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# Lazy-CFR: a fast regret minimization algorithm for extensive games with imperfect information

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### Abstract

In this paper, we focus on solving two-player zero-sum extensive games with imperfect information. Counterfactual regret minimization (CFR) is the most popular algorithm on solving such games 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 of CFR is that CFR has to traverse the whole game tree in each round, which is not tolerable in large scale games. In this paper, we present a novel technique, lazy update, which can avoid traversing the whole game tree in CFR. Further, we present a novel analysis on the CFR with lazy update. Our analysis can also be applied to the vanilla CFR, which results in a much tighter regret bound than that proved 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 the regret bound almost keep 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.

### Most cited references5

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### An analog of the minimax theorem for vector payoffs

(1956)
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### The complexity of two-person zero-sum games in extensive form

(1992)
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### Lossless abstraction of imperfect information games

(2007)
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### Author and article information

###### Journal
10 October 2018
###### Article
1810.04433