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

Deep Patel, Emmanouil-Vasileios Vlatakis-Gkaragkounis

Keywords: game theory, min-max optimization, neural networks, learning theory

TL;DR: We provide the first convergence guarantees and overparameterization bounds for alternating gradient methods in a special class of nonconvex-nonconcave zero-sum games where both the players are shallow neural networks.

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 \emph{hidden convexity} and \emph{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 alternating gradient-descent-ascent scheme in min-max games; and (b) we show that games with hidden convex–concave geometry reduce to settings satisfying two-sided Polyak–Łojasiewicz (PL) and smoothness conditions, which hold with high probability under overparameterization, using tools from random matrix theory.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 13652