Develop a theoretically grounded and practically robust minimax solver

Develop a minimax optimization solver that is both theoretically grounded and practically robust, particularly for the non-convex and non-concave settings characteristic of deep-learning applications.

Background

The paper describes persistent instability in minimax optimization, including oscillations, limit cycles, and divergence, especially when standard gradient descent-ascent is applied to non-convex and non-concave deep-learning objectives. Existing approaches either depend on restrictive curvature or timescale assumptions, introduce inductive biases through regularization, or incur substantial computational overhead without guaranteeing stability in general cases.

Against this background, the authors identify the development of a solver combining rigorous theoretical guarantees with practical robustness as an unresolved challenge. The paper proposes Controlled-Minimax as one response by modeling training dynamics as a controlled second-order ordinary differential equation and proving local exponential stability, but the broader challenge is stated explicitly in the introduction.