Thu-Le Tran

Clément Elvira

Hong-Phuong Dang

AND Cédric Herzet

ONE TO BEAT THEM ALL: "RYU" -A UNIFYING FRAMEWORK FOR THE CONSTRUCTION OF SAFE BALLS

Keywords: Convex optimization, screening rules, safe regions MSC codes. 68Q25, 68U05

In this paper, we put forth a novel framework (named "RYU") for the construction of "safe" balls, i.e., regions that provably contain the dual solution of a target optimization problem. We concentrate on the standard setup where the cost function is the sum of two terms: a closed, proper, convex Lipschitz-smooth function and a closed, proper, convex function. The RYU framework is shown to generalize or improve upon all the results proposed in the last decade for the considered family of optimization problems.

Introduction.

1.1. Context and state of the art. In this paper, we consider the following family of optimization problems: (P) find x ⋆ ∈ arg min

x∈R n P (x) f (Ax) + g(x)
where A ∈ R m×n is some known matrix, f : R m → R , g : R n → R ∪ {+∞} are proper, closed, convex functions and f is moreover "α -1 -Lipschitz smooth over R m ", that is f is differentiable everywhere on R m and its gradient obeys the following regularity condition for some positive scalar α > 0:

∀z, z ′ ∈ R m : ∇f (z) -∇f (z ′ ) 2 ≤ α -1 z -z ′ 2 . (1.1)
Under such hypotheses, problem (P) is well-posed in the sense that at least one minimizer x ⋆ exists (see Section 3). Instances of problems satisfying these hypotheses are common in the literature of machine learning, statistics or signal processing, and include (among many others) least-squares sparse regression [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF], logistic sparse regression [START_REF] Koh | An interior-point method for large-scale l1-regularized logistic regression[END_REF] or the "Elastic Net" problem [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF].

The focus of this paper is on the construction of "safe regions", i.e., subsets of R m provably containing the unique solution of the dual problem of (P). More specifically, our goal is to identify some subset S ⊆ R m such that

u ⋆ ∈ S (1.2) where (D) u ⋆ = arg max u∈R m D(u) -f * (-u) -g * (A T u)
and f * , g * denote the convex conjugates of f , g. The construction of safe regions has become an active field of research during the last decade (see e.g., [START_REF] Dai | An ellipsoid based, two-stage screening test for BPDN[END_REF][START_REF] Fercoq | Mind the duality gap: safer rules for the Lasso[END_REF][START_REF] Herzet | Region-free safe screening tests for ℓ 1 -penalized convex problems[END_REF][START_REF] Herzet | Safe screening tests for Lasso based on firmly non-expansiveness[END_REF][START_REF] Ndiaye | Gap safe screening rules for sparsity enforcing penalties[END_REF][START_REF] Pan | A safe feature elimination rule for l1-regularized logistic regression[END_REF][START_REF] Tran | Beyond gap screening for lasso by exploiting new dual cutting half-spaces[END_REF][START_REF] Wang | Sequential safe feature elimination rule for l1-regularized regression with kullback-leibler divergence[END_REF][START_REF] Wang | Lasso screening rules via dual polytope projection[END_REF][START_REF] Wang | A safe screening rule for sparse logistic regression[END_REF][START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF]) and has been triggered by the so-called "safe feature elimination" technique (also referred to as "safe screening"), a procedure to accelerate the resolution of (P), first proposed in [START_REF] Ghaoui | Safe feature elimination for the Lasso and sparse supervised learning problems[END_REF] and further extended in many contributions, see e.g., [START_REF] Elvira | Safe squeezing for antisparse coding[END_REF][START_REF] Elvira | Short and squeezed: accelerating the computation of antisparse representations with safe squeezing[END_REF][START_REF] Guyard | Screen & relax: accelerating the resolution of Elastic-Net by safe identification of the solution support[END_REF][START_REF] Ndiaye | Screening Rules and its Complexity for Active Set Identification[END_REF]. One central element in the effectiveness of these acceleration methods is the identification (preferably at low computational cost) of small safe regions with some specific geometry (e.g., ball, ellipsoid, dome, etc). In this paper, we focus on safe regions having a "ball" geometry, that is

(1.3) S = B(c, r) {u ∈ R m | u -c 2 ≤ r}
for some c ∈ R m and r > 0. In this respect, the state-of-the-art safe ball for the general family of optimization problems considered in this paper is indubitably the so-called "GAP ball" proposed in [START_REF] Fercoq | Mind the duality gap: safer rules for the Lasso[END_REF][START_REF] Ndiaye | Gap safe screening rules for sparsity enforcing penalties[END_REF]. It is defined for any couple (x, u) ∈ dom(P )× dom(-D) as

B GAP (x, u) B(c GAP , r GAP ) (1.4)
where

c GAP u (1.5a) r GAP 2GAP(x, u) α (1.5b)
and GAP(x, u) P (x) -D(u) is the so-called duality gap. The popularity of the GAP ball is due to the two following assets:

1. The construction of the ball is valid for any problem satisfying our minimal hypotheses, that is f , g are proper, closed, convex and f is α -1 -Lipschitz smooth over R m . 2. A GAP ball can be constructed from any primal-dual feasible couple (x, u).

In particular, under strong duality assumption and continuity of the duality gap over its domain, the radius of the ball can be made arbitrarily small by choosing (x, u) sufficiently close to some optimal couple (x ⋆ , u ⋆ ).

These features have led the GAP ball to be widely applied and to allow for substantial acceleration performance in many setups, see e.g., [START_REF] Dantas | Stable safe screening and structured dictionaries for faster ℓ 1 regularization[END_REF][START_REF] Elvira | Safe squeezing for antisparse coding[END_REF][START_REF] Elvira | Safe rules for the identification of zeros in the solutions of the slope problem[END_REF][START_REF] Guyard | Screen & relax: accelerating the resolution of Elastic-Net by safe identification of the solution support[END_REF][START_REF] Herzet | Gather and conquer: Region-based strategies to accelerate safe screening tests[END_REF].

Other constructions of safe balls, requiring either additional hypotheses on f and g or the knowledge of some specific primal-dual couple (x, u), have also been proposed in the literature. All the safe ball constructions (to the best of our knowledge) falling into the optimization framework considered in this paper are gathered in Table 1 and will be reviewed in greater details in Section 4. We note that, although requiring additional assumptions, some of these works put to the forth that the construction of safe balls smaller than the GAP ball is possible. For example in [START_REF] Herzet | Safe screening tests for Lasso based on firmly non-expansiveness[END_REF] the authors proposed a safe ball (referred to as "FNE ball") for the LASSO problem and proved that it is a subset of the GAP ball. More recently, the authors of [START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF] introduced the so-called "dynamic EDPP ball" and emphasized that the latter has a smaller radius then the GAP ball constructed with the same primal-dual pair [START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF]Theorem 10].

In this paper, we provide a new mathematical framework gathering and extending these results to the general family of optimization problems (P).

1.2. Contributions. The contribution of this paper is two-fold. We first introduce a new safe-ball referred to as "RYU ball" and defined ∀(x, u) ∈ dom(P ) × dom(-D) as

(1.6) B RYU (x, u) B(c RYU , r RYU )
Safe region Relation Cstr. on (x, u) Hyp. on f and g GAP [START_REF] Fercoq | Mind the duality gap: safer rules for the Lasso[END_REF][START_REF] Ndiaye | Gap safe screening rules for sparsity enforcing penalties[END_REF][START_REF] Ndiaye | Screening Rules and its Complexity for Active Set Identification[END_REF] ⊇ BRYU(x, u) feasible (H1)-(H2) x-GAP [START_REF] Herzet | Region-free safe screening tests for ℓ 1 -penalized convex problems[END_REF] ⊇ BRYU(x, u) feasible (H1)-(H2), g = λ • 1 Dyn. EDPP [START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF] The first column provides the name and the references associated to the safe ball, the second describes its connection with the proposed RYU ball, the third indicates the constraints on the primal-dual couple (x, u) used in the construction. The last column specifies the setup considered by the authors in their work.

