Adaptive extra-gradient methods for min-max optimization and games

Introduction

The surge of recent breakthroughs in generative adversarial networks , robust reinforcement learning , and other adversarial learning models has sparked renewed interest in the theory of min-max optimization problems and games. In this broad setting, it has become empirically clear that, ceteris paribus, the simultaneous training of two (or more) antagonistic models faces drastically new challenges relative to the training of a single one. Perhaps the most prominent of these challenges is the appearance of cycles and recurrent (or even chaotic) behavior in min-max games. This has been studied extensively in the context of learning in bilinear games, in both continuous and discrete time , and the methods proposed to overcome recurrence typically focus on mitigating the rotational component of min-max games.

The method with the richest history in this context is the extra-gradient (EG) algorithm of Korpelevich and its variants. The extra-gradient (EG) algorithm exploits the Lipschitz smoothness of the problem and, if coupled with a Polyak–Ruppert averaging scheme, it achieves an $\operatorname{\mathcal{O}}(1/T)$ rate of convergence in smooth, convex-concave min-max problems . This rate is known to be tight but, in order to achieve it, the original method requires the problem’s Lipschitz constant to be known in advance. If the problem is not Lipschitz smooth (or the algorithm is run with a vanishing step-size schedule), the method’s rate of convergence drops to $\operatorname{\mathcal{O}}(1/\sqrt{T})$.

Our aim in this paper is to provide an algorithm that automatically adapts to smooth / non-smooth min-max problems and games, and achieves order-optimal rates in both classes without requiring any prior tuning by the optimizer. In this regard, we propose a flexible algorithmic scheme, which we call AdaProx, and which exploits gradient data observed at earlier iterations to perform more informative extra-gradient steps in later ones. Thanks to this mechanism, and to the best of our knowledge, AdaProx is the first algorithm that simultaneously achieves the following:

An \operatorname{\mathcal{O}}\big{(}1/\sqrt{T}\big{)} convergence rate in non-smooth problems and $\operatorname{\mathcal{O}}(1/T)$ in smooth ones.

Applicability to min-max problems and games where the standard boundedness / Lipschitz continuity conditions required in the literature do not hold.

Convergence without prior knowledge of the problem’s parameters (e.g., whether the problem’s defining vector field is smooth or not, its smoothness modulus if it is, etc.).

Our proposed method achieves the above by fusing the following ingredients: \edefnit\selectfonta\edefnn) a family of local norms – a Finsler metric – capturing any singularities in the problem at hand; \edefnit\selectfonta\edefnn) a suitable mirror-prox template; and \edefnit\selectfonta\edefnn) an adaptive step-size policy in the spirit of Rakhlin & Sridharan . We also show that, under a suitable coherence assumption, the sequence of iterates generated by the algorithm converges, thus providing an appealing alternative to iterate averaging in cases where the method’s “last iterate” is more appropriate (for instance, if using AdaProx to solve non-monotone problems).

Related work.

There have been several works improving on the guarantees of the original extra-gradient/mirror-prox template. We review the most relevant of these works below; for convenience, we also tabulate these contributions in Table 1. Because many of these works appear in the literature on variational inequalities , we also use this language in the sequel.

In unconstrained problems with an operator that is locally Lipschitz continuous (but not necessarily globally so), the golden ratio algorithm (GRAAL) achieves convergence without requiring prior knowledge of the problem’s Lipschitz parameter. However, golden ratio algorithm (GRAAL) provides no rate guarantees for non-smooth problems – and hence, a fortiori, no interpolation guarantees either. By contrast, such guarantees are provided in problems with a bounded domain by the generalized mirror-prox (GMP) algorithm of Stonyakin et al. under the umbrella of Hölder continuity. Still, nothing is known about the convergence of GRAAL / generalized mirror-prox (GMP) in problems with singularities (i.e., when the problem’s defining vector field blows up at a boundary point of the problem’s domain).

Another method that simultaneously achieves an $\operatorname{\mathcal{O}}(1/\sqrt{T})$ rate in non-smooth problems and an $\operatorname{\mathcal{O}}(1/T)$ rate in smooth ones is the recent algorithm of Bach & Levy . The Bach–Levy (BL) algorithm employs an adaptive, AdaGrad-like step-size policy which allows the method to interpolate between the two regimes – and this, even with noisy gradient feedback. On the negative side, the BL algorithm requires a bounded domain with a (Bregman) diameter that is known in advance; as a result, its theoretical guarantees do not apply to problems with an unbounded domain. In addition, the BL algorithm makes crucial use of operator boundedness and Lipschitz continuity; extending the BL method beyond this standard framework is a highly non-trivial endeavor which formed a big part of this paper’s motivation.

Operators with singularities were treated in a recent series of papers by means of a “Bregman continuity” or “Lipschitz-like” condition in the spirit of Bauschke et al. and Lu et al. . Albeit different, the adaptive methods presented in are both order-optimal in the smooth case, without requiring any knowledge of the problem’s smoothness modulus. On the other hand, like GRAAL – but unlike GMP – they do not provide any rate interpolation guarantees between smooth and non-smooth problems. Finally, the method of provides an “inexact model” framework that unifies the approach of and , providing rate interpolation in the Hölder case and convergence in problems with singularities;Personal communication with P. Dvurechensky suggests that the method of can be further adapted to problems with singularities under the metric boundedness framework presented in this paper. however, in problems with an unbounded domain, it still requires an initial guess of a compact set containing a solution.

Problem Setup and Blanket Assumptions

We begin in this section by reviewing some basics for min-max problems and games.

A min-max game is a saddle-point problem of the form

By the minimax theorem of von Neumann , Nash equilibria are guaranteed to exist when $\Theta,\Phi$ are compact and $\mathcal{L}$ is convex-concave (i.e., convex in $\theta$ and concave in $\phi$). Much of our paper is motivated by the question of calculating a Nash equilibrium $(\theta^{\ast},\phi^{\ast})$ of (SP) in the context of von Neumann’s theorem; we expand on this below.

2. Games

In this context, a Nash equilibrium is any action profile $x^{\ast}\in\mathcal{K}$ that is unilaterally stable, i.e.,

If there is no continuous vector field $V_{i}(x)$ satisfying (2), the game is called non-smooth.

If there is a continuous vector field $V_{i}(x)$ satisfying (2), the game is called smooth.

3. Resource allocation and equilibrium problems

The notion of a Nash equilibrium captures the unilateral minimization of the players’ individual loss functions. In many pratical cases of interest, a notion of equilibrium is still relevant, even though it is not necessarily attached to the minimization of individual loss functions. Such problems are known as “equilibrium problems” ; to avoid unnecessary generalities, we focus here on a relevant problem that arises in distributed computing architectures (such as GPU clusters and the like).

To state the problem, consider a distributed computing grid consisting of $N$ parallel processors that serve demands arriving at a rate of $\rho$ per unit of time (measured e.g., in flop/s). If the maximum processing rate of the $i$-th node is $\mu_{i}$ (without overclocking), and jobs are buffered and served on a first-come, first-served (FCFS) basis, the mean time required to process a unit demand at the $i$-th node is given by the Kleinrock M/M/1 response function $\tau_{i}(x_{i})=1/(\mu_{i}-x_{i})$, where $x_{i}$ denotes the node’s load . Accordingly, the set of feasible loads that can be processed by the grid is $\mathcal{X}\coloneqq\{(x_{1},\dotsc,x_{N}):0\leq x_{i}<\mu_{i},x_{1}+\dotsm+x_{N}=\rho\}$.

In this context, a load profile $x^{\ast}\in\mathcal{X}$ is said to be balanced if no infinitesimal process can be better served by buffering it at a different node ; formally, this amounts to the so-called Wardrop equilibrium condition

We note here a crucial difference between (WE) and (NE): if we view the grid’s computing nodes as “players”, the constraint $\sum_{i}x_{i}=\rho$ means that there is no allowable unilateral deviation $(x_{i}^{\ast};x_{-i}^{\ast})\mapsto(x_{i};x_{-i}^{\ast})$ with $x_{i}\neq x_{i}^{\ast}$. As a result, (NE) is meaningless as a requirement for this equilibrium problem.

As we discuss below, this resource allocation problem will require the full capacity of our framework.

4. Variational inequalities

Importantly, all of the above problems can be restated as a variational inequality of the form

5. Merit functions and monotonicity

A widely used assumption in the literature on equilibrium problems and variational inequalities is the monotonicity condition

In single-player games, monotonicity is equivalent to convexity of the optimizer’s loss function; in min-max games, it is equivalent to $\mathcal{L}$ being convex-concave ; etc. In the absence of monotonicity, approximating an equilibrium is PPAD-hard , so we will state most of our results under (MC).

Now, to assess the quality of a candidate solution $\hat{x}\in\mathcal{X}$, we will employ the restricted merit function

where the “test domain” $\mathcal{C}$ is a nonempty convex subset of $\mathcal{X}$ . The motivation for this is provided by the following proposition:

Let $\mathcal{C}$ be a nonempty convex subset of $\mathcal{X}$. Then: \edefitit\selectfonta\edefitn) $\operatorname{Gap}_{\mathcal{C}}(\hat{x})\geq 0$whenever $\hat{x}\in\mathcal{C}$; and \edefitit\selectfonta\edefitn) if $\operatorname{Gap}_{\mathcal{C}}(\hat{x})=0$ and $\mathcal{C}$ contains a neighborhood of $\hat{x}$, then $\hat{x}$ is a solution of (VI).

Proposition 1 generalizes an earlier characterization by Nesterov and justifies the use of $\operatorname{Gap}_{\mathcal{C}}(x)$ as a merit function for (VI); to streamline our presentation, we defer the proof to the paper’s supplement. Moreover, to avoid trivialities, we will also assume that the solution set $\mathcal{X}^{\ast}$ of (VI) is nonempty and we will reserve the notation $x^{\ast}$ for solutions of (VI). Together with monotonicity, this will be our only blanket assumption.

The Extra-Gradient Algorithm and its Limits

Perhaps the most widely used solution method for games and variational inequalities is the extra-gradient (EG) algorithm of Korpelevich and its variants . This algorithm has a rich history in optimization, and it has recently attracted considerable interest in the fields of machine learning and AI, see e.g., and references therein.

In its simplest form, for problems with closed domains, the algorithm proceeds recursively as

where $\operatorname{\Pi}(x)=\operatorname*{arg\,min}_{x^{\prime}\in\mathcal{X}}\lVert x^{\prime}-x\rVert$ is the Euclidean projection on $\mathcal{X}$, $V_{t}\coloneqq V(X_{t})$ for $t=1,3/2,\dotsc$, and $\gamma_{t}>0$, is the method’s step-size. Then, running (EG) for $T$ iterations, the algorithm returns the “ergodic average”

In this setting, the main guarantees for (EG) date back to and can be summarized as follows:

For non-smooth problems (discontinuous $V$): Assume $V$ is bounded, i.e., there exists some $M>0$ such that

Then, if (EG) is run with a step-size of the form $\gamma_{t}\propto 1/\sqrt{t}$, we have

For smooth problems (continuous $V$): Assume $V$ is $L$-Lipschitz continuous, i.e.,

Then, if (EG) is run with a constant step-size $\gamma<1/L$, we have

In the above, $\lVert\cdot\rVert$ is tacitly assumed to be the standard Euclidean norm. Non-Euclidean considerations will play a crucial role in the sequel, but they are not necessary for the moment.

Importantly, the distinction between smooth and non-smooth problems cannot be lifted: the bounds (5) and (6) are tight in their respective problem classes and they cannot be improved without further assumptions . Moreover, we should also note the following:

The algorithm changes drastically from the non-smooth to the smooth case: non-smoothness requires $\gamma_{t}\propto 1/\sqrt{t}$, but such a step-size cannot achieve a fast $\operatorname{\mathcal{O}}(1/T)$ rate.

If (EG) is run with a constant step-size, $L$ must be known in advance; otherwise, running (EG) with an ill-adapted step-size ($\gamma>1/L)$ could lead to non-convergence.

We illustrate this failure of (EG) in Fig. 1. As we discussed in the introduction, our aim in the sequel will be to provide a single, adaptive algorithm that simultaneously achieves the following: \edefnit\selectfonta\edefnn) an order-optimal \operatorname{\mathcal{O}}\big{(}1/\sqrt{T}\big{)} convergence rate in non-smooth problems and $\operatorname{\mathcal{O}}(1/T)$ in smooth ones; \edefnit\selectfonta\edefnn) convergence in problems where the boundedness / Lipschitz continuity conditions (BD) / (LC) no longer hold; and \edefnit\selectfonta\edefnn) achieves all this without prior knowledge of the problem’s parameters.

Rate Interpolation: the Euclidean Case

As a prelude to our main result, we provide in this section an adaptive version of (EG) that achieves the “best of both worlds” in the Euclidean setting of Section 3, i.e., an \operatorname{\mathcal{O}}\big{(}1/\sqrt{T}\big{)} convergence rate in problems satisfying (BD), and an $\operatorname{\mathcal{O}}(1/T)$ rate in problems satisfying (LC). Our starting point is the observation that, if the sequence $X_{t}$ produced by (EG) converges to a solution of (VI), the difference

The intuition behind (8) is as follows: If $V$ is not smooth and $\liminf_{t\to\infty}\delta_{t}>0$, then $\gamma_{t}$ will vanish at a \Theta\big{(}1/\sqrt{t}\big{)} rate, which is the optimal step-size schedule for problems satisfying (BD) but not (LC). Instead, if $V$ satisfies (LC) and $X_{t}$ converges to a solution $x^{\ast}$ of (VI), it is plausible to expect that the infinite series $\sum_{t}\delta_{t}^{2}$ is summable, in which case the step-size $\gamma_{t}$ will not vanish as $t\to\infty$. Furthermore, since $\delta_{t}$ is defined in terms of successive gradient differences, it automatically exploits the variation of the gradient data observed up to time $t$, so it can be expected to adjust to the “local” Lipschitz constant of $V$ around a solution $x^{\ast}$ of (VI).

Our step-size policy and motivation are similar in spirit to the “predictable sequence” approach of . color=DodgerBlue!20!LightGray,author=PM]Is this ok? However, making our reasoning precise (especially the summability of $\sum_{t}\delta_{t}^{2}$ in the smooth case) involves considerable conceptual and technical difficulties that we present in detail in the supplement. For now, we only state (without proof) our main result for problems satisfying (BD) or (LC).

Suppose $V$ satisfies (MC), let $\mathcal{C}$ be a compact neighborhood of a solution of (VI), and let $0pt=\sup_{x\in\mathcal{C}}\lVert X_{1}-x\rVert^{2}$. If (EG) is run with the adaptive step-size policy (8), we have:

Finsler Regularity

To motivate our analysis outside the setting of (BD)/(LC), consider the vector field

which corresponds to the distributed computing problem of Section 2.3 plus a regularization term designed to limit the activation of computing nodes at low loads. Clearly, we have $\lVert V(x)\rVert\to\infty$ whenever $x_{i}\to 0^{+}$, so (BD) and (LC) both fail (the latter even if $\lambda=0$). On the other hand, if we consider the “local” norm $\lVert v\rVert_{x,\ast}=\sum_{i=1}^{d}(\mu_{i}-x_{i})\,\lvert v_{i}\rvert$, we have $\lVert V(x)\rVert_{x,\ast}\leq d+\lambda\sum_{i=1}^{d}\mu_{i}$, so $V$ is bounded relative to $\lVert\cdot\rVert_{x,\ast}$. This observation motivates the use of a local – as opposed to global – norm, which we define formally as follows:

Subadditivity: $F(x;z+z^{\prime})\leq F(x;z)+F(x;z^{\prime})$.

Positive-definiteness: $F(x;z)\geq 0$ with equality if and only if $z=0$.

Given a Finsler metric on $\mathcal{X}$, the induced primal / dual local norms on $\mathcal{X}$ are respectively defined as

When $\mathcal{X}$ is equipped with a regular Finsler metric as above, we will say that it is a Finsler space.

Hence, by dividing by $\lVert v\rVert_{x,\ast}$, we readily get $\lVert v\rVert_{x^{\prime},\ast}/\lVert v\rVert_{x,\ast}\leq 1+\lVert x-x^{\prime}\rVert_{x}$ i.e., $\lVert\cdot\rVert_{x}$ is regular in the sense of Definition 1. As we discuss in the sequel, this metric plays an important role for distributed computing problems of the form presented in Section 2.3. $\blacktriangleleft$

Metrically bounded if there exists some $M>0$ such that

Metrically smooth if there exists some $L>0$ such that

The notion of metric boundedness/smoothness extends that of ordinary boundedness/Lipschitz continuity to a Finsler context; note also that, even though neither side of (MS) is unilaterally symmetric under the change $x\leftrightarrow x^{\prime}$, the condition (MS) as a whole is. Our next example shows that this extension is proper, i.e., (BD)/(LC) may both fail while (MB)/(MS) both hold:

For all $\lambda\geq 0$, $V$ satisfies (MB) with $M=d(1+\lambda)$: $\lVert V(x)\rVert_{x,\ast}\leq\sum_{i=1}^{d}x_{i}\cdot(1/x_{i}+\lambda)=d(1+\lambda)$.

For $\lambda=0$, $V$ satisfies (MS) with $L=d$: indeed, for all $x,x^{\prime}\in\mathcal{X}$, we have

The AdaProx Algorithm and its Guarantees

We are now in a position to define a family of algorithms that is capable of interpolating between the optimal smooth/non-smooth convergence rates for solving (VI) without requiring either (BD) or (LC). To do so, the key steps in our approach will be to (\edefnit\selectfonti \edefnn) equip $\mathcal{X}$ with a suitable Finsler structure (as in Section 5); and (\edefnit\selectfonti \edefnn) replace the Euclidean projection in (EG) with a suitable “Bregman proximal” step that is compatible with the chosen Finsler structure on $\mathcal{X}$.

We begin with the latter (assuming that $\mathcal{X}$ is equipped with an arbitrary Finsler structure):

$h$ is convex, lower semi-continuous (l.s.c.), $\operatorname{cl}(\operatorname{dom}h)=\operatorname{cl}(\mathcal{X})$, and $\operatorname{dom}\partial h=\mathcal{X}$.

The subdifferential of $h$ admits a continuous selection $\nabla h(x)\in\partial h(x)$ for all $x\in\mathcal{X}$.

$h$ is strongly convex, i.e., there exists some $K>0$ such that

for all $x\in\mathcal{X}$ and all $x^{\prime}\in\operatorname{dom}h$.

The Bregman divergence induced by $h$ is defined for all $x\in\mathcal{X}$, $x^{\prime}\in\operatorname{dom}h$ as

Definition 2 is fairly technical, so some clarifications are in order. First, to connect this definition with the Euclidean setup of Section 4, the prox-mapping (16) should be seen as the Bregman equivalent of a Euclidean projection step, i.e., $\operatorname{\Pi}(x+y)\leftrightsquigarrow P_{x}(y)$. Second, a key difference between Definition 2 and other definitions of Bregman functions in the literature is that $h$ is assumed strongly convex relative to a local norm – not a global norm. This “locality” will play a crucial role in allowing the proposed methods to adapt to the geometry of the problem. For concreteness, we provide below an example that expands further on Examples 5.2 and 5.3:

Consider the local norm $\lVert z\rVert_{x}=\max_{i}\lvert z_{i}\rvert/x_{i}$ on $\mathcal{X}=(0,1]^{d}$ and let $h(x)=\sum_{i=1}^{d}1/x_{i}$ on $(0,1]^{d}$. We then have

i.e., $h$ is $1$-strongly convex relative to $\lVert\cdot\rVert_{x}$ on $\mathcal{X}$. $\blacktriangleleft$

With all this is in place, the extra-gradient method can be adapted to our current setting as follows:

with $V_{t}=V(X_{t})$, $t=1,3/2,\dotsc$, as in Section 3. In words, this method builds on the template of (EG) by (\edefnit\selectfonti \edefnn) replacing the Euclidean projection with a mirror step; (\edefnit\selectfonti \edefnn) replacing the global norm in (8) with a dual Finsler norm evaluated at the algorithm’s leading state $X_{t+1/2}$. The first of these two steps is the main ingredient of the mirror-prox (MP) algorithm of Nemirovski ; the name “AdaProx” has beeen chosen precisely because the proposed method can be seen as a mirror-prox method that adapts between the smooth and non-smooth regimes.

Convergence speed.

With all this in hand, our main result for AdaProx can be stated as follows:

Suppose $V$ satisfies (MC), let $\mathcal{C}$ be a compact neighborhood of a solution of (VI), and set $0pt=\sup_{x\in\mathcal{C}}D(x,X_{1})$ Then, the AdaProx algorithm enjoys the guarantees:

For the constants that appear in Eq. 18, we refer the reader to the discussion following Theorem 1 (of course, since Theorem 1 is a special case of Theorem 2, it is not surprising that the same remarks apply). As for the proof of Theorem 2, it is quite intricate, so we defer it to the paper’s supplement. We only mention here that its key element is the determination of the asymptotic behavior of the adaptive step-size policy $\gamma_{t}$ in the non-smooth and smooth regimes, i.e., under (MB) and (MS) respectively. At a very high level, (MB) guarantees that the difference sequence $\delta_{t}$ is bounded, which implies in turn that $\sum_{t=1}^{T}\gamma_{t}=\Omega(\sqrt{T})$ and eventually yields the bound (18a) for the algorithm’s ergodic average $\bar{X}_{T}$. On the other hand, if (MS) kicks in, we have the following finer result:

Assume $V$ satisfies (MS). Then, \edefitit\selectfonta\edefitn) $\gamma_{t}$decreases monotonically to a strictly positive limit $\gamma_{\infty}=\lim_{t\to\infty}\gamma_{t}>0$; and \edefitit\selectfonta\edefitn) the sequence $\delta_{t}$ is square summable: in particular, $\sum_{t=1}^{\infty}\delta_{t}^{2}=1/\gamma_{\infty}^{2}-1$.

By means of this lemma (which we prove in the paper’s supplement), it follows that $\sum_{t=1}^{T}\gamma_{t}\geq\gamma_{\infty}T=\Omega(T)$. Because the algorithm’s rate of convergence is controlled by this quantity, it ultimately follows that AdaProx enjoys an $\operatorname{\mathcal{O}}(1/T)$ rate of convergence under (MS). However, the details of the ensuing calculations are quite complicated, so we defer them to the supplement.

Trajectory convergence.

In complement to Theorem 2, we also provide a trajectory convergence result that governs the actual iterates of the AdaProx algorithm:

Suppose that $\langle V(x),x-x^{\ast}\rangle<0$ whenever $x^{\ast}$ is a solution of (VI) and $x$ is not. If, in addition, $V$ satisfies (MB) or $\eqref{eq:MS}$, the iterates $X_{t}$ of AdaProx converge to a solution of (VI).

The importance of this result is that, in many practical applications (especially in non-monotone problems), it is more common to harvest the “last iterate” of the method ($X_{t}$) rather than its ergodic average ($\bar{X}_{T}$); as such, Theorem 3 provides a certain justification for this design choice.

The proof of Theorem 3 relies on non-standard arguments, so we relegate it to the supplement. Structurally, the first step is to show that $X_{t}$ visits any neighborhood of a solution point $x^{\ast}\in\mathcal{X}^{\ast}$ infinitely often (this is where the coherence assumption $\langle V(x),x-x^{\ast}\rangle$ is used). The second is to use this trapping property in conjunction with a suitable “energy inequality” to establish convergence via the use of a quasi-Fejér technique as in ; this part is detailed in a separate appendix.

Numerical Experiments

We conclude in this section with a numerical illustration of the convergence properties of AdaProx in two different settings: \edefnit\selectfonta\edefnn) bilinear min-max games; and \edefnit\selectfonta\edefnn) a simple Wasserstein GAN in the spirit of Daskalakis et al. with the aim of learning an unknown covariance matrix.

Covariance matrix learning.

Going a step further, we also considered the covariance learning game

The goal here is to generate data drawn from a centered Gaussian distribution with unknown covariance $\Sigma$; in particular, this model follows the Wasserstein GAN formulation of Daskalakis et al. with generator and discriminator respectively given by $G(z)=\theta z$ and $D(x)=x^{\top}\phi x$ (no clipping). For the experiments, we took $d=100$, a mini-batch of $m=128$ samples per update, and we ran the EG, BL and AdaProx algorithms as above, tracing the square norm of $V$ as a measure of convergence. Since the problem is non-monotone, there are several disjoint equilibrium components so the algorithms’ behavior is considerably more erratic; however, after this initial warm-up phase, AdaProx again gave the faster convergence rates.

Acknowledgments

This research was partially supported by the COST Action CA16228 “European Network for Game Theory” (GAMENET) and the French National Research Agency (ANR) in the framework of the grants ORACLESS (ANR–16–CE33–0004–01) and ELIOT (ANR-18-CE40-0030), the “Investissements d’avenir” program (ANR-15-IDEX-02), the LabEx PERSYVAL (ANR-11-LABX-0025-01), and MIAI@Grenoble Alpes (ANR-19-P3IA-0003).

Appendix A Properties of the restricted gap function

In this appendix, we discuss the basic properites of the restricted merit function $\operatorname{Gap}_{\mathcal{C}}$ introduced in (3). For completeness, we provide the proof of Proposition 1,which itself is an extension of a similar result by Nesterov :

Let $x^{\ast}\in\mathcal{X}$ be a solution of (VI) so $\langle V(x^{\ast}),x-x^{\ast}\rangle\geq 0$ for all $x\in\mathcal{X}$. Then, by monotonicity, we get:

so $\operatorname{Gap}_{\mathcal{C}}(x^{\ast})\leq 0$. On the other hand, if $x^{\ast}\in\mathcal{C}$, we also get $\operatorname{Gap}(x^{\ast})\geq\langle V(x^{\ast}),x^{\ast}-x^{\ast}\rangle=0$, so we conclude that $\operatorname{Gap}_{\mathcal{C}}(x^{\ast})=0$.

For the converse statement, assume that $\operatorname{Gap}_{\mathcal{C}}(\hat{x})=0$ for some $\hat{x}\in\mathcal{C}$ and suppose that $\mathcal{C}$ contains a neighborhood of $\hat{x}$ in $\mathcal{X}$. First, we claim that the following inequality holds:

Indeed, assume to the contrary that there exists some $x_{1}\in\mathcal{C}$ such that

which is a contradiction. Now, we further claim that $\hat{x}$ is a solution of (VI),i.e.,:

If we suppose that there exists some $z_{1}\in\mathcal{X}$ such that $\langle V(\hat{x}),z_{1}-\hat{x}\rangle<0$, then, by the continuity of $V$, there exists a neighborhood $U^{\prime}$ of $\hat{x}$ in $\mathcal{X}$ such that

Hence, assuming without loss of generality that $U^{\prime}\subset U\subset\mathcal{C}$ (the latter assumption due to the assumption that $\mathcal{C}$ contains a neighborhood of $\hat{x}$), and taking $\lambda>0$ sufficiently small so that $x=\hat{x}+\lambda(z_{1}-\hat{x})\in U^{\prime}$, we get that $\langle V(x),x-\hat{x}\rangle=\lambda\langle V(x),z_{1}-\hat{x}\rangle<0$, in contradiction to (A.2). We conclude that $\hat{x}$ is a solution of (VI), as claimed. ∎

Appendix B Properties of Bregman functions and proximal mappings

In this appendix, we present some basic facts about Bregman functions and proximal mappings. Similar results exist in the literature in different contexts (see e.g., and references therein), but given that many of our results rely on the use of local – as opposed to global – norms, we provide here complete statements and proofs. We then have the following basic lemma connecting the above notions:

Let $h$ be a Bregman function on $\mathcal{X}$. Then, for all $p\in\operatorname{dom}h$, $x\in\operatorname{dom}\partial h$ and all $y\in\partial h(x)$, we have

By a simple continuity argument, it is sufficient to show that the inequality holds for the relative interior $\operatorname{ri}\mathcal{X}$ of $\mathcal{X}$. In order to show this, pick a base point $p\in\operatorname{ri}\mathcal{X}$, and let

Since, $h$ is strongly convex and $y\in\partial h(x)$ due to the first equivalence, it follows that $\phi(t)\geq 0$ with equality if and only if $t=0$. Since, $\psi(t)=\langle\nabla h(x+t(p-x))-y,p-x\rangle$ is a continuous selection of subgradients of $\phi$ and both $\phi$ and $\psi$ are continuous over $$, it follows that$\phi$is continuously differentiable with$\phi^{\prime}=\psi$on$$. Hence, with$\phi$convex and$\phi(t)\geq 0=\phi(0)$for all$t\in$, we conclude that$\phi^{\prime}(0)=\langle\nabla h(x)-y,p-x\rangle\geq 0$ and thus we obtain the result. ∎

The basic ingredient for establishing connections in the Bregman framework is a generalization of the rule of cosines which is known in the literature as the “three-point identity” and will be the main tool for deriving the main estimations for our analysis. Being more precise, we have the following lemma:

Let $h$ be a Bregman function on $\mathcal{X}$. Then, for all $p\in\mathcal{X}$ and all $x,x^{\prime}\in\mathcal{X}^{\circ}$, we have:

The proof of this lemma follows as in the classic Bregman case so we omit it and proceed to derive some key bounds for the Bregman divergence before and after a mirror step:

By the three-point identity established in Lemma B.2, we get:

Due to (B.1) and the fact that $x^{+}=P_{x}(v)$ so $\nabla h(x)+v\in\partial h(x^{+})$, we get the result. ∎

Thanks to the above estimations, we obtain the following inequalities relating the Bregman divergence between two prox-steps:

Let $h$ be a Bregman function compatible on $\mathcal{X}$. Letting $x^{+}_{1}=P_{x}(v_{1})$ and $x^{+}_{2}=P_{x}(v_{2})$, we have:

For the first inequality, by applying Proposition B.1 for $x^{+}_{2}=P_{x}(v_{2})$, we get:

For the second inequality, we need to bound $\langle v_{2},x^{+}_{2}-x^{+}_{1}\rangle-D_{h}(x^{+}_{2},x)$. In particular, applying again Proposition B.1 for $p=x^{+}_{2}$, we get:

So, combining the above inequalities we get:

and thus we get the second inequality as well. ∎

Appendix C Main bounds and energy inequality

In this appendix, we shall provide the bound of the variation of the operators, i.e.,

that lies in the core of our analysis. To begin with, we recall that $(\mathcal{X},\lVert\cdot\rVert_{x})$ is a regular Finsler space, i.e., $\lVert v\rVert_{x,\ast}/\lVert v\rVert_{x^{\prime},\ast}=1+\operatorname{\mathcal{O}}(\lVert x-x^{\prime}\rVert_{x})$. However, in what follows we shall assume the more general condition:

It is straightforward for one to observe that a regular Finsler space satisfies (C.2) for $\beta=1$.

Owning this regularity geometrical property for the problem’s domain we shall proceed into showing that

is uniformly bounded. More precisely, we have the following lemma.

Suppose that $V$ satisfies (MB). Then, the sequence $\lVert V(X_{t+1/2})-V(X_{t})\rVert^{2}_{X_{t},\ast}$ is bounded. In particular, the following inequality holds:

It suffices to show that: $\lVert V(X_{t+1/2})-V(X_{t})\rVert_{X_{t+1/2},\ast}$ is bounded. More precisely, by the triangle inequality we have:

Let us now bound the (RHS) part of (C.5) term by term. In particular, we have:

For the first term $\lVert V(X_{t+1/2})\rVert_{X_{t+1/2},\ast}$ we readily get due to (MB):

For the second term $\lVert V(X_{t})\rVert_{X_{t+1/2},\ast}$, we have:

Therefore, it suffices to show that the quantity $\lVert X_{t}-X_{t+1/2}\rVert_{X_{t}}+\lVert X_{t}-X_{t+1/2}\rVert_{X_{t+1/2}}$ is bounded from above. Indeed, we have:

where the last inequality is obtained due to (MB). Moreover, due to (14) we get:

Hence, due to the local strong convexity (14) of $h$, we get:

Moreover, by combining (C.7) and (C.11) we get:

Summarizing, (C.5) combined with (C.7) and (C.12) yields:

We now proceed to prove the energy inequality stated in Lemma C.2.

For all $x\in\mathcal{X}$, the iterates $X_{t}$ of AdaProx satisfy the recursive bound:

The result follows directly by setting $X_{1}^{+}=X_{t+1/2}$, $X_{2}^{+}=X_{t+1}$, $x=X_{t}$, $v_{1}=-\gamma_{t}V(X_{t})$ and $v_{2}=-\gamma_{t}V(X_{t+1/2})$ in Proposition B.2. ∎

Appendix D Rate interpolation guarantees

In this appendix, we provide the proof of the the regime-agnostic rate interpolation guarantees of the UniProx. In order, to provide the necessary the respective rates we shall provide an intermediate result concerning the case of (MS). Formally, we have the following lemma.

Assume $V$ satisfies (MS) and $X_{t},X_{t+1/2}$ are the iterates of AdaProx. Then, the following hold:

The sequence $\lVert V(X_{t+1/2})-V(X_{t})\rVert^{2}_{X_{t+1/2},\ast}$ is summable. In particular, we have:

Since $\gamma_{t}$ is decreasing and bounded from below ($\gamma_{t}\geq 0$), then we readily obtain that its limit exists and more precisely we have:

Let us now assume that $\gamma_{\infty}=0$. Then, by recalling (C.14):

By rearranging the above and telescoping $t=1,\dotsc,T$ we get:

whereas, by applying Fenchel-Young inequality to the above we readily get:

Therefore, by the definition (MS) we have:

Now, by setting $p=x^{\ast}$ with $x^{\ast}$ being a solution of (VI) and using the fact that $\langle V(X_{t+1/2}),X_{t+1/2}-x^{\ast}\rangle\geq 0$ and $D(x^{\ast},X_{1})\leq D^{\prime}$ (by the compatibility of $h$), we obtain:

In addition, since $1/\gamma_{T+1}\to+\infty$, by the fact that $\gamma_{t}\to 0$, this yields that:

which is a contradiction. Hence, we get that:

In order to prove our second claim, we first recall the definition of $\gamma_{t}$:

whereas by developing and rearranging we have:

Hence, by taking limits on both sides we get:

where $0\leq\frac{1}{\gamma_{\infty}^{2}}-1<+\infty$, since $0<\gamma_{\infty}\leq 1$ and therefore the result follows. ∎

We start our analysis rearranging (C.14). In particular, by telescoping $t=1,\dotsc,T$ we get:

On the other hand, since $V$ is monotone, we readily get:

Thus, combining (D.20) and (D.19), dividing by $\sum_{t=1}^{T}\gamma_{t}$ and setting $\bar{X}_{T}=\left[\sum_{t=1}^{T}\gamma_{t}\right]^{-1}\sum_{t=1}^{T}\gamma_{t}X_{t+1/2}$ we get:

whereas, by applying Fenchel-Young inequality to the above we readily get:

Thus, if $\mathcal{C}$ is a compact neighbourhood of the solution set $\mathcal{X}^{\ast}$, considering that by (14):

and taking suprema on both sides, yields:

Therefore, in order to determine the convergence speed of $\overline{X}_{T}$ under (MB), we shall examine the asymptotic behaviour of each term of the nominator on the (RHS) of (D.31). In particular, we have the following:

For the first term: we readily get by the compactness of $\mathcal{C}$,

by the compatibility of the regularizer $h$.

For the second term: $\sum_{t=1}^{T}\gamma_{t}^{2}\lVert V(X_{t+1/2})-V(X_{t})\rVert_{X_{t+1/2}}^{2}$, we have:

Hence, $\gamma_{t}$ is non-increasing and therefore $(\gamma_{t}^{2}-\gamma_{t+1}^{2}\geq 0)$, and $\gamma_{t}\leq 1$ the above becomes:

with the last inequality being obtained by Lemma F.1 which combined with (MB) yields:

Finally, for $\sum_{t=1}^{T}\gamma_{t}$, we have the following upper-bound

Now, by combining (D.25), (D.27) and (D.29) we readily get that under (MB) we get that:

Case 2: Convergence under (MS).

We now suppose that $V$ satisfies (MS) condition. By applying Lemma D.1 along with :

by examining the asymptotic behaviour term by term, we get:

For the first term $D(x^{\ast},X_{1})$, since $x^{\ast}\in\operatorname{dom}V=\operatorname{dom}h$ and $X_{1}\in\operatorname{dom}\partial h$, we have:

For the second term $\sum_{t=1}^{T}\gamma_{t}^{2}\lVert V(X_{t+1/2})-V(X_{t})\rVert_{X_{t+1/2},\ast}^{2}$we have:

with $\gamma_{\infty}=\inf_{t}\gamma_{t}>0$.

Appendix E Last iterate’s convergence analysis

In this appendix, we establish the convergence of the sequence generated by (AdaProx), i.e., its so-called last iterate. In particular, we show that the actual iterates (before averaging) of AdaProx converge towards the solution set $\mathcal{X}^{\ast}$.This result comprises of two parts: first we extract convergent subsequences of $X_{t},X_{t+1/2}$ to the said set; then we apply the "trapping" argument described in Section 6 .

Suppose that $V$ satisfies (MB) (respectively (MS)) and $X_{t},X_{t+1/2}$ are the iterates of AdaProx. Then, the following hold:

$\lVert X_{t+1/2}-X_{t}\rVert\to 0$ while $t\to+\infty$

$\max\{D(X_{t+1/2},X_{t}),D(X_{t},X_{t+1/2})\}\leq\frac{2M^{2}}{K}\gamma_{t}^{2}$

For the proof of the first claim, we shall treat the cases of (MB) and (MS) individually.

Since $\gamma_{t}$ is decreasing and bounded from below, then we readily obtain that. its limit exists and more precisely:

We shall distinguish two individual cases:

$\gamma_{\infty}>0$: By recalling the definition of the adaptive step-size:

whereas by rearranging and developing we have:

Therefore, by taking limits on both sides:

which in turn by (E.4) yields $\sum_{t=1}^{+\infty}D(X_{t+1/2},X_{t})<+\infty$ and hence $D(X_{t+1/2},X_{t})\to 0$. Moreover, by applying (14):

Now, by recalling $\mu\lVert\cdot\rVert\leq\lVert\cdot\rVert_{x}$, we get:

$\gamma_{\infty}=0$: By the prox-step, we get:

where the last inequality is obtained due to (MB); which in turn yields:

Now, by recalling $\mu\lVert\cdot\rVert\leq\lVert\cdot\rVert_{x}$, we get:

and the result follows since we assumed that $\gamma_{t}\to 0$.

Case 2: Under (MS) condition.

Following similar reasoning as above, we have:

which by taking limits on both sides and by applying Lemma D.1 we get that:

Therefore, $D(X_{t+1/2},X_{t})\to 0$, whereas by applying (14) we obtain:

Now, by recalling $\mu\lVert\cdot\rVert\leq\lVert\cdot\rVert_{x}$, we get:

and the result follows. On the other hand, for the second claim, we have by the prox-step:

Therefore, by following the same reasoning with the first claim, we get:

and hence since $D(\cdot,\cdot)\geq 0$, we have:

Suppose that $V$ satisfies (MB) (respectively (MS)). Then, the iterates $X_{t},X_{t+1/2}$ of AdaProx possess convergent subsequences towards the equilibrium set $\mathcal{X}^{\ast}$.

By Lemma E.1, it suffices to show that $X_{t+1/2}$ possesses such a subsequence. Assume to the contrary that it does not. That implies that:

Now, by setting $p=x^{\ast}$ for some $x^{\ast}\in\mathcal{X}^{\ast}$ in (C.14), we get:

whereas by telescoping $t=1,\dotsc,T$ we obtain:

Having this established this general setting, we shall examine the asymptotic behaviour term by term for each regularity case individually, which in both cases shall lead to a contradiction.

Case 1: Under (MB) condition.

For the first term: $\sum_{t=1}^{T}\gamma_{t}$, due to (AdaProx) we have by (D.29) that:

For the second term $\dfrac{\sum_{t=1}^{T}\gamma_{t}^{2}\lVert V(X_{t+1/2})-V(X_{t})\rVert^{2}_{X_{t+1/2},\ast}}{\sum_{t=1}^{T}\gamma_{t}}$, we first examine the denominator. In particular, due to (AdaProx) we get:

So, by combining (E.21) and (E.23) we readily obtain:

Therefore, by letting $T\to+\infty$, the inequality (E.20) yields $D(x^{\ast},X_{T})\to-\infty$, contradiction.

Case 2: Under (MS) condition.

Examining the asymptotic behaviour of (E.20) term by term under the light of (MS) condition we get the following:

For $\sum_{t=1}^{T}\gamma_{t}$, (MS) guarantees by (D.36):

For $\frac{\sum_{t=1}^{T}\gamma_{t}^{2}\lVert V(X_{t+1/2})-V(X_{t})\rVert^{2}_{X_{t+1/2,\ast}}}{\sum_{t=1}^{T}\gamma_{t}}$, (D.1) guarantees:

Therefore, y letting $T\to+\infty$, the inequality (E.20) yields that $D(x^{\ast},X_{T})\to-\infty$, a contradiction. ∎

Having all this at hand, we are finally in the position to prove the main result of this section; namely the convergence of the actual iterates of the method. For that we will need an intermediate lemma that shall allow us to pass from a convergent subsequence to global convergence (see also , ).

Hence, $\limsup_{t}\alpha_{t}\leq\liminf_{t}\alpha_{t}+\delta$. Since, $\delta$ is chosen arbitrarily the result follows. ∎

Once more, we shall treat each regularity class individually.

Case 1: Under (MB) condition.

For the (MB), b y denoting $\lim_{t\to+\infty}\gamma_{t}=\gamma_{\infty}$ case we shall consider two cases for the asymptotic behaviour of the step-size $\gamma_{t}$.

$\gamma_{\infty}>0$: By recalling the definition of $\gamma_{t}$:

Therefore, by recalling (C.14), we have for solution of (VI), $x^{\ast}\in\mathcal{X}$

which enables us to directly apply Lemma E.2 for $\alpha_{t}=D(x^{\ast},X_{t})$, $\beta_{t}=\gamma_{t}\langle V(X_{t+1/2}),X_{t+1/2}-x^{\ast}\rangle$ and $\varepsilon_{t}=\gamma_{t}^{2}\lVert V(X_{t+1/2})-V(X_{t})\rVert_{X_{t+1/2},\ast}^{2}$.

$\gamma_{\infty}=0$: Fix an equilibrium $x^{\ast}\in\mathcal{X}^{\ast}$ and consider the "Bregman zone":

By the assumption for the regularizer $h$, it follows that there exists some $\delta>0$ such that:

is contained in $D_{\varepsilon}$. Hence, by regularity assumption for the (3), it follows that:

whereas by Lemma B.2 and after rearranging we get:

$X_{t}\in D_{2\varepsilon}\setminus D_{\varepsilon}$: Then, $\langle V(X_{t}),X_{t}-x^{\ast}\rangle\geq c>0$. So,

Now, provided that $\frac{2M^{2}\gamma_{t}^{2}}{K}\leq c\gamma_{t}$ or equivalently $\gamma_{t}\leq\frac{cK}{2M^{2}}$. we get: $D(x^{\ast},X_{t+1/2})\leq 2\varepsilon$.

$X_{t}\in D_{\varepsilon}$: Then, in this case we have:

Again, provided that $\frac{2M^{2}}{K}\gamma_{t}^{2}\leq\varepsilon$ or equivalently $\gamma_{t}\leq\frac{\sqrt{2\varepsilon K}}{2M}$ we get $D(x^{\ast},X_{t+1/2})\leq 2\varepsilon$

Therefore, by summarizing the above we get that if $\gamma_{t}\leq\min\{\frac{\sqrt{2\varepsilon K}}{2M},\frac{cK}{2M^{2}}\}$, we have that $X_{t+1/2}\in D_{2\varepsilon}$ whenever $X_{t}\in D_{2\varepsilon}$. Going further, due to Proposition B.2 by setting $p=x^{\ast}$, $x_{1}=X_{t+1/2}$, $x_{2}^{+}=X_{t+1}$, $x=X_{t}$, $v_{1}=-\gamma_{t}V(X_{t+1/2})$ and $v_{2}=-\gamma_{t}V(X_{t+1/2})$ we get:

whereas by applying Fenchel’s inequality we obtain:

Now, since $\frac{K}{2}\lVert X_{t+1}-X_{t+1/2}\rVert_{X_{t+1/2}}^{2}-D(X_{t+1},X_{t+1/2})\leq 0$ by (14) we get:

which, in turn, by (C.13) the above yields:

with $C=2M+\beta\frac{4M}{K}$. Recall that $X_{t+1/2}\in D_{2\varepsilon}$ by our previous claim. We now consider the following two cases:

$X_{t+1/2}\in D_{2\varepsilon}\setminus D_{\varepsilon}$: In this case: $\langle V(X_{t+1/2}),X_{t+1/2}-x^{\ast}\rangle\geq c>0$, so,

which holds provided that $\frac{C^{2}\gamma_{t}^{2}}{2K}\leq c\gamma_{t}$ or equivalently $\gamma_{t}\leq\frac{2cK}{C^{2}}$,

$X_{t+1/2}\in D_{\varepsilon}$: First recall that:

Clearly, $D_{\varepsilon}(\alpha)$ is continuous relative to $\alpha$ and $\lim_{\alpha\to 0^{+}}D_{\varepsilon}(\alpha)=\varepsilon$. Therefore, we have:

Moreover, due to (E.48), we conclude that $D(x^{\ast},X_{t+1})\leq 2\varepsilon$, provided that $\gamma_{t}\leq\frac{\alpha^{\ast}}{2\mu C}{K}$.

We conclude that $X_{t+1}\in U_{2\varepsilon}$ provided that $X_{t}\in D_{2\varepsilon}$ and $\gamma_{t}\leq\min\{\frac{2cK}{M^{2}},\frac{\sqrt{2\varepsilon K}}{2M},\frac{\alpha^{\ast}}{2\mu C}{K}\}$. Since, $\gamma_{t}\to 0$ and $X_{t}\in D_{2\varepsilon}$ infinitely often (due to Proposition E.1) we conclude that $X_{t}\in D_{2\varepsilon}$ for all sufficiently large $t$. With $\varepsilon>0$ being arbitrary, the result follows.

Case 2: Under (MS) condition.

By plugging in $\alpha_{t}=D(x^{\ast},X_{t})$, $\beta_{t}=\gamma_{t}\langle V(X_{t+1/2},X_{t+1/2}-x^{\ast}\rangle$ and $\varepsilon_{t}=\gamma_{t}^{2}\lVert V(X_{t+1/2})-V(X_{t})\rVert^{2}_{X_{t+1/2},\ast}$ in Lemma E.2 and combine it with Lemma D.1, we get $\inf_{x^{\ast}\in\mathcal{X}^{\ast}}\lVert x^{\ast},X_{t}\rVert$ converges. Thus, the result follows by applying Proposition E.1 ∎

Appendix F Properties of Numerical Sequences

In this appendix, we provide the necessary inequality of numerical sequences. This inequality is due to Bach & Levy and Levy et al. and will play an indispensable role for establishing the last iterate convergence and universality of our method.

For all non-negative numbers $\alpha_{1},\dotsc\alpha_{t}$, the following inequality holds:

The lemma will proved by induction. The induction base $T=1$ holds, since:

Assume now that the lemma holds for $T-1$. Then, we are left to show that it also holds for $T$. Indeed, by the induction hypothesis, we get:

Thus, in order to complete the induction it suffices to show that:

By denoting $x=\alpha_{T}/(1+\sum_{t=1}^{T-1}\alpha_{t})$, the above equation is equivalent:

which can be straighforwardly checked since $H(x)=\log(x+1)-\frac{x}{1+x}\geq 0$ for all $x\geq 0$. Therefore, the result follows. ∎

References