Game Theoretic Optimization via Gradient-based Nikaido-Isoda Function

TL;DR

This paper introduces the Gradient-based Nikaido-Isoda (GNI) function, a novel merit function for efficiently computing Nash equilibria in multi-player games, with convergence guarantees and practical effectiveness demonstrated through experiments.

Contribution

The paper proposes the GNI function as a new tool for Nash equilibrium computation, providing convergence analysis and applicability to bilinear and quadratic games.

Findings

Gradient descent on GNI converges sublinearly to stationary points.

In bilinear and quadratic games, GNI is convex, enabling linear convergence.

Numerical experiments show GNI always converges to stationary points.

Abstract

Computing Nash equilibrium (NE) of multi-player games has witnessed renewed interest due to recent advances in generative adversarial networks. However, computing equilibrium efficiently is challenging. To this end, we introduce the Gradient-based Nikaido-Isoda (GNI) function which serves: (i) as a merit function, vanishing only at the first-order stationary points of each player's optimization problem, and (ii) provides error bounds to a stationary Nash point. Gradient descent is shown to converge sublinearly to a first-order stationary point of the GNI function. For the particular case of bilinear min-max games and multi-player quadratic games, the GNI function is convex. Hence, the application of gradient descent in this case yields linear convergence to an NE (when one exists). In our numerical experiments, we observe that the GNI formulation always converges to the first-order stationary point of each player's optimization problem.