Minimax-Optimal Policy Regret in Partially Observable Markov Games

Problem
This work addresses the gap in sequential decision-making under uncertainty in partially observable environments, specifically in the context of partially observable Markov games (POMGs). The challenge arises from the need to learn latent dynamics from incomplete observations while contending with strategic, adaptive opponents whose actions depend on the learner’s strategy. Existing regret notions are inadequate for this setting, necessitating a new approach to quantify policy regret effectively. This paper is a preprint and has not undergone peer review.

Method
The authors propose an epoch-based optimistic maximum-likelihood algorithm that achieves a policy regret of $\tilde{O}(\sqrt{T})$ for fixed problem parameters. The algorithm operates by selecting one policy per geometrically growing epoch, utilizing confidence sets that are cumulatively built from past observations. This design allows for efficient comparison of adversary responses across policies, maintaining a logarithmic cost in terms of the time horizon $T$. The algorithm’s performance is explicitly dependent on several factors: the horizon length, the memory of the adversary, the confidence radius, and the aggregate Eluder dimension of the observable-operator class. Additionally, the authors establish a matching lower bound for the regret, demonstrating that the $\sqrt{T}$ dependence is optimal, up to logