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Counterfactual regret minimization (CFR) is the most popular algorithm onsolving two-player zero-sum extensive games with imperfect information andachieves state-of-the-art performance in practice. However, the performance ofCFR is not fully understood, since empirical results on the regret are muchbetter than the upper bound proved in \cite{zinkevich2008regret}. Another issueis that CFR has to traverse the whole game tree in each round, which istime-consuming in large scale games. In this paper, we present a noveltechnique, 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 analysiscan also be applied to the vanilla CFR, resulting in a much tighter regretbound than that in \cite{zinkevich2008regret}. Inspired by lazy update, wefurther present a novel CFR variant, named Lazy-CFR. Compared to traversing$O(|\mathcal{I}|)$ information sets in vanilla CFR, Lazy-CFR needs only totraverse $O(\sqrt{|\mathcal{I}|})$ information sets per round while keeping theregret bound almost the same, where $\mathcal{I}$ is the class of allinformation sets. As a result, Lazy-CFR shows better convergence resultcompared with vanilla CFR. Experimental results consistently show that Lazy-CFRoutperforms the vanilla CFR significantly.

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