Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent

TL;DR

This paper analyzes gradient descent ascent dynamics in a special class of non-convex non-concave zero-sum games called hidden zero-sum games, proving convergence to the von-Neumann equilibrium under certain conditions and extending results with regularization.

Contribution

It introduces the concept of hidden zero-sum games and proves convergence of GDA to the von-Neumann solution in this setting, including cases lacking strict convexity.

Findings

GDA converges to von-Neumann equilibrium in strictly convex-concave hidden zero-sum games.

Regularization ensures convergence in non-strict convexity cases.

Non-local convergence guarantees are established for non-convex non-concave games.

Abstract

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.