Solving Neural Min-Max Games: The Role of Architecture, Initialization & Dynamics

TL;DR

This paper provides a theoretical framework explaining why simple gradient methods often converge in neural min-max games, highlighting the roles of architecture, initialization, and overparameterization in guaranteeing global convergence.

Contribution

It introduces the first theoretical results for convergence in two-layer neural network min-max games, linking hidden convexity and overparameterization to convergence guarantees.

Findings

Global convergence to Nash equilibrium under certain conditions

Path-length bounds for gradient descent-ascent in min-max games

High-probability reduction to convex-concave geometry via overparameterization

Abstract

Many emerging applications - such as adversarial training, AI alignment, and robust optimization - can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions - spanning initialization, training dynamics, and network width - that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-{\L}ojasiewicz (P{\L}) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.