= BRYU(t ⋆ x, u) feasible f = 1 2 y -• 2 2 , g = gauge FNE [15] = BRYU(x, u) A T u ∈ ∂g(x) f = 1 2 y -• 2 2 , g = λ • 1 SASVI [17] = BRYU(0n, u) u ∈ dom(-D) f = 1 2 y -• 2 2 , g = λ • 1 EDPP [23] = BRYU(x, u) (4.7)-(4.8) f = 1 2 y -• 2 2 , g = λ • 1 DPP [23] ⊇ BRYU(x, u) (4.7)-(4.8) f = 1 2 y -• 2 2 , g = λ • 1 SAFE [7] ⊇ BRYU(0n, u) u ∈ dom(-D) f = 1 2 y -• 2 2 , g = λ • 1 SLORES [24] ⊇ BRYU(x, u) (4.7)-(4.8) f = logistic, g = λ • 1 SFER [20] = BRYU(x, u) (4.7)-(4.8) f = logistic, g = λ • 1
where

c RYU 1 2 (u -∇f (Ax)) (1.7a) r RYU GAP(x, u) α - u + ∇f (Ax) 2 2 4 , (1.7b)
see Theorem 4.1. The name "RYU" stems from "Refined Fenchel-Young inequality" and refers to the fact that the safeness of the ball is a consequence of the (double) application of a refined version of the well-known Fenchel-Young inequality (see Appendix B). Our ball construction is valid under the same generic assumptions as the GAP ball (see Section 3 for a detailed discussion about our working hypotheses). In particular: 1. it can be applied to any problem (P) involving a proper, closed, convex function f which is α -1 -Lipschitz smooth over R m , and a proper, closed, convex function g; 2. primal-dual feasibility is the only assumption required for the pair (x, u).

Second, we show that our safe ball construction generalizes or improves over all the existing results of the literature. More specifically, we prove that all the existing safe balls correspond to particular cases or supersets of the proposed ball. These results are summarized in the second column of Table 1 and correspond to Propositions 4.4 to 4.9 of the paper. Interestingly, we note that the GAP ball is always included in the RYU ball, that is:

(1.8) B RYU (x, u) ⊆ B GAP (x, u),
where the inclusion is shown to be strict as long as (x, u) is not a primal-dual optimal couple, see Proposition 4.4. In fact, a rapid inspection of (1.5b) and (1.7b) shows that the squared radius of the RYU ball is never greater than half the squared radius of the GAP ball. Since our construction is valid for any feasible 1 couple (x, u) and holds under minimal assumptions on f , g, the results in Table 1 therefore emphasize that the 1 Strictly speaking, the proposed construction in fact applies to any couple (x, u) ∈ R n × R m but (similarly to the GAP ball) leads to a ball with infinite radius when (x, u) / ∈ dom(P ) × dom(-D). This is the reason why we restrict our construction to dom(P ) × dom(-D) in the paper.

proposed framework unifies and generalizes all the methodologies previously proposed in the literature.

1.3. Paper organization. The rest of the paper is organized as follows. In Section 2, we detail the notational conventions used in the paper. In Section 3, we describe the working hypotheses considered in our derivations and discuss some of their implications. Section 4 is dedicated to the presentation of our new safe ball and its connection with the previous results of the literature. Most of the technical details are deferred to Appendices A to C.

Notations.

Unless mentioned explicitly, we will use the following notational conventions throughout the paper. Vectors are denoted by lowercase bold letters (e.g., x, z) and matrices by uppercase bold letters (e.g., A). We use the symbol " T " to denote the transpose of a vector or a matrix. The "all-zero" vector of dimension n is written 0 n . z | z ′ denotes the standard inner product between z and z ′ . We use the notation x j to refer to the jth entry of a vector x. For matrices, we use a j to denote the jth column of A. We let R = R ∪ {-∞, +∞} where R refers to the set of real numbers. Given an extended real-valued function h : R d → R , we let

(2.1) dom(h) z ∈ R d | h(z) < +∞ .
The subdifferential set of h : R d → R is defined as

(2.2) ∂h(z) g ∈ R d | ∀z ′ ∈ R d : h(z ′ ) ≥ h(z) + g | z ′ -z
and we refer to its elements as "subgradients", see [2, Definition 3.2]. Finally, (2.3)

h * : R d → R z * → sup z∈R d z * , z -h(z) ,
is the convex conjugate of h, see [2, Definition 4.1].

3. Optimization framework. In this paper, we consider problem (P) with the following minimal assumptions: f and g are proper, closed and convex functions. (H1)

f is α -1 -Lipschitz smooth over R m . (H2)
We note that (H1) and (H2) correspond to the general hypotheses needed for the construction of the GAP ball, see [START_REF] Ndiaye | Screening Rules and its Complexity for Active Set Identification[END_REF]. In the rest of this section we elaborate on some properties of problems (P)-(D) induced by these hypotheses.

First, since f (resp. g) is proper, closed and convex from (H1), its convex conjugate f * (resp. g * ) is proper, closed and convex, see [2, Theorems 4.13 and 4.5]. Moreover, the convexity and α -1 -Lipschitz smoothness on R m of f in (H2) implies that f * is α-strongly convex, see [START_REF] Beck | First-Order Methods in Optimization[END_REF]Theorem 5.26], that is:

(3.1) ∀z, z ′ ∈ dom(f * ) and g ∈ ∂f * (z) : f * (z ′ ) ≥ f * (z) + g | z ′ -z + α 2 z ′ -z 2 2 .
Under the properness assumption in (H1), the duality gap, defined as

(3.2) GAP : R n × R m → R (x, u) → P (x) -D(u)
is always a nonnegative quantity, see [1, Item (i) of Proposition 15.21]. Moreover, GAP(x, u) < +∞ if and only if (x, u) ∈ dom(P ) × dom(-D). Hypotheses (H1)-(H2) also imply that strong duality holds for some primal-dual couple as emphasized by the following lemma:

Lemma 3.1. If (H1)-(H2) hold, then there exists (x ⋆ , u ⋆ ) ∈ R n × R m such that (3.3) GAP(x ⋆ , u ⋆ ) = 0.
Proof. If (H1) is verified, we have from [2, Theorem 4.15] that strong duality holds and a primal-dual optimal couple exists provided that

(3.4) relint(dom( f )) ∩ relint(dom(g)) = ∅,
where f (•) f (A•) and relint(•) denotes the relative interior of a set. Now, under our α -1 -Lipschitz smoothness assumption (H2), we have that dom( f ) = R m so that condition (3.4) reduces to relint(dom(g)) = ∅. Since g is a proper convex function, its domain dom(g) is non-empty (by definition) and convex [2, Section 2.3.1]. Therefore, the relative interior of dom(g) is non-empty, see [2, Theorem 3.17].

We note that any couple verifying (3.3) must obviously be a primal-dual solution of (P)-(D), so that at least one minimizer of (P) and one maximizer of (D) exist. Moreover, the α-strong convexity of f * implies that the maximizer of (D) is unique, see [START_REF] Beck | First-Order Methods in Optimization[END_REF]Theorem 5.25]. Finally, since strong duality holds, the following conditions must be satisfied by any primal-dual optimal couple (x ⋆ , u ⋆ ), see [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF]Theorem 19.1]:

u ⋆ = -∇f (Ax ⋆ ) (3.5) A T u ⋆ ∈ ∂g(x ⋆ ). (3.6)
4. The RYU framework and its connection to the state of the art. The main theoretical result of this paper is the following new safe ball: Theorem 4.1 (RYU ball). Assume (H1)-(H2) hold true. Then, we have for any (x, u) ∈ dom(P ) × dom(-D):

(4.1) u ⋆ ∈ B RYU (x, u) B(c RYU , r RYU )
where

c RYU 1 2 (u -∇f (Ax)) (4.2a) r RYU GAP(x, u) α - u + ∇f (Ax) 2 2 4 . (4.2b)
The name "RYU" stands for "Refined Fenchel-Young inequality", a central element appearing in the proof of the safeness of the proposed region, see Appendix B.1.

A close inspection of the hypotheses of Theorem 4.1 reveals that the construction of the RYU ball is applicable under exactly the same assumptions as the GAP ball, that is: i) it holds for any problem satisfying minimal hypotheses (H1)-(H2) on f and g; ii) it is valid for any primal-dual feasible couple (x, u). Despite of its generality, a careful examination of the definition of the radius of the GAP and RYU balls in (1.5b) and (1.7b) indicates that -given a feasible primal-dual pair (x, u)-the squared radius of the RYU ball is always at least twice as small as the squared radius of the GAP ball. In fact, as emphasized in Subsection 4.1 below, the GAP ball is a strict subset of the proposed RYU ball for any feasible primal-dual (x, u) different from (x ⋆ , u ⋆ ).

In the rest of this section, we explain how the safe balls previously-proposed in the literature relate to the RYU ball. More specifically, we emphasize that the previous results of the state of the art can be seen as either particular cases or supersets of the proposed ball. Our results are contained in Propositions 4.4 to 4.9 and summarized in the second column of Table 1. In particular, the safe balls of the literature presented hereafter rely on different assumptions on: i) the functions f and g defining (P); ii) the primal-dual couple (x, u) used in the construction. In the sequel, the working hypotheses made for the construction of each safe ball are recalled before stating its connection with the proposed RYU ball. These hypotheses are summarized in the third and fourth columns of Table 1 for each region.

Before proceedings to the connection between the RYU ball and the existing results of the literature, we make two important remarks regarding the choice the couple (x, u) involved in the construction of the safe ball.

First, an approach which have been considered (often implicitly) in many contributions of the literature consists in choosing a feasible pair (x, u) such that

(4.3) A T u ∈ ∂g(x).
Interestingly, when (4.

If (4.3) is verified, then ∀(x, u) ∈ dom(P ) × dom(-D): (4.6) GAP(x, u) = Fen(x, u) = Breg(x, u).
We refer the reader to Appendix C.1 for a proof of this result. The connection between the duality gap and the Fenchel/Bregman divergences is of interest in two respects.

On the one hand, these divergences are sometimes more straightforward to express than the duality gap and thus give an alternative formulation to the proposed RYU ball under the particular assumption (4.3). On the other hand, some contributions of the literature (see Subsection 4.5) have directly expressed their safe ball as a function of Breg(x, u). The connection established in Lemma 4.2 will thus allow us to make a direct link between these works and the RYU framework proposed in this paper. Second, we mention that the following definition of primal-dual couple (x, u) has been considered in many contributions of the literature (see e.g., [START_REF] Liu | Safe screening with variational inequalities and its application to lasso[END_REF][START_REF] Pan | A safe feature elimination rule for l1-regularized logistic regression[END_REF][START_REF] Wang | Lasso screening rules via dual polytope projection[END_REF][START_REF] Wang | A safe screening rule for sparse logistic regression[END_REF]):

(x, u) x ⋆ γ , -γ -1 ∇f (Ax ⋆ γ ) (4.7)
where γ > 0 and

x ⋆ γ ∈ arg min x∈R n f (Ax) + γg(x). (4.8)
This type of construction appears for example in "sequential" settings where one wants to solve (P) for g(•) = λ • and the solution of a similar problem with g(•) = λ 0 • has already been computed previously. The solution of the latter problem can then be expressed as in Proof. Using the optimality condition (3.5) and the definition of u in (4.7), we see that γu corresponds to the optimal solution of the dual problem of (4.8). Using optimality condition (3.6), we then have that

γA T u ∈ ∂γg(x ⋆ γ ) = γ∂g(x) (4.9)
where we used the homogeneity of subdifferentials (see [START_REF] Beck | First-Order Methods in Optimization[END_REF]Theorem 3.35]) and the definition of x in (4.7). We finally obtain the result by simplifying both sides by γ.

Since many safe balls proposed in the literature rely on the particular construction (4.7)-(4.8), the result in Lemma 4.3 emphasizes that these constructions in fact consider a primal-dual feasible pair (x, u) verifying (4.3) and that (from Lemma 4.2) the connection (4.6) between the duality gap and the Fenchel/Bregman divergences is thus in force.

GAP balls.

The GAP ball first proposed in [START_REF] Fercoq | Mind the duality gap: safer rules for the Lasso[END_REF] and later generalized in [START_REF] Ndiaye | Gap safe screening rules for sparsity enforcing penalties[END_REF][START_REF] Ndiaye | Screening Rules and its Complexity for Active Set Identification[END_REF] is defined in (1.5a)-(1.5b). Its construction is valid under assumptions (H1)-(H2) and can take any primal-dual (x, u) as input. The next result shows that the RYU ball is always a subset of the GAP ball: Proposition 4.4 (The GAP ball contains the RYU ball). Assume (H1)-(H2) hold. Then, for any primal-dual pair (x, u) ∈ dom(P ) × dom(-D):

(4.10) B RYU (x, u) ⊆ B(c GAP , r GAP ).
Moreover, the inclusion is strict as soon as the primal-dual pair (x, u) is not optimal.

In [14, Section IV.B] a variant of the GAP ball for the specific case where g = λ • 1 was proposed. Although the construction of the ball presented in [START_REF] Herzet | Region-free safe screening tests for ℓ 1 -penalized convex problems[END_REF] holds in a slightly more general setup, 3 we focus hereafter on the case where f is proper, closed, convex and satisfies (H2). The center and radius of this ball (referred to as x-GAP since its center depends on x instead of u as in the standard GAP ball) reads as follows:

c x-GAP -∇f (Ax) (4.11a) r x-GAP 2GAP(x, u) α (4.11b)
where (x, u) can be any primal-dual feasible couple. Similarly to Proposition 4.4, the next result shows that the x-GAP ball is also a superset of the proposed RYU region: Proposition 4.5 (The x-GAP ball contains the RYU ball). Assume (H1)-(H2) hold. Then, for any primal-dual pair (x, u) ∈ dom(P ) × dom(-D):

(4.12) B RYU (x, u) ⊆ B(c x-GAP , r x-GAP ).
Moreover, the inclusion is strict as soon as the primal-dual pair (x, u) is not optimal.

The proof of Propositions 4.4 and 4.5 is given in Appendix C.2.

4.2. Dynamic EDDP ball. In [START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF], the authors focused on the particular family of problems where

f = 1 2 y -• 2 2 (4.13) g = λ • (4.14)
for some vector y ∈ R m , scalar λ > 0 and norm • . 4 They introduced a new safe ball (see [START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF]Theorem 9]), dubbed "dynamic EDPP ball" and valid for any primaldual feasible couple (x, u). The center and radius of the dynamic EDPP ball reads as follows:

c dyn.EDPP = 1 2 (y + u -t ⋆ Ax) (4.15a) r dyn.EDPP = 1 2 y -u 2 2 -t ⋆ Ax 2 2 (4.15b)
where t ⋆ is defined as (with the conventions 0/0 = 0 and 1/0 = +∞):

(4.16) t ⋆ = max 0, Ax | y + u -2λ x Ax 2 2 .
The connection between the RYU and dynamic EDPP balls is established in the following proposition:

Proposition 4.6 (Dynamic EDPP ball is a special case of the RYU ball). Assume (4.13)-(4.14) holds. Then, for any (x, u) ∈ dom(P ) × dom(-D):

i) The quantity t ⋆ defined in (4.16) verifies

t ⋆ ∈ arg min t≥0 GAP(tx, u) - 1 4 u + ∇f (tAx) 2 2 .
ii) We have

B(c dyn.EDPP , r dyn.EDPP ) = B RYU (t ⋆ x, u).
A proof of this result is available in Appendix C.3. In particular, the dynamic EDPP ball was shown to exhibit a smaller radius as compared to the GAP ball constructed with the (feasible) primal-dual pair (x, u), see [START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF]Theorem 10]. Item ii) of Proposition 4.6 (combined with Proposition 4.4) elucidates this connection by showing that the dynamic EDPP ball is in fact a subset of the GAP ball B GAP (t ⋆ x, u). Finally, item i) of Proposition 4.6 provides a novel interpretation of the definition of t ⋆ , that is t ⋆ corresponds to a nonnegative rescaling of the primal vector x minimizing the radius of the RYU ball.

4.3. FNE, EDDP, DPP and SASVI balls. FNE [START_REF] Herzet | Safe screening tests for Lasso based on firmly non-expansiveness[END_REF], (E)DDP [START_REF] Wang | Lasso screening rules via dual polytope projection[END_REF] and SASVI [START_REF] Liu | Safe screening with variational inequalities and its application to lasso[END_REF] balls are safe regions designed for the same problem where

f = 1 2 y -• 2 2
(4.17)

g = λ • 1 . (4.18)
They all assume (explicitly or implicitly) that the primal-dual couple (x, u) used in the construction verifies (4.3). We start with the description of the FNE ball which corresponds to the most general construction. We address the EDDP, DPP and SASVI balls at the end of the section as particular cases or relaxation of the FNE region.

The FNE ball is defined by the following center and radius:

c FNE = u + 1 2 (y -Ax -u) (4.19a) r FNE = 1 2 y -Ax -u 2 . (4.19b)
In [15, Theorem 1], the authors showed that the FNE ball is safe for any primal-dual feasible pair (x, u) satisfying

(4.20) u | Ax = λ x 1 .
The following result shows that the FNE ball in fact corresponds to a particular case of the RYU ball when (4.17 A proof of this result is available in Appendix C.4. We note that the authors also showed in [15, Lemma 1] that the FNE ball is a strict subset of the GAP ball as long as (x, u) = (x ⋆ , u ⋆ ). Interestingly, in view of Proposition 4.7, this result turns out to be a particular case of Proposition 4.4 in the more general framework of the RYU ball.

The EDPP and SASVI balls represent specific instances of the FNE ball, resulting from particular choices of the pair (x, u). On the one hand, the SASVI ball (see [START_REF] Liu | Safe screening with variational inequalities and its application to lasso[END_REF]Section 2.2]) corresponds to the case where (x, u) = (0 n , u) for some dual feasible point u, i.e.,

c SASVI = 1 2 (y + u) (4.22a) r SASVI = 1 2 y -u 2 . (4.22b)
It is easy to see this couple trivially verifies (4.20). Using Proposition 4.7 with (x, u) = (0 n , u), u ∈ dom(-D), then directly leads to

B(c SASVI , r SASVI ) = B RYU (0 n , u).
On the other hand, the center and radius of the EDDP ball (see [START_REF] Wang | Lasso screening rules via dual polytope projection[END_REF]Theorem 13]) obeys the same definition (4.19a)-(4.19b) as those of the FNE ball but for (x, u) defined as in (4.7)-(4.8) for some γ > 0. Taking into account that

∂ x 1 = {z ∈ R n | z | x = x 1 , z ∞ ≤ 1}, (4.23)
it is easy to see that (4.20) together with feasibility of (x, u) is in fact an equivalent rewriting of (4.3) for g = λ • 1 . Hence, in view of Lemma 4.3, the couple (x, u) considered in the EDDP construction verifies (4.20) and this ball is nothing but a particular instance of FNE ball. Proposition 4.7 thus applies for the EDDP ball as well.

Finally, it was shown in [START_REF] Wang | Lasso screening rules via dual polytope projection[END_REF]Theorem 13] that the DPP ball is always a superset of the EDPP ball. This directly leads to the inclusion reported in Table 1.

SAFE ball.

The SAFE ball is the first safe region proposed in the seminal paper [START_REF] Ghaoui | Safe feature elimination for the Lasso and sparse supervised learning problems[END_REF]. It applies in the case where

f = 1 2 y -• 2 2 (4.24) g = λ • 1 . (4.
r SFER = 1 4 Breg(x, u) - 1 4 (1 -γ)u 2 2 . (4.32b)
In Appendix C.6, we show that SFER is in fact a particular instance of the RYU ball whereas the SLORE ball corresponds to a superset of the latter. More specifically, we prove that the following result holds: Proposition 4.9. Assume (4.29)-(4.30) hold and (x, u) is defined as (4.7)-(4.8) for some γ > 0. Then, we have

(4.33) B(c SFER , r SFER ) = B RYU (x, u) ⊆ B(c SLORES , r SLORES ).

Conclusion.

In this paper we introduced a new framework to devise safe balls. Our construction applies to any instance of a general family of optimization problems where the cost function is the sum of a closed, proper, convex, Lipschitzsmooth function and a closed, proper, convex function, and leverages solely the knowledge of a primal-dual feasible pair. Our new safe ball is shown to encompass as particular cases or outperform all the previous constructions proposed in the literature during the last decade, thus acting as a unifying framework for the construction of safe balls for the general family of problems considered in the paper. Proposition B.1 can be proved by applying Lemma B.2 for two different choices of couple (x, u). We remind the reader that, from Lemma 3.1, at least one primal-dual couple (x ⋆ , u ⋆ ) exists. A first application of Lemma B.2 with (x ⋆ , u) then leads to

(B.3) u + ∇f (Ax ⋆ ) 2 2 ≤ 2(P (x ⋆ ) -D(u)) α .
Since Lemma 3.1 also ensures that strong duality holds, we have P (x ⋆ ) = D(u ⋆ ) and u ⋆ = -∇f (Ax ⋆ ). Therefore, the previous inequality can also be rewritten as

(B.4) u -u ⋆ 2 2 ≤ 2(D(u ⋆ ) -D(u)) α .
A second application of Lemma B.2 with the pair (x, u ⋆ ) yields

(B.5) u ⋆ + ∇f (Ax) 2 2 ≤ 2(P (x) -D(u ⋆ )) α .
Summing up (B.4) and (B.5) leads to the desired result (B.1).

B.2. Proof of Lemma B.2. If (x, u) / ∈ dom(P ) × dom(-D), then (B.1) is trivially satisfied since the left-hand side is finite whereas the right-hand side is equal to +∞. In the rest of the proof, we thus assume that (x, u) ∈ dom(P ) × dom(-D).

As discussed in Section 3, hypotheses (H1)-(H2) imply that f * is proper, closed and α-strongly convex. Applying the refined Fenchel-Young inequality (A.3) with h = f * , z = -u and z * = Ax then leads to

(B.6) f * (-u) + f * * (Ax) ≥ -u | Ax + α 2 u + ∇f * * (Ax) 2 2 .
Using [2, Theorem 4.8], we have that f * * = f since f is proper, closed and convex by (H1). Therefore, (B.6) can be equivalently rewritten as:

(B.7) f (Ax) + f * (-u) ≥ -u | Ax + α 2 u + ∇f (Ax) 2 2 .
In order to conclude the proof, we need to add g(x) + g * (A T u) to both sides of this inequality. Prior to this operation, we have nevertheless to ensure that g(x) + g * (A T u) < +∞. To that end, we notice that f (x) > -∞ and f * (-u) > -∞ since f and f * are proper. Hence,

P (x) < +∞ =⇒ g(x) < +∞ (B.8) -D(u) < +∞ =⇒ g * (A T u) < +∞. (B.9)
Since we assume that (x, u) ∈ dom(P ) × dom(-D), the left-hand sides of these implications are satisfied, so that g(x) + g * (A T u) < +∞.

Adding g(x) + g * (A T u) to both sides of (B.7) then leads to (B.10)

P (x) -D(u) ≥ g(x) + g * (A T u) -u | Ax + α 2 u + ∇f (Ax) 2 2 .
Finally, since g is proper, closed and convex from (H1), we can apply the Fenchel-Young inequality (A.2) with h = g, z = x and z * = A T u to obtain (B.2).

Appendix C. Proofs of the connections with existing results. This appendix gathers all proofs related to the comparison of the proposed RYU ball with state-of-the-art safe regions presented in Subsection 4.1 to Subsection 4.5.

C.1. Proof of Lemma 4.2. Let (x, u) ∈ dom(P ) × dom(-D). Using the definition of the primal and dual cost functions, we obtain:

GAP(x, u) = f (Ax) + g(x) + f * (-u) + g * (A T u) = f (Ax) + f * (-u) + Ax | u + g(x) + g * (A T u) -Ax | u = Fen(x, u) + g(x) + g * (A T u) -Ax | u . If (4.
3) holds, we then have from Lemma A.2 that

g(x) + g * (A T u) -Ax | u = 0.
This shows the first equality in (4.6).

The second inequality can be obtained by noticing that (from Lemma A.2)

f (Ax) + f * (∇f (Ax)) = ∇f (Ax) | Ax
since f is convex, proper and differentiable and ∂f (Ax) = {∇f (Ax)}. Hence,

Fen(x, u) = f (Ax) + f * (-u) + Ax | u = f * (-u) -f * (∇f (Ax)) + Ax | u + ∇f (Ax) = Breg(x, u).
B(c RYU , r RYU ) = u ′ ∈ R m | u ′ -u 2 2 + u ′ + ∇f (Ax) 2 2 ≤ 2GAP(x, u) α , (C.1)
whereas the definitions of the GAP and x-GAP balls lead to: Finally, to prove strict inclusion it is then sufficient to note that r RYU ≤ GAP(x,u) α < r GAP r RYU ≤ GAP(x,u) α < r x-GAP whenever GAP(x, u) = 0.

B(c GAP , r GAP ) = u ′ ∈ R m | u ′ -u 2 2 ≤ 2GAP(x, u) α (C.2) B(c x-GAP , r x-GAP ) = u ′ ∈ R m | u ′ + ∇f (Ax)
C.3. Comparison with dynamic EDDP ball: Proof of Proposition 4.6. We first note that the RYU ball in Theorem 4.1 is well-defined. Indeed, functions f and g in (4.17)-(4.18) are closed, proper and convex, so that (H1) holds. Moreover, (H2) is verified with α = 1. For any feasible (tx, u) with t ≥ 0, we thus have by definition: It is straightforward from the definition of f and g in (4.17)-(4.18) that (H1)-(H2) are satisfied with α = 1. The RYU ball in Theorem 4.1 is therefore well-defined. We next show that the center and radius of the RYU and FNE balls coincide.

c RYU (t) =
First, using the definition of f in (4.17 Second, using the definition of g in (4.18) and (4.23), it can be seen that condition " u | Ax = λ x 1 " in (4.20) together with feasibility of (x, u) is equivalent to "A T u = ∂g(x)" in (4.3) so that Lemma 4.2 applies. In particular, we have:

GAP(x, u) = Fen(x, u) = f (Ax) + f * (-u) + u | Ax . Since f * (u) = 1 2 u 2 2 + u | y ,
the duality gap can thus also be written as

GAP(x, u) = 1 2 y -Ax 2 2 + 1 2 u 2 2 -u | y + u | Ax = 1 2 y -Ax -u 2 2 . (C.17)
Going back to the definition of the radius of the RYU ball (with α = 1), we finally obtain:

r 2 RYU = GAP(x, u) -1 4 u + ∇f (Ax) 2 2 = 1 2 y -Ax -u 2 2 -1 4 y -Ax -u 2 2 = r 2 FNE .
where we have used (C. [START_REF] Koh | An interior-point method for large-scale l1-regularized logistic regression[END_REF]) and (C.17) in the second equality.

3 )Lemma 4 . 2 .

 342 is satisfied the function GAP can be related to two other wellknown quantities, namely the Fenchel divergence of f (see [3, Definition 2]) and the Bregman divergence of f * (-•) (see [2, Definition 9.2]). In particular, the following lemma holds: Assume (H1)-(H2) hold and let 2 Fen(x, u) f (Ax) + f * (-u) + u | Ax (4.4) Breg(x, u) f * (-u)f * (∇f (Ax)) + Ax | u + ∇f (Ax) .(4.5)

  (4.8) with γ = λ0 λ and g(•) = λ • . The next lemma emphasizes that (4.7)-(4.8) correspond in fact to a particular strategy to build primal-dual couples obeying (4.3): Lemma 4.3. If (x, u) is defined as in (4.7)-(4.8), it verifies (4.3).

  )-(4.18) and (4.20) hold: Proposition 4.7 (FNE ball is a special case of the RYU ball). Assume (4.17)-(4.18) holds. Then, for any (x, u) ∈ dom(P ) × dom(-D) satisfying (4.20): (4.21) B(c FNE , r FNE ) = B RYU (x, u).

25 )

 25 Its center and radius are defined ∀u ∈ dom(-D) as c SAFE = y (4.26) r SAFE = yu 2 . (4.27) In Appendix C.5, we show that the SAFE ball is a relaxation of the proposed RYU ball for (x, u) = (0 n , u) with u ∈ dom(-D). More specifically, we prove that the following result holds: Proposition 4.8. Assume (4.24)-(4.25) holds. Then, for any u ∈ dom(-D):(4.28) B(c SAFE , r SAFE ) ⊇ B RYU (0 n , u).4.5. SLORE and SFER balls. We end up this section by considering the SLORES and SFER balls respectively proposed in [24, Theorem 2] and [20, Corollary 1]. The focus of these papers is on problem (P) with the following definitions for f and g:f (z) = m i=1 log(1 + e -zi ), (4.29) g(z) = λ z 1 . (4.30)The construction of these balls is moreover based on the knowledge of a primal-dual couple (x, u) verifying (4.7)-(4.8) for some γ > 0.The expression of the center and radius of the SLORE and SFER balls respectively read as c SLORES = γu (4.31a)

C. 2 .

 2 Comparison with GAP ball: Proof of Proposition 4.4. Using Proposition B.1, we have

3 )

 3 Since the membership conditions in (C.2) and (C.3) are relaxations of the inequality defining the RYU ball in (C.1), inclusions (4.10) and (4.12) necessarily hold.

  ), we have∇f (Ax) = -(y -Ax). (C.16) Hence, c RYU = 1 2 (u -∇f (Ax)) = u + 1 2 (y -Axu) = c FNE .

Table 1

 1 Summary of the main safe-ball constructions proposed in the literature during the last decade.

  On the one hand, since item i) of Proposition 4.6 is true, we directly have from the expression of c RYU (t) in (C.4) that (C.13) holds. On the other hand, (C.14) can be shown by examining the following two cases.If t ⋆ = 0, the equality in (C.14) follows directly from the definition (C.6). Ift ⋆ > 0, we have (r 2 RYU (t ⋆ )) ′ = 0, i.e., RYU (t ⋆ ) = 1 2 t ⋆ Ax | yu -1 2 t ⋆ Ax 2 2 + 1 4 t ⋆ Axy + u 2

	(C.15)	λ x = 1 2 Ax | u + y -t ⋆ 2 Ax 2 2 .
	Plugging this equality into (C.6) then leads to
	r 2	2
		= 1 4 y -u 2 2 -1 4 t ⋆ Ax 2 2
		= r dyn.EDPP .

1 2 (uy + tAx) (C.4) r 2 RYU (t) = GAP(tx, u) -1 4 uy + tAx 2 2 . (C.

5) C.4. Proof of Proposition 4.7.

Fen(x, u) corresponds to the Fenchel divergence of f evaluated at (Ax, -u); Breg(x, u) is the Bregman divergence of f * (-•) evaluated at (u, -∇f (Ax)).

In particular, a weaker version of (H2) is considered and up to a modification of (4.11b).

More precisely, the authors of[START_REF] Yamada | Dynamic Sasvi: Strong safe screening for norm-regularized least squares[END_REF] considered gauge functions instead of norms for the definition of g. Although the results presented in this section results still hold in this more general setup, we stick to norms to simplify the exposition.

Appendix A. Convex analysis. This appendix reminds two standard results from convex analysis. The first result relates the subdifferential of a function to the subdifferential of its convex conjugate: Lemma A.1 (Subdifferential inversion). Let h : R d → R be a proper, closed and convex function. Then, for all z, z * ∈ R d :

A proof of this lemma can be found in [START_REF] Beck | First-Order Methods in Optimization[END_REF]Theorem 4.20].

The second result recalls two Fenchel-Young inequalities:

Lemma A.2 (Fenchel-Young inequalities). Let h : R d → R be a proper and convex function. Then, for all z, z * ∈ R d :

with equality if and only if z * ∈ ∂h(z).

If h is moreover closed and α-strongly convex, then for all z, z * ∈ R d :

We note that (A.2) corresponds to the standard formulation of the well-known Fenchel-Young inequality. A proof of this result follows from [2, Theorem 4.6] and [START_REF] Beck | First-Order Methods in Optimization[END_REF]Theorem 4.20]. (A.3) is a refined version of the Fenchel-Young inequality which applies to closed and strongly convex functions. Since the latter is less common in the literature, a proof is provided hereafter.

Proof of (A.3). Assume that h is α-strongly convex and let z, z * ∈ R d . First note that since h is proper, closed and α-strongly convex, we have from [2, Item (b) of Theorem 5.26] that h * is α -1 -Lipschitz smooth over R m . In particular, dom(h

Second, if z / ∈ dom(h), then the left-hand side of (A.3) is infinite and the inequality trivially holds true since the right-hand side is finite. We conclude the proof by showing that (A.3) is also valid for z ∈ dom(h). Using the fact that h is proper, closed and convex, we obtain from Lemma A.1 with z = ∇h * (z * ) that

Invoking the first-order characterization of α-strong convexity of h at ∇h * (z * ) ∈ dom(h) (see [START_REF] Beck | First-Order Methods in Optimization[END_REF]Theorem 5.24]) then leads to

Finally, considering the standard Fenchel-Young inequality (A.2) with z = ∇h * (z * ) and using (A.4), we have that the following equality holds:

We obtain the desired result (A.3) by re-injecting (A.8) into (A.7).

In the rest of this section, we thus concentrate on the proof of Proposition B.1. Our arguments leverage the following lemma whose proof is postponed to Appendix B.2:

Lemma B.2. If hypotheses (H1)-(H2) hold true, then the following inequality is satisfied for any

Using the definitions of f and g in (4.13)-(4.14), we note that the duality gap can be expressed as

We distinguish between three cases. First, if Ax = 0 n , then (r 2 RYU (t)) ′ = λ x ≥ 0 so that (C.10) 0 ∈ arg min t≥0 r 2 RYU (t).

In this case, the definition of t ⋆ in (4.16) also leads to t ⋆ = 0 (by using the conventions 0/0 = 0 and 1/0 = +∞). Second, if Ax = 0 n and

we easily have that

. Finally, if Ax = 0 n and t < 0, we then have that (r 2 RYU (0)

t and t < 0. In this case, 0 is a minimizer since it verifies (C.8) and this corresponds again to the definition of t ⋆ in (4.16). It is easy to see that definition of f and g in (4.17)-(4.18) verifies (H1)-(H2) with α = 1, so that the RYU ball in Theorem 4.1 is well-defined.

Showing item ii) of Proposition

On the one hand, using the definition of f in (4.17) with x = 0 n , we have

On the other hand, noticing that the couple (0 n , u) verifies (4.3) and using the same reasoning as in the proof of Proposition 4.7, we obtain from (C.17):

Finally, using Proposition B.1, we have

whereas the SAFE ball is defined as

. This shows the equality in (4.33).

The inclusion in (4.33) can be shown as follows. On the one hand, particularizing the result in Proposition B.1 with (C.22)-(C.23), we have:

On the other hand, the SLORES ball defined in (4.31a)-(4.31b) can be written as

Since the membership inequality in (C.25) is a relaxation of the one in (C.24), the inclusion between the SLORES and RYU balls stated in (4.33) holds.


Cédric Villain

Sandra Wagner

Laetitia Koppe
email: laetitia.koppe@chu-lyon.fr

MD Pr Laetitia Koppe

Low protein diets for older patients: is consensus growing ?

Recent data estimated that nearly 90% of individuals aged 65 years and older have at least one chronic disease, with about 75% having two or more. Therefore, older adults represent a growing proportion of patients with chronic kidney disease (CKD). The nutritional care of this specific and expanding population is of significant importance, in order to avoid malnutrition or sarcopenia and their related poor prognosis, as well as CKD progression to kidney failure. The recommended daily protein intake for adults is 0.8 g/kg/day. However, the European Society for Clinical Nutrition and Metabolism (ESPEN) recommends a higher daily protein intake of at least 1 g/kg and 30 kcal/kg/day, for older adults. 1 On the other hand, the 2020 Kidney Disease Outcomes Quality

Initiative (KDOQI) guidelines suggest a lower protein intake of 0.55-0.6 g/kg/day (or 0.28-0.43 g/kg/day with keto-acid analogs) for stable patients with non-dialysis CKD stages 3-5, and 0.6-0.8 g/kg/day for those with diabetes. [START_REF] Ikizler | KDOQI Clinical Practice Guideline for Nutrition in CKD: 2020 Update[END_REF] The energy intake should be within the range of 25-35 kcal/kg/day. It is important to note that ESPEN does not specifically address the case of CKD, and KDOQI does not specifically focus on older individuals. These seemingly conflicting positions arise from the limited knowledge of the actual protein needs of frail and ill older individuals, especially those with CKD. Currently, scientific evidence, particularly from intervention trials, is insufficient to provide concrete figures regarding protein requirements in this population. Thus, proposing recommendations for protein intake in older adults with CKD poses a significant challenge.

Recently in Clinical Nutrition, ESPEN and ERN-ERA jointly published for the first time a critical review paper regarding the optimal protein and energy intake for older adults with varying degrees of CKD severity. These recommendations are based on expert opinions and are intended to aid clinicians in designing personalized diets in older patients with non-dialysis CKD. Additionally, they provide guidance on the effective implementation of these dietary approaches. [START_REF] Piccoli | Nutritional status and the risk of malnutrition in older adults with chronic kidney disease -implications for low protein intake and nutritional care: A critical review endorsed by ERN-ERA and ESPEN[END_REF] This critical review aims to address several significant questions: 1) What constitutes the definition of an older individual with or without CKD ? 2) Which criteria can be used to define malnutrition? 3) Is there any evidence supporting the effectiveness of low-protein diets (LPD) in older patients with CKD? Older adults constitute a heterogeneous group, exhibiting a wide range of health conditions and functional status, which are not reflected by chronological age. Thus, not all older adults with CKD require dietary modifications with the implementation of a LPD. In fact, those with stable or slowly progressing CKD, especially those at early stages of the disease, may not derive any benefits from an LPD, as they will probably never reach kidney failure. Or at the opposite patients, patients with or at high risk of malnutrition, with impairment in functional status or a limited life expectancy may also not benefit from an LPD. This review provides a list of simple criteria to help clinicians choose between: "an elderly and fragile patient with CKD" or "a patient with CKD who is also elderly". (Table 1) The authors also point out that management of this population is multidisciplinary, person-centered, with a stepwise approach with initial evaluation and individualized assessment and monitoring. They finally underline that the dietary advice must take into account many other factors (socio-economic, eating habits, life expectancy, co-morbidity, dependency, cognitive status, etc.). Overall, this review complements other nutritional recommendations that target specific populations, such as the recently proposed guidelines for hospitalized patients with kidney disease. [START_REF] Fiaccadori | ESPEN guideline on clinical nutrition in hospitalized patients with acute or chronic kidney disease[END_REF] While this review has addressed several inquiries, numerous unresolved issues still remain. Specifically, the potential benefits of LPD in terms of enhancing survival, quality of life and nephroprotection for older adults have yet to be conclusively demonstrated through well-designed and specific randomized controlled trials (RCT). Within the existing literature, large clinical trials that contribute data to meta-analyses and guidelines often have limited recruitment of older and frail CKD patients. [START_REF] Hahn | Low protein diets for non-diabetic adults with chronic kidney disease[END_REF] There are only one RCT in selected older adults with eGFR (glomerular filtration rate) of 5 to 7 ml/min/1,73m² found that LPD permits to postpone dialysis start without increasing the risk of death. [START_REF] Brunori | Efficacy and safety of a very-low-protein diet when postponing dialysis in the elderly: a prospective randomized multicenter controlled study[END_REF] This suggests that a LPD could be an effective treatment for conservative management of CKD, which is often chosen by patients over 80 years old. However, other observational studies have yielded conflicting results regarding the risk of kidney failure, and GFR decline associated with the amount protein intake in older adults with CKD. [START_REF] Narasaki | Protein intake and renal function in older patients[END_REF] In terms of survival, in the general population without kidney disease, a LPD is associated with a reduction in mortality. However, this association does not hold true for adults aged over 65 years. [START_REF] Levine | Low protein intake is associated with a major reduction in IGF-1, cancer, and overall mortality in the 65 and younger but not older population[END_REF] Similar observations have been found in patients with CKD. [START_REF] Watanabe | Age Modifies the Association of Dietary Protein Intake with All-Cause Mortality in Patients with Chronic Kidney Disease[END_REF] Nevertheless, as highlighted in previous reviews on the same topic, pathophysiological considerations support and make the reduction of protein intake in the presence of CKD very attractive. [START_REF] Kalantar-Zadeh | Nutritional Management of Chronic Kidney Disease[END_REF] LPD have been shown to reduce glomerular hyperfiltration, decrease the production of uremic toxins, and mitigate acidosis, inflammation and metabolic complications, all of which are factors involved in the progression of CKD. Are these nephroprotective mechanisms still applicable in older adults with CKD? Indeed, the aging process and comorbidities commonly seen in the aged population may influence the effectiveness of these mechanisms. There are arguments suggesting that the kidneys' ability to compensate for increased protein intake by increasing GFR appears to be preserved up to the age of 80. [START_REF] Fliser | Renal functional reserve in healthy elderly subjects[END_REF] This implies that a LPD may be theoretically effective in at least the septuagenarians. Observational studies have found that adherence to a LPD in old men (aged 70-71 years) with CKD stages 3-5 is associated with improved insulin sensitivity and decreased inflammation. [START_REF] González-Ortiz | Plant-based diets, insulin sensitivity and inflammation in elderly men with chronic kidney disease[END_REF] Further specific studies in this population are required.

Another crucial aspect to consider is the high risk of malnutrition related to ageing and CKD.

Preliminary, defining malnutrition accurately is of utmost importance in order to closely monitor these patients and identify individuals at risk of nutritional inadequacy. Major clinical nutrition societies worldwide, under the Global Leadership Initiative on Malnutrition (GLIM), have collaborated to establish a consensus definition for the diagnosis of malnutrition in adults. [START_REF] Bahat | Sarcopenia: revised European consensus on definition and diagnosis[END_REF] This definition is independent of etiology and requires the presence of at least one phenotypic criterion (such as non-volitional weight loss, low body mass index, or reduced muscle mass) and one etiological criterion (such as reduced food intake or assimilation, and disease burden/inflammation).

No study has validated the application of these criteria specifically in patients with CKD who are over 70 years old. Furthermore, in the context of CKD, the term "protein-energy wasting" (PEW) is preferred, and although the criteria used are similar, there are significant differences. The present recommendations highlight the fact that standardizing the definition is desirable, as supported elsewhere, but do not discuss in depth the available tools for nutritional assessment. PEW is a strong predictor of mortality in patients with CKD, and it is also associated with sarcopenia, characterized by a decline in muscle mass and strength. [START_REF] Gungor | Dietary protein and muscle wasting in chronic kidney disease: new insights[END_REF] Protein intake plays a crucial role as an anabolic trigger for muscle, promoting a positive muscle balance [START_REF] Gungor | Dietary protein and muscle wasting in chronic kidney disease: new insights[END_REF] Both ageing and CKD are associated with an anabolic resistance to proteins, resulting in a decrease in muscle protein synthesis. This phenomenon partly explains the high prevalence of sarcopenia in older CKD adults, which is linked to falls, impaired functional status, and increased mortality. While LPD has been found associated with muscle wasting in non-dialysis CKD patients, the relationship between protein intake and muscle functional status has been rarely studied in older patients with CKD. Recently, a short pilot RCT conducted in older patients with CKD (mean age 81 years) showed that LPD had a favorable impact on nutritional status. [START_REF] Caldiroli | Possible Benefits of a Low Protein Diet in Older Patients With CKD at Risk of Malnutrition: A Pilot Randomized Controlled Trial[END_REF] A long-term study and the utilization of muscle quality assessment tools are necessary to ensure the safety of a LPD in older patients.

Further investigation is also needed about the role of protein type. Animal protein, in comparison to plant-based protein, may pose greater risks to kidney health. [START_REF] Ko | The Effects of High-Protein Diets on Kidney Health and Longevity[END_REF] There is some evidence suggesting that the lower anabolic properties of plant-based foods may potentially impact muscle mass and strength in older adults. [START_REF] Domić | Perspective: Vegan Diets for Older Adults? A Perspective On the Potential Impact On Muscle Mass and Strength[END_REF] Although no studies have specifically examined this issue extensively in CKD patients, observational studies indicate that a diet rich in plant-based proteins is associated with a reduced risk of developing new-onset CKD in middle-aged and older adults. [START_REF] Kwon | Low-protein diet is inversely related to the incidence of chronic kidney disease in middle-aged and older adults: results from a community-based prospective cohort study[END_REF] In conclusion, for the first time, this critical review synthesizes the existing body of knowledge on the impact of dietetics on nephroprotection in older adults, focusing on the specific challenges faced. They conclude that a one-size-fits-all approach may be inappropriate in this population, and they explain that it is not possible to propose a single threshold for protein intake. It emphasizes the importance of offering nephroprotective diets only to a specific population of older individuals. [START_REF] Piccoli | Nutritional status and the risk of malnutrition in older adults with chronic kidney disease -implications for low protein intake and nutritional care: A critical review endorsed by ERN-ERA and ESPEN[END_REF] While there are still many aspects that require further in-depth investigation, this review provides a starting point for collective reflection. It presents an opportunity for future studies to address these remaining questions and contribute to our understanding of the topic. Without such studies, it will be difficult to convincingly advocate for the usefulness of LPD regimens, and to overcome the barriers associated with their implementation in older adults for whom they might be beneficial.

Funding:

No funding

Word count :1513

Number of figures, tables, boxes, references : 18

Authors' contributions:

LK, CV and SW wrote and revised the editorial

Conflict of interest statement:

No conflict of interest related to this topic. L.K. received lecture fees and travel grants from Fresenus Kabi, AstraZeneca, Bayer, Dr Shäre.


Andreia Valente

Philippe Zinck
email: philippe.zinck@ensc-lille.fr

Andre Mortreux

Marc Visseaux

André Mortreux

Catalytic chain transfer (co-)polymerization using borohydridohalflanthanidocene / dilakylmagnesium combinations: polyisoprene CCG and co-transmetalation as a new concept to tune the composition of a copolymer

Keywords: Catalysis, transfer, diene, copolymerization

published or not. The documents may come

Introduction

Chain transfer polymerization presents various advantages over single component polymerization catalysis, including atom economy, molecular weight control and terminal functionalization. [1] Such a process is particularly interesting if the transmetalation between the transition metal and the chain transfer agent shows a rapid and reversible character, and involves all alkyl groups from the main group metal in the absence of other termination reactions. Such a behavior is known as a catalyzed chain growth (CCG), about which our laboratory described more than one decade ago the first example of reversible and living transfer polymerization of ethylene involving lanthanidocene and magnesium dialkyls for the synthesis of long chain dialkylmagnesiums. [2] We reported recently halflanthanidocenes / dialkylmagnesium catalytic systems that show good potentialities for styrene transfer polymerization, [3] and notably the first polystyrene CCG. The same catalytic systems had also shown also good potentialities for the polymerization of isoprene [4] and its statistical copolymerization with styrene. [5] . Furthermore,, the literature devoted to the transmetalation of a growing polydiene chain between a transition metal and aluminum, magnesium or zinc show that so far, the transfer generally occurs with a moderate efficiency, [6]- [12] apart from a tris(aryloxy) lanthanide-dialkylmagnesium mediated butadiene oligomerization from our group. [11] On the other hand, chain transfer in the course of a statistical copolymerization has been moderately dealt with so far and mainly focused on the chain end functionalization of the resulting copolymers. [1]; [13] We report in this contribution preliminary results of our investigations on polymerization reactions using half lantanidocene-dialkymagnesium combinations, namely (i) an exceptionnal dual transfer efficiency in the course of the polymerization of both isoprene and styrene and (ii) the application of this versatile catalytic system as a new and easy route to tune the composition of the resulting styrene-isoprene copolymers.

Experimental

Materials

All operations were performed under dry argon by using Schlenk techniques. Toluene was dried over sodium / benzophenone and distilled over molecular sieves before use. Styrene and isoprene (99% from Aldrich) were dried over calcium hydride, distilled twice over molecular sieves and once just before use. n-butylethylmagnesium (20% wt. in heptane from Texas Alkyl) was used as received. The complex Cp*La(BH4)2(THF)2.5 (1) (Cp*=C5Me5) was synthesized by the reaction between 1 eq. La(BH4)3(THF)3 (synthesized after [16] ) and 1 eq.

