LEAD: Least-Action Dynamics for Min-Max Optimization

Adversarial formulations such as generative adversarial networks (GANs) have rekindled interest in two-player min-max games. A central obstacle in the optimization of such games is the rotational dynamics that hinder their convergence. Existing methods typically employ intuitive, carefully hand-designed mechanisms for controlling such rotations. In this paper, we take a novel approach to address this issue by casting min-max optimization as a physical system. We leverage tools from physics to introduce LEAD (Least-Action Dynamics), a second-order optimizer for min-max games. Next, using Lyapunov stability theory and spectral analysis, we study LEAD's convergence properties in continuous and discrete-time settings for bilinear games to demonstrate linear convergence to the Nash equilibrium. Finally, we empirically evaluate our method on synthetic setups and CIFAR-10 image generation to demonstrate improvements over baseline methods.