**Keywords:**two-player zero-sum Markov game, last-iterate convergence, path convergence, learning in games

**TL;DR:**This paper presents uncoupled algorithms for learning in two-player zero-sum Markov games with non-asymptotic convergence rates under bandit feedback.

**Abstract:**We revisit the problem of learning in two-player zero-sum Markov games, focusing on developing an algorithm that is *uncoupled*, *convergent*, and *rational*, with non-asymptotic convergence rates to Nash equilibrium. We start from the case of stateless matrix game with bandit feedback as a warm-up, showing an $\tilde{\mathcal{O}}(t^{-\frac{1}{8}})$ last-iterate convergence rate. To the best of our knowledge, this is the first result that obtains finite last-iterate convergence rate given access to only bandit feedback. We extend our result to the case of irreducible Markov games, providing a last-iterate convergence rate of $\tilde{\mathcal{O}}(t^{-\frac{1}{9+\varepsilon}})$ for any $\varepsilon>0$. Finally, we study Markov games without any assumptions on the dynamics, and show a *path convergence* rate, a new notion of convergence we defined, of $\tilde{\mathcal{O}}(t^{-\frac{1}{10}})$. Our algorithm removes the synchronization and prior knowledge requirement of Wei et al. (2021), which pursued the same goals as us for irreducible Markov games. Our algorithm is related to Chen et al. (2021) and Cen et al. (2021)'s and also builds on the entropy regularization technique. However, we remove their requirement of communications on the entropy values, making our algorithm entirely uncoupled.

**Submission Number:**5037

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