Game Theoretic Optimization via Gradient-based Nikaido-Isoda Function
This work addresses the problem of equilibrium computation in game theory, relevant for applications like generative adversarial networks, but it is incremental as it builds on existing Nikaido-Isoda function concepts.
The paper tackles the challenge of efficiently computing Nash equilibria in multi-player games by introducing the Gradient-based Nikaido-Isoda (GNI) function, which provides error bounds and enables gradient descent to converge sublinearly to first-order stationary points, with linear convergence in specific cases like bilinear min-max games.
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.