the variance/covariance matrix is given by:

∀ i = j : Cov[X i , X j ] = - α i α j ᾱ2 (1 + ᾱ) and ∀ 1 ≤ i ≤ d , Var[X i ] = α i ( ᾱ -α i ) ᾱ2 (1 + ᾱ) .
Eventually, if we define n = (n 1 , . . . , n d ) ∈ N d , then the moments of the Dirichlet distribution are given by:

E d i=1 X ni i = B(α + n) B(α) .
This is equivalent to say that the moment-generating function of the Dirichlet distribution is:

E[exp(λ X)] = ∞ m=0 n1+•••+n d =m B(α + n) B(α) λ n1 1 . . . λ n d d (n 1 )! . . . (n d )! , = ∞ m=0 n1+•••+n d =m λ n1 1 . . . λ n d d (n 1 )! . . . (n d )! Γ( ᾱ) Γ( ᾱ + n) d i=1 Γ(α i + n i ) Γ(α i ) , = ∞ m=0 n1+•••+n d =m λ n1 1 . . . λ n d d (n 1 )! . . . (n d )! 1 ( ᾱ) n d i=1 (α i ) ni ,
where we have defined λ

= (λ 1 , . . . , λ d ) ∈ R d and n = d i=1 n i .
Let us define e i the i th canonical vector of R d and X = (X 1 , . . . , X d ) ∼ Dir(α). From Definition 1 and the results regarding the Beta(α, β) distribution obtained in Section 2.3, we immediately get that

e i X = X i is σ 2 i -sub-Gaussian with σ 2 i def = σ 2 opt (α i , ᾱ -α i ) defined from Theorem 1. Moreover, in direction e i , σ 2
i is the optimal proxy variance. Therefore, the remaining issue is to generalize these results for arbitrary unit vectors on S d-1 . We obtain the following result:

Theorem 3 (Optimal proxy variance for the Dirichlet distribution). For any parameter α, the Dirichlet distribution Dir(α) is sub-Gaussian with optimal proxy variance σ 2 opt (α) given from Theorem 1 and:

σ 2 opt (α) = σ 2 opt (α max , ᾱ -α max ) where α max = max 1≤i≤d {α i }.

Proof. We first observe that the computations of σ 2 opt (α i , β i = ᾱ -α i ) correspond to cases where the sum α i + β i is fixed to ᾱ and thus independent of i. Therefore, σ 2 opt (α i , ᾱ -α i ) is maximal when |( ᾱ -α i ) -α i | is minimal, i.e. when the distance from α i to 1 2 ᾱ is minimal. It is easy to see that this corresponds to choosing α i = α max = max{α i , 1 ≤ i ≤ d} (by looking at the two possible cases α max ≤ 1 2 ᾱ and α max > 1 2 ᾱ). We then observe that σ 2 max cannot be improved. Indeed, let us denote i 0 one of the components for which the maximum is obtained. Then, if we take u = e i0 , the discussion presented above shows that σ 2 i0 = σ 2 max is the optimal proxy variance in this direction. Hence the optimal proxy variance cannot be lower than σ 2 max . Let us now prove that X is σ 2 max -sub-Gaussian. Let u = (u 1 , . . . , u d ) be a unit vector on S d-1 and λ ∈ R. We define for clarity λ = λu. We have: Note that we also have: because both sides have the same number of terms in the product (i.e. n) but those of the right hand side are always greater or equal to those of the left hand side. Hence, from ( 24) and ( 25), we find:

E exp λu X = E exp λ X = E exp d i=1 λ i X i , = ∞ m=0
d i=1 E [exp(λ i X i )] = d i=1   ∞ j=0 (α i ) j (j!)( ᾱ) j λ j i   = ∞ m=0
E exp λ (X -µ) ≤ d i=1 E [exp(λ i (X i -µ i ))] .
Using the optimal proxy variance of the Beta distribution proven in Theorem 1, we find:

E exp λ (X -µ) ≤ d i=1 E [exp(λ i (X i -µ i ))] ≤ d i=1 exp λ 2 i σ 2 i 2 ≤ d i=1 exp λ 2 i σ 2 max 2 = exp σ 2 max λ 2 2 ,
thus showing that X is σ 2 max -sub-Gaussian and concluding the proof.

Note that using Theorem 3, we obtain the following corollary:

Corollary 2. For any integer d ≥ 2, the Dirichlet distribution Dir(α 1 , . . . , α d ) is strictly sub-Gaussian if and only if d = 2 and α 1 = α 2 .

Indeed, we first need to require α 1 = • • • = α d def = α so that all directions have the same optimal proxy variance. Then, each component satisfies X i = e i X ∼ Beta(α, (d -1)α) and Theorem 1 shows that σ 2 i is the optimal proxy variance for X i if and only if α = (d -1)α, i.e. if and only if d = 2.

Figure 1 :

 1 Figure 1: Left: curves of Var[Beta(α, β)] (green), σ 2 opt (α, β) (purple) and

  n1+•••+n d =m λ n1 1 . . . λ n d d (n 1 )! . . . (n d )! 1 ( ᾱ) n d i=1 (α i ) ni . (24)

  ni ≤ ( ᾱ) n,

Note indeed that Equation (1), together with Markov inequality, imply P(X -µ > ) ≤ e -2

2σ 2 .2 The implied predicate is actually stronger than sub-Gaussianity, but it is not useful for our purposes.

Acknowledgments

We wish to thank Stéphane Boucheron for an enlightening discussion of our results and an anonymous referee for insightful suggestions. O.M. would like to thank Université de Lyon, Université Jean Monnet and Institut Camille Jordan for financial support. This work was supported by the LABEX MI-LYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). J.A. would like to thank Inria Grenoble Rhône-Alpes and Laboratoire Jean Kuntzmann, Université Grenoble Alpes for financial support. J.A. is also member of Laboratoire de Statistique, CREST, Paris. This work was partially conducted during a scholar visit of J.A. at the Department of Statistics & Data Science of the University of Texas at Austin.