Composability of Regret Minimizers

Regret minimization is a powerful tool for solving large-scale problems; it was recently used in breakthrough results for large-scale extensive-form-game solving. This was achieved by composing simplex regret minimizers into an overall regret-minimization framework for extensive-form-game strategy spaces. In this paper we study the general composability of regret minimizers. We derive a calculus for constructing regret minimizers for complex convex sets that are constructed from convexity-preserving operations on simpler convex sets. In particular, we show that local regret minimizers for the simpler sets can be composed with additional regret minimizers into an aggregate regret minimizer for the complex set. As an application of our framework we show that the CFR framework can be constructed easily from our framework. We also show how to construct a CFR variant for extensive-form games with strategy constraints. Unlike a recently proposed variant of CFR for strategy constraints by Davis, Waugh, and Bowling (2018), the algorithm resulting from our calculus does not depend on any unknown constants and thus avoids binary search.