🤖 AI Summary
This paper addresses the computation of mixed Nash equilibria in two-player zero-sum games with continuous strategy spaces. We propose a particle-based parametric algorithm that models mixed strategies as atomic measures and jointly optimizes particle locations and weights. Our key theoretical contribution is the first establishment of an explicit connection between the particle update rule and the implicit time discretization of the Wasserstein–Fisher–Rao (WFR) gradient flow; under non-degeneracy conditions, we prove local exponential convergence. Unlike conventional grid-based discretizations—which suffer from curse-of-dimensionality in high-dimensional strategy spaces—our method avoids spatial discretization entirely, requiring only first-order oracle access to the payoff function. Experiments demonstrate its efficacy in solving joint adversarial distribution–model parameter optimization problems, notably in maximum-margin classification and distributionally robust neural network training.
📝 Abstract
We consider the problem of computing mixed Nash equilibria of two-player zero-sum games with continuous sets of pure strategies and with first-order access to the payoff function. This problem arises for example in game-theory-inspired machine learning applications, such as distributionally-robust learning. In those applications, the strategy sets are high-dimensional and thus methods based on discretisation cannot tractably return high-accuracy solutions. In this paper, we introduce and analyze a particle-based method that enjoys guaranteed local convergence for this problem. This method consists in parametrizing the mixed strategies as atomic measures and applying proximal point updates to both the atoms' weights and positions. It can be interpreted as a time-implicit discretization of the"interacting"Wasserstein-Fisher-Rao gradient flow. We prove that, under non-degeneracy assumptions, this method converges at an exponential rate to the exact mixed Nash equilibrium from any initialization satisfying a natural notion of closeness to optimality. We illustrate our results with numerical experiments and discuss applications to max-margin and distributionally-robust classification using two-layer neural networks, where our method has a natural interpretation as a simultaneous training of the network's weights and of the adversarial distribution.
