  As one of the most popular methods in the field of reinforcement learning,
Q-learning has received increasing attention. Recently, there have been more
theoretical works on the regret bound of algorithms that belong to the
Q-learning class in different settings. In this paper, we analyze the
cumulative regret when conducting Nash Q-learning algorithm on 2-player
turn-based stochastic Markov games (2-TBSG), and propose the very first gap
dependent logarithmic upper bounds in the episodic tabular setting. This bound
matches the theoretical lower bound only up to a logarithmic term. Furthermore,
we extend the conclusion to the discounted game setting with infinite horizon
and propose a similar gap dependent logarithmic regret bound. Also, under the
linear MDP assumption, we obtain another logarithmic regret for 2-TBSG, in both
centralized and independent settings.
