Lazy-CFR: fast and near optimal regret minimization for extensive games with imperfect information

TL;DR

Lazy-CFR introduces a novel lazy update technique that reduces computation in extensive games with imperfect information, achieving near-optimal regret bounds and significantly faster convergence than traditional CFR methods.

Contribution

The paper proposes Lazy-CFR, a new variant of CFR that reduces traversal complexity and tightens regret bounds through a novel lazy update technique and analysis.

Findings

Lazy-CFR needs only O(√|I|) information set traversals per round.

Lazy-CFR achieves almost the same regret bounds as vanilla CFR.

Experimental results show Lazy-CFR outperforms vanilla CFR significantly.

Abstract

Counterfactual regret minimization (CFR) is the most popular algorithm on solving two-player zero-sum extensive games with imperfect information and achieves state-of-the-art performance in practice. However, the performance of CFR is not fully understood, since empirical results on the regret are much better than the upper bound proved in \cite{zinkevich2008regret}. Another issue is that CFR has to traverse the whole game tree in each round, which is time-consuming in large scale games. In this paper, we present a novel technique, lazy update, which can avoid traversing the whole game tree in CFR, as well as a novel analysis on the regret of CFR with lazy update. Our analysis can also be applied to the vanilla CFR, resulting in a much tighter regret bound than that in \cite{zinkevich2008regret}. Inspired by lazy update, we further present a novel CFR variant, named Lazy-CFR. Compared to traversing $O(|\mathcal{I}|)$ information sets in vanilla CFR, Lazy-CFR needs only to traverse $O(\sqrt{|\mathcal{I}|})$ information sets per round while keeping the regret bound almost the same, where $\mathcal{I}$ is the class of all information sets. As a result, Lazy-CFR shows better convergence result compared with vanilla CFR. Experimental results consistently show that Lazy-CFR outperforms the vanilla CFR significantly.