KCp* (synthesized from the reaction between HCp* and KH) in THF. The reaction mixture was stirred at room temperature for 4 days, filtered and evaporated to dryness. Yield: 91%. The signals at 2.30 observed for the crude compound was thus attributed to a bridged THF adduct oligomeric form that dissociates in the presence of THF. [15] After drying under vacuum 48h at 60 °C, 1 was obtained in the solvent-free form Cp*La(BH4)2. Anal. Calcd for C10H23B2La. C,39.53; H,7.63. Found: C,40.12; H,7.48. The complex Cp*Nd(BH4)2(THF)2 (2) was synthesized according to literature procedure. [14] Polymerization In a glove box, the borohydrido complex was weighed into a glass vessel. Toluene, the monomer(s) and n-butylethylmagnesium were added in this order using syringes. The mixture was magnetically stirred at the reaction temperature for a given time, quenched with methanol, eventually diluted in toluene and poured into methanol, leading to the precipitation of the polymer. Solvents were evaporated under partial vacuum, and the product was dried under vacuum until constant weight.

Measurements

NMR analyses of polyisoprene and copolymer samples were conducted on a AC 300 Brücker at room temperature in CdCl3 and tetrachloroethane-d2, respectively. The chemical shifts were calibrated using the residual resonances of the solvent. Quantitative 13 C NMR was realized using the zgig sequence (Brücker library). The relaxation time d1 was set to 2,5 and 8s and it was found that a relaxation time of 5s is necessary to allow the relaxation of all carbon nucleus. Size exclusion chromatography (SEC) was performed in THF as eluent at 40°C using a Waters SIS HPLC-pump, a Waters 410 refractometer and Waters styragel column (HR2, HR3, HR4, HR5E) calibrated with polystyrene standards. A correction factor of 0.5 was applied for the determination of the true number-average molecular weight of polyisoprene. [17] DSC measurements were conducted under nitrogen on a Setaram 141 apparatus at a heating rate of 10°C.min -1 from -120°C to 300°C using around 20 mg samples in aluminum crucibles.

Results

Isoprene transfer polymerization

Representative results of isoprene polymerization using 1 combined to nbutylethylmagnesium are presented in Table 1, entries 1-5. The observed molecular weight distributions are monomodal, and the number average molecular weight is close to the calculated one considering two growing chains per magnesium atom. Reasonable polydispersities are obtained, with PDI in the range 1.3-1.6. This highlights a behavior that is close to a lanthanide catalyzed polyisoprene chain growth on magnesium. The transmetalation is accompanied by a modification of the selectivity of the reaction: a 98.5 % 1,4-trans polyisoprene is obtained in the presence of 1 eq. n-butylethylmagnesium while 39% isoprene units are in a 3,4 enchainment in the macromolecules obtained using 10 eq. chain transfer agent. Using the neodymium homologue 2 for the transmetalation lead to a broader distribution (PDI 1.9-entry 7), together with a slightly higher amount of 3,4 units. with the repetition of two 3,4 units between a 1,4-trans backbone, which would not lead to a unique signal on the 13 C NMR spectra in the -CH3 zone. These observations, together with the monomodal character of the chromatograms and the presence of a unique glass transition temperature indicates that the 3,4 units are inserted into a 1,4-trans backbone, and that the polymer is not a mix between poly(1,4-trans isoprene) and poly(3,4-isoprene).

The increase of the proportion of 3,4 units can be due to steric hindrance in the presence of an excess Mg, inducing a higher propensity to insert isoprene in the 3,4 configuration. This latter consideration would be a consequence of a transmetalation occuring in an olefin-like configuration, similar to that reported for styrene. Mechanistic investigations will be dealt with more deeply in a forthcoming study.

Co-transmetalation in the course of a statistical copolymerization

Results obtained on statistical copolymerization of isoprene and styrene starting from a 50/50 molar ratio in the presence of 1 and increasing amount of dialkylmagnesium are reported in Table 1 (entries 8-12). The number-average molecular weight of the copolymer decreases from 45200 to 8100 g/mol as the amount of dialkylmagnesium increases from 1 to 10 equivalents, highlighting the occurrence of a co-transmetalation of the growing macromolecular chain. The resulting molecular weight distributions are monomodal in all cases, and the polydispersity index increases from 1.3 to 1.8, highlighting the probable occurrence of other chain termination pathways and/or a partially irreversible character of the transfer. The amount of styrene inserted in the copolymer increases continuously in the presence of an excess dialkylmagnesium, from 12 to 32% mol, and the content of 3,4 isoprene units increases from 2 to 16% at the expense of the trans-selectivity. The glass transition temperature increases from around -57°C to around -31°C with the amount of styrene inserted. Similar trends are obtained starting from the neodymium half sandwich 2, with a slightly higher polydispersity however (PDI 2.0-entry 14 vs. 1.8 -entry 12).

Concluding remarks

The influence of an excess organometallic compound on the polymerization of dienes has been studied in the literature. The sole high efficiency reported for diene as far as we know concerns a neodymium catalyzed oligobutadiene chain growth on magnesium (1100 g/mol). [11; To the best of our knowledge, if the changes in selectivity observed here have already been reported in the literature [4a];[10a;c]; [18] , the polyisoprene transfer efficiency close to 100% obtained with our systems is noteworthy and unprecedented for molecular weight as high as 6000 -30 000 g/mol. Furthermore, the influence of reversible transmetalation of the growing macromolecular chain on the composition of a copolymer has never been studied systematically as far as we know (although some experimental results were reported, [13] the changes in copolymer composition remained unnoticed by the authors). In this study, we found that due tyo the dual CGC charcter of the half lanthanidocene catalyst , the composition of the copolymer can be easily changed by modifying the ratio of chain transfer agent in the reactive mixture. The comonomer content in a copolymer is usually tailored by reactor feed, catalyst modification or process considerations. [19] The co-transmetalation can thus be viewed as a new and original route for the control of the composition of a copolymer. The influence of experimental parameters including feed is currently under studies, together with a deeper structural analysis of the polymers as well as mechanistic insights. Reactions conducted at 50°C in 1.5 ml toluene using 1.5 ml isoprene (1.5 10 -2 mol) for entries 1-7 and 1.1 ml styrene (10 -2 mol) / 1 ml isoprene (10 -2 mol) for entries 8-14. Monomer / Ln = 2000 b) Number-average molecular weight determined by size exclusion chromatography corrected for polyisoprene (entries 1-4), in equivalent PS for entries 5-8 5,6 et 7 sont des manips faites avec uniquement l'isoprène: pourquoi en eq PS? c) Calculated polyisoprene number average molecular weight considering the yield and two growing chains per magnesium atom d) Styrene content of the copolymer from 1 H NMR measurements [3c] e) From NMR measurements. The 1,4-cis content was found to be <1% in all cases f) from DSC measurements at 10°C/m

Figure 1 . 13 C

 113 Figure 1. 13 C NMR spectra of polyisoprene (entry 4) in the methyl zone.

Table 1 .

 1 Isoprene and isoprene / styrene chain transfer (co-)polymerization

	Entry a)	Metal Ip/St Mg	Time	Yield	n M b)	PDI	n M calc. c)	St d)	microstructure e) Isoprene	Tg f)
				Eq.	(h)	(%)	(g/mol)		(g/mol)	(%)	1,4-trans (%)	3,4 (%)	(°C)
	1	La	1/0	1	2	73	64400 1.4	50500	-	98.5	1.5	-66
	2	La	1/0	2	5	95	30800 1.3	32700	-	97	3	n.d
	3	La	1/0	3	8	84	18500 1.5	19200	-	91	9	n.d
	4	La	1/0	5	20	78	8800	1.6	10800	-	85	15	n.d
	5	La	1/0 10	40	86	6200	1.6	5800	-	61	39	-48
	6	Nd	1/0	1	2	76	43400 1.3	51700	-	98	2	-66
	7	Nd	1/0 10	40	54	4300	1.9	3700	-	54	46	-47
	8	La	1/1	1	2	48	45200 1.3	-	12	98	2	-57
	9	La	1/1	2	5	54	41600 1.5	-	14	97	3	-53
	10	La	1/1	3	8	56	25500 1.7	-	16	95	5	-53
	11	La	1/1	5	20	79	18900 1.8	-	27	90	10	-36
	12	La	1/1 10	40	58	8100	1.8	-	32	84	16	-31
	13	Nd	1/1	1	2	30	54600 1.4	-	11	98	2	-59
	14	Nd	1/1 10	40	39	7100	2.0	-	29	84.5	15.5 -35

a)

Acknowledgments

The authors thank Marc Bria and Pierre Watkin for NMR studies, Aurélie Malfait for GPC measurements and Nora Djelal for calorimetry measurements.