  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.
