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

arXiv: https://arxiv.org/abs/2512.00389 · HTML: https://mlsanity.com/paper/2512.00389/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2025.

## Built on by (most cited first, 1 of 1)

- [Online Learning on Hidden-Convex Losses via Algorithmic Equivalence: Optimal Regret, Geometric Barrier, and Bandit Feedback](https://mlsanity.com/paper/2605.26373/) (2026)

## Abstract (arXiv)

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-Lojasiewicz (PL) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.