A Variational Inequality Approach to Bayesian Regression Games
This work addresses computational difficulties in Bayesian games for machine learning applications, offering a more general alternative to minimax formulations, though it is incremental in extending known results to broader classes of games.
The paper tackled the computational and theoretical limitations of Bayesian regression games by proving the existence and uniqueness of a Bayesian equilibrium for convex and smooth games using a variational inequality approach, and provided algorithms with strong convergence guarantees validated on real datasets.
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.