  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.
