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.
翻译:Bayesian 回归游戏是双玩者通用和 Bayesian 游戏的特殊类别, 学习者通过Bayesian 之前的一个 Bayesian 游戏部分了解对手的目标。 这种配方可以捕捉对手的不确定性, 并且对于学习者和对手可能相互冲突, 但不一定完全敌对目标的问题很有用。 虽然Bayesian 方法是标准迷你轴配方的一种比较普通的替代方法, 但Bayesian 回归游戏的应用由于计算困难而受到限制, 而Bayesian 平衡的存在和独特性只为二次曲线成本函数所知道。 首先, 我们证明Bayesian 平衡对于一类锥体和平滑的游戏的存在和独特性, 我们把它视为Hilbert 空间无限差异性不平等(VI) 的解决方案。 我们考虑了两个特殊案例, 无限维六降低为高维六或非凝固度优化, 并且提供了两种简单的算法, 以强烈的趋同保证来解决这些问题。 在真实数据设置上, 数值结果显示了这一方法的前景。