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

AI summary

Overview

Research area: theoretical machine learning at the intersection of nonconvex min-max optimization, game theory, and neural network training. Technical level: Advanced. Scope: the paper develops convergence guarantees for alternating gradient descent-ascent in hidden convex-concave zero-sum games where each player is parameterized by a two-layer neural network, using overparameterization, hidden convexity, two-sided Polyak–Łojasiewicz conditions, and random matrix theory. The arXiv paper is 2512.00389v1 [cs.LG], dated 29 Nov 2025.

What This Paper Is About

Many AI systems use zero

Authors’ 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-Łojasiewicz (PŁ) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.