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Register a new userSolving Non-Convex Non-Differentiable Min-Max Games using Proximal Gradient Method
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Added by
Babak Barazandeh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Min-max saddle point games appear in a wide range of applications in machine leaning and signal processing. Despite their wide applicability, theoretical studies are mostly limited to the special convex-concave structure. While some recent works generalized these results to special smooth non-convex cases, our understanding of non-smooth scenarios is still limited. In this work, we study special form of non-smooth min-max games when the objective function is (strongly) convex with respect to one of the players decision variable. We show that a simple multi-step proximal gradient descent-ascent algorithm converges to $epsilon$-first-order Nash equilibrium of the min-max game with the number of gradient evaluations being polynomial in $1/epsilon$. We will also show that our notion of stationarity is stronger than existing ones in the literature. Finally, we evaluate the performance of the proposed algorithm through adversarial attack on a LASSO estimator.
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Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex non-concave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the hidden convex-concave game. We prove that if the hidden game is strictly convex-concave then vanilla GDA converges not merely to local Nash, but typically to the von-Neumann solution. If the game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence guarantees are non-local, which as far as we know is a first-of-its-kind type of result in non-convex non-concave games. Finally, we discuss connections of our framework with generative adversarial networks.
We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem gradient-descent-ascent dynamics can exhibit a variety of behaviors antithetical to convergence to the game theoretically meaningful min-max solution. Specifically, different forms of recurrent behavior (including periodicity and Poincare recurrence) are possible as well as convergence to spurious (non-min-max) equilibria for a positive measure of initial conditions. At the technical level, our analysis combines tools from optimization theory, game theory and dynamical systems.
The min-max optimization problem, also known as the saddle point problem, is a classical optimization problem which is also studied in the context of zero-sum games. Given a class of objective functions, the goal is to find a value for the argument which leads to a small objective value even for the worst case function in the given class. Min-max optimization problems have recently become very popular in a wide range of signal and data processing applications such as fair beamforming, training generative adversarial networks (GANs), and robust machine learning, to just name a few. The overarching goal of this article is to provide a survey of recent advances for an important subclass of min-max problem, where the minimization and maximization problems can be non-convex and/or non-concave. In particular, we will first present a number of applications to showcase the importance of such min-max problems; then we discuss key theoretical challenges, and provide a selective review of some exciting recent theoretical and algorithmic advances in tackling non-convex min-max problems. Finally, we will point out open questions and future research directions.
We introduce SPRING, a novel stochastic proximal alternating linearized minimization algorithm for solving a class of non-smooth and non-convex optimization problems. Large-scale imaging problems are becoming increasingly prevalent due to advances in data acquisition and computational capabilities. Motivated by the success of stochastic optimization methods, we propose a stochastic variant of proximal alternating linearized minimization (PALM) algorithm cite{bolte2014proximal}. We provide global convergence guarantees, demonstrating that our proposed method with variance-reduced stochastic gradient estimators, such as SAGA cite{SAGA} and SARAH cite{sarah}, achieves state-of-the-art oracle complexities. We also demonstrate the efficacy of our algorithm via several numerical examples including sparse non-negative matrix factorization, sparse principal component analysis, and blind image deconvolution.
We study decentralized non-convex finite-sum minimization problems described over a network of nodes, where each node possesses a local batch of data samples. In this context, we analyze a single-timescale randomized incremental gradient method, called GT-SAGA. GT-SAGA is computationally efficient as it evaluates one component gradient per node per iteration and achieves provably fast and robust performance by leveraging node-level variance reduction and network-level gradient tracking. For general smooth non-convex problems, we show the almost sure and mean-squared convergence of GT-SAGA to a first-order stationary point and further describe regimes of practical significance where it outperforms the existing approaches and achieves a network topology-independent iteration complexity respectively. When the global function satisfies the Polyak-Lojaciewisz condition, we show that GT-SAGA exhibits linear convergence to an optimal solution in expectation and describe regimes of practical interest where the performance is network topology-independent and improves upon the existing methods. Numerical experiments are included to highlight the main convergence aspects of GT-SAGA in non-convex settings.
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