A Variational Inequality Approach to Bayesian Regression Games

TL;DR

This paper introduces a variational inequality framework for Bayesian regression games, establishing existence and uniqueness of equilibrium, and providing algorithms with convergence guarantees, demonstrated through real data experiments.

Contribution

It develops a variational inequality approach to Bayesian regression games, proving equilibrium properties and offering scalable algorithms with convergence guarantees.

Findings

Proved existence and uniqueness of Bayesian equilibrium for certain convex games.

Reduced infinite-dimensional VI to high-dimensional VI or stochastic optimization in special cases.

Numerical experiments show the effectiveness of the proposed algorithms on real datasets.

Abstract

Bayesian regression games are a special class of two-player general-sum Bayesian games in which the learner is partially informed about the adversary's objective through a Bayesian prior. This formulation captures the uncertainty in regard to the adversary, and is useful in problems where the learner and adversary may have conflicting, but not necessarily perfectly antagonistic objectives. Although the Bayesian approach is a more general alternative to the standard minimax formulation, the applications of Bayesian regression games have been limited due to computational difficulties, and the existence and uniqueness of a Bayesian equilibrium are only known for quadratic cost functions. First, we prove the existence and uniqueness of a Bayesian equilibrium for a class of convex and smooth Bayesian games by regarding it as a solution of an infinite-dimensional variational inequality (VI) in Hilbert space. We consider two special cases in which the infinite-dimensional VI reduces to a high-dimensional VI or a nonconvex stochastic optimization, and provide two simple algorithms of solving them with strong convergence guarantees. Numerical results on real datasets demonstrate the promise of this approach.