This paper considers the problem of designing optimal algorithms for reinforcement learning in two-player zero-sum games. We focus on self-play algorithms which learn the optimal policy by playing against itself without any direct supervision. In a tabular episodic Markov game with $S$ states, $A$ max-player actions and $B$ min-player actions, the best existing algorithm for finding an approximate Nash equilibrium requires $\tilde{\mathcal{O}}(S^2AB)$ steps of game playing, when only highlighting the dependency on $(S,A,B)$. In contrast, the best existing lower bound scales as $\Omega(S(A+B))$ and has a significant gap from the upper bound. This paper closes this gap for the first time: we propose an optimistic variant of the \emph{Nash Q-learning} algorithm with sample complexity $\tilde{\mathcal{O}}(SAB)$, and a new \emph{Nash V-learning} algorithm with sample complexity $\tilde{\mathcal{O}}(S(A+B))$. The latter result matches the information-theoretic lower bound in all problem-dependent parameters except for a polynomial factor of the length of each episode. Towards understanding learning objectives in Markov games other than finding the Nash equilibrium, we present a computational hardness result for learning the best responses against a fixed opponent. This also implies the computational hardness for achieving sublinear regret when playing against adversarial opponents.

本文提出了楽观的Nash Q-learning算法，并使用了新的Nash V-learning算法，解决了在马尔可夫博弈环境中的奖励学习优化问题，且这个算法的采样复杂度比现有算法还要低.

使用自我博弈的近乎最优强化学习