On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach

Abstract

Many tasks in modern machine learning can be formulated as finding equilibria in \emph{sequential} games. In particular, two-player zero-sum sequential games, also known as minimax optimization, have received growing interest. It is tempting to apply gradient descent to solve minimax optimization given its popularity and success in supervised learning. However, it has been noted that naive application of gradient descent fails to find some local minimax and can converge to non-local-minimax points. In this paper, we propose \emph{Follow-the-Ridge} (FR), a novel algorithm that provably converges to and only converges to local minimax. We show theoretically that the algorithm addresses the notorious rotational behaviour of gradient dynamics, and is compatible with preconditioning and \emph{positive} momentum. Empirically, FR solves toy minimax problems and improves the convergence of GAN training compared to the recent minimax optimization algorithms.

21 pages

References in corpus (2)

• GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium
• On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Games

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• Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization
• Local Stochastic Gradient Descent Ascent: Convergence Analysis and Communication Efficiency
• Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization
• Delta-STN: Efficient Bilevel Optimization for Neural Networks using Structured Response Jacobians