  We consider the problem of finding Nash equilibrium for two-player turn-based
zero-sum games. Inspired by the AlphaGo Zero (AGZ) algorithm, we develop a
Reinforcement Learning based approach. Specifically, we propose
Explore-Improve-Supervise (EIS) method that combines "exploration", "policy
improvement"' and "supervised learning" to find the value function and policy
associated with Nash equilibrium. We identify sufficient conditions for
convergence and correctness for such an approach. For a concrete instance of
EIS where random policy is used for "exploration", Monte-Carlo Tree Search is
used for "policy improvement" and Nearest Neighbors is used for "supervised
learning", we establish that this method finds an $\varepsilon$-approximate
value function of Nash equilibrium in $\widetilde{O}(\varepsilon^{-(d+4)})$
steps when the underlying state-space of the game is continuous and
$d$-dimensional. This is nearly optimal as we establish a lower bound of
$\widetilde{\Omega}(\varepsilon^{-(d+2)})$ for any policy.
