Zeroth-Order Algorithms for Nonconvex Minimax Problems with Improved Complexities

Abstract

In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (\texttt{ZO-GDA}) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step Ascent (\texttt{ZO-GDMSA}) algorithm that significantly improves the oracle complexity of \texttt{ZO-GDA}. We then consider stochastic versions of \texttt{ZO-GDA} and \texttt{ZO-GDMSA}, to handle stochastic nonconvex minimax problems. For this case, we provide oracle complexity results under two assumptions on the stochastic gradient: (i) the uniformly bounded variance assumption, which is common in traditional stochastic optimization, and (ii) the Strong Growth Condition (SGC), which has been known to be satisfied by modern over-parametrized machine learning models. We establish that under the SGC assumption, the complexities of the stochastic algorithms match that of deterministic algorithms. Numerical experiments are presented to support our theoretical results.

To appear in the Journal of Global Optimization

References in corpus (9)

• Practical Bayesian Optimization of Machine Learning Algorithms
• ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models
• Delving into Transferable Adversarial Examples and Black-box Attacks
• Evolutionary Algorithms for Reinforcement Learning
• Connecting Generative Adversarial Networks and Actor-Critic Methods
• Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems
• Efficient Algorithms for Smooth Minimax Optimization
• Online and Bandit Algorithms for Nonstationary Stochastic Saddle-Point Optimization
• Poincaré Recurrence, Cycles and Spurious Equilibria in Gradient-Descent-Ascent for Non-Convex Non-Concave Zero-Sum Games

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• New First-Order Algorithms for Stochastic Variational Inequalities
• Zeroth-Order Methods for Convex-Concave Minmax Problems: Applications to Decision-Dependent Risk Minimization
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