  Learning strategies for imperfect information games from samples of
interaction is a challenging problem. A common method for this setting, Monte
Carlo Counterfactual Regret Minimization (MCCFR), can have slow long-term
convergence rates due to high variance. In this paper, we introduce a variance
reduction technique (VR-MCCFR) that applies to any sampling variant of MCCFR.
Using this technique, per-iteration estimated values and updates are
reformulated as a function of sampled values and state-action baselines,
similar to their use in policy gradient reinforcement learning. The new
formulation allows estimates to be bootstrapped from other estimates within the
same episode, propagating the benefits of baselines along the sampled
trajectory; the estimates remain unbiased even when bootstrapping from other
estimates. Finally, we show that given a perfect baseline, the variance of the
value estimates can be reduced to zero. Experimental evaluation shows that
VR-MCCFR brings an order of magnitude speedup, while the empirical variance
decreases by three orders of magnitude. The decreased variance allows for the
first time CFR+ to be used with sampling, increasing the speedup to two orders
of magnitude.
