Online Learning for Uninformed Markov Games: Empirical Nash-Value Regret and Non-Stationarity Adaptation
Junyan Liu, Haipeng Luo, Zihan Zhang, Lillian J. Ratliff
TL;DR
This paper tackles online learning in two-player uninformed Markov games where the opponent's actions are hidden, introducing empirical Nash-value regret (Enr), a stronger metric than Nr that reduces to external regret under a fixed opponent. It analyzes epoch-V-learning to obtain a bound of $\tilde{O}(\eta C + \sqrt{K/\eta})$ for Enr and then designs a parameter-free meta-algorithm that adaptively restarts epoch-V-learning to achieve $\tilde{O}(\min\{ \sqrt{K}+(CK)^{1/3}, \sqrt{LK} \})$ for Enr, where $C$ captures non-stationarity and $L$ counts opponent policy switches. This approach recovers the known $\tilde{O}(\sqrt{K})$ external regret in stationary settings and avoids the worst-case $\tilde{O}(K^{2/3})$ regret by automatically adapting to the level of non-stationarity. The results provide a principled interpolation between regimes and open pathways for further tightening bounds or extending to broader multi-agent online learning scenarios.
Abstract
We study online learning in two-player uninformed Markov games, where the opponent's actions and policies are unobserved. In this setting, Tian et al. (2021) show that achieving no-external-regret is impossible without incurring an exponential dependence on the episode length $H$. They then turn to the weaker notion of Nash-value regret and propose a V-learning algorithm with regret $O(K^{2/3})$ after $K$ episodes. However, their algorithm and guarantee do not adapt to the difficulty of the problem: even in the case where the opponent follows a fixed policy and thus $O(\sqrt{K})$ external regret is well-known to be achievable, their result is still the worse rate $O(K^{2/3})$ on a weaker metric. In this work, we fully address both limitations. First, we introduce empirical Nash-value regret, a new regret notion that is strictly stronger than Nash-value regret and naturally reduces to external regret when the opponent follows a fixed policy. Moreover, under this new metric, we propose a parameter-free algorithm that achieves an $O(\min \{\sqrt{K} + (CK)^{1/3},\sqrt{LK}\})$ regret bound, where $C$ quantifies the variance of the opponent's policies and $L$ denotes the number of policy switches (both at most $O(K)$). Therefore, our results not only recover the two extremes -- $O(\sqrt{K})$ external regret when the opponent is fixed and $O(K^{2/3})$ Nash-value regret in the worst case -- but also smoothly interpolate between these extremes by automatically adapting to the opponent's non-stationarity. We achieve so by first providing a new analysis of the epoch-based V-learning algorithm by Mao et al. (2022), establishing an $O(ηC + \sqrt{K/η})$ regret bound, where $η$ is the epoch incremental factor. Next, we show how to adaptively restart this algorithm with an appropriate $η$ in response to the potential non-stationarity of the opponent, eventually achieving our final results.