← 研踪 PaperTrailarXiv:2611.01649 · v1
Near-Optimal $\sqrt{K}$-Regret for Online Learning in Uninformed Markov Games
首次提交 2026-11-01 14:56
版本更新 2026-11-01 14:56
雷达观测 2026-11-04 23:43
摘要
We study online learning in two-player uninformed Markov games, where the opponent may follow an arbitrary, unknown policy in each episode and the learner does not observe the opponent's actions. [TWYS21] showed that achieving sublinear external regret requires an exponential dependence on the episode length $H$. Motivated by this limitation, they studied the weaker notion of Nash-value regret and obtained an $\widetilde{O}(K^{2/3})$ regret bound, where $K$ is the number of episodes. More recently, [LLZR26] introduced a stronger notion called empirical Nash-value regret and achieved an $\widetilde{O}(\sqrt{K})$ bound when the opponent's non-stationarity is moderate, including the case where the opponent follows a fixed policy. However, whether the $\sqrt{K}$ rate is achievable in the worst case remained open. In this paper, we resolve this open question by developing an algorithm that achieves $\widetilde{O}(H^{3/2}S\sqrt{AK})$ empirical Nash-value regret against arbitrary opponent policies, where $S$ is the number of states and $A$ is the number of learner's actions. Specifically, our algorithm combines global and local planning by maintaining a collection of candidate predictions of future losses, each associated with a policy. Global planning learns which candidates to favor across episodes, while local planning combines their policies differently at each state to make decisions. A key ingredient is to coordinate these two components to control the discrepancy between local loss estimates and the total loss of the executed policy, using only the learner's observed trajectories.
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v1 · 2026-11-01 14:56 · 最新观测版本
Near-Optimal $\sqrt{K}$-Regret for Online Learning in Uninformed Markov Games
We study online learning in two-player uninformed Markov games, where the opponent may follow an arbitrary, unknown policy in each episode and the learner does not observe the opponent's actions. [TWYS21] showed that achieving sublinear external regret requires an exponential dependence on the episode length $H$. Motivated by this limitation, they studied the weaker notion of Nash-value regret and obtained an $\widetilde{O}(K^{2/3})$ regret bound, where $K$ is the number of episodes. More recently, [LLZR26] introduced a stronger notion called empirical Nash-value regret and achieved an $\widetilde{O}(\sqrt{K})$ bound when the opponent's non-stationarity is moderate, including the case where the opponent follows a fixed policy. However, whether the $\sqrt{K}$ rate is achievable in the worst case remained open. In this paper, we resolve this open question by developing an algorithm that achieves $\widetilde{O}(H^{3/2}S\sqrt{AK})$ empirical Nash-value regret against arbitrary opponent policies, where $S$ is the number of states and $A$ is the number of learner's actions. Specifically, our algorithm combines global and local planning by maintaining a collection of candidate predictions of future losses, each associated with a policy. Global planning learns which candidates to favor across episodes, while local planning combines their policies differently at each state to make decisions. A key ingredient is to coordinate these two components to control the discrepancy between local loss estimates and the total loss of the executed policy, using only the learner's observed trajectories.
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