Stochastic gradient descent with momentum (SGDM) is the dominant algorithm in many optimization scenarios, including convex optimization instances and non-convex neural network training. Yet, in the stochastic setting, momentum interferes with gradient noise, often leading to specific step size and momentum choices in order to guarantee convergence, set aside acceleration. Proximal point methods, on the other hand, have gained much attention due to their numerical stability and elasticity against imperfect tuning. Their stochastic accelerated variants though have received limited attention: how momentum interacts with the stability of (stochastic) proximal point methods remains largely unstudied. To address this, we focus on the convergence and stability of the stochastic proximal point algorithm with momentum (SPPAM), and show that SPPAM allows a faster linear convergence to a neighborhood compared to stochastic proximal point algorithm (SPPA) with a better contraction factor, under proper hyperparameter tuning. In terms of stability, we show that SPPAM depends on problem constants more favorably than SGDM, allowing a wider range of step size and momentum that lead to convergence.
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Between Stochastic and Adversarial Online Convex Optimization: Improved Regret Bounds via Smoothness
Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing adversarially poisoned rounds or shifts in the data distribution. To accomplish this goal, we introduce two key quantities associated with the loss sequence, that we call the cumulative stochastic variance and the adversarial variation. Our upper bounds are attained by instances of optimistic follow the regularized leader, and we design adaptive learning rates that automatically adapt to the cumulative stochastic variance and adversarial variation. In the fully i.i.d. case, our bounds match the rates one would expect from results in stochastic acceleration, and in the fully adversarial case they gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes for the cumulative stochastic variance and the adversarial variation.
Stochastic Gradient Descent-Ascent (SGDA) is one of the most prominent algorithms for solving min-max optimization and variational inequalities problems (VIP) appearing in various machine learning tasks. The success of the method led to several advanced extensions of the classical SGDA, including variants with arbitrary sampling, variance reduction, coordinate randomization, and distributed variants with compression, which were extensively studied in the literature, especially during the last few years. In this paper, we propose a unified convergence analysis that covers a large variety of stochastic gradient descent-ascent methods, which so far have required different intuitions, have different applications and have been developed separately in various communities. A key to our unified framework is a parametric assumption on the stochastic estimates. Via our general theoretical framework, we either recover the sharpest known rates for the known special cases or tighten them. Moreover, to illustrate the flexibility of our approach we develop several new variants of SGDA such as a new variance-reduced method (L-SVRGDA), new distributed methods with compression (QSGDA, DIANA-SGDA, VR-DIANA-SGDA), and a new method with coordinate randomization (SEGA-SGDA). Although variants of the new methods are known for solving minimization problems, they were never considered or analyzed for solving min-max problems and VIPs. We also demonstrate the most important properties of the new methods through extensive numerical experiments.
This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an $\ell_p$ norm ($0<p<1$) of the variables, which may appear in either the objective or the constraint. This kind of problems have strong applicability to a wide range of areas since usually the $\ell_p$ norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the $\ell_p$ norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the $\ell_p$ norm and the normal cones of the $\ell_p$ ball. For both problems, we derive the first-order necessary conditions under various constraint qualifications. We also derive the sequential optimality conditions for both problems and study the conditions under which these conditions imply the first-order necessary conditions. We point out that the sequential optimality conditions can be easily satisfied for iteratively reweighted algorithms and show that the global convergence can be easily derived using sequential optimality conditions.
Plug-and-Play methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Although Plug-and-Play methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data terms. In this work, we propose a new type of Plug-and-Play methods, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep neural network. Exploiting convergence results for proximal gradient descent algorithms in the non-convex setting, we show that the proposed Plug-and-Play algorithm is a convergent iterative scheme that targets stationary points of an explicit global functional. Besides, experiments show that it is possible to learn such a deep denoiser while not compromising the performance in comparison to other state-of-the-art deep denoisers used in Plug-and-Play schemes. We apply our proximal gradient algorithm to various ill-posed inverse problems, e.g. deblurring, super-resolution and inpainting. For all these applications, numerical results empirically confirm the convergence results. Experiments also show that this new algorithm reaches state-of-the-art performance, both quantitatively and qualitatively.
In this paper, we study smooth stochastic multi-level composition optimization problems, where the objective function is a nested composition of $T$ functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an $\epsilon$-stationary point of the problem. We show that the first algorithm, which is a generalization of \cite{GhaRuswan20} to the $T$ level case, can achieve a sample complexity of $\mathcal{O}(1/\epsilon^6)$ by using mini-batches of samples in each iteration. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to $\mathcal{O}(1/\epsilon^4)$. {\color{black}This modification not only removes the requirement of having a mini-batch of samples in each iteration, but also makes the algorithm parameter-free and easy to implement}. To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multi-level setting, obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle.
Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice, the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart~(Sim-GDA) in many settings. We prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a much slower rate. To our knowledge, this is the \emph{first} result of any setting showing that Alt-GDA converges faster than Sim-GDA by more than a constant. We further adapt the theory of integral quadratic constraints (IQC) and show that Alt-GDA attains the same rate \emph{globally} for a subclass of SCSC minimax problems. Empirically, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.
Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky, which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build a comprehensive Coq library of measure-theoretic probability theory and stochastic processes.
We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.
Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) is one of the most popular algorithms in distributed machine learning. However, its convergence properties for these complicated nonconvex problems is still largely unknown, because of the current technical limit. Therefore, in this paper, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problem - streaming PCA, which helps us to understand Aync-MSGD better even for more general problems. Specifically, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA by diffusion approximation. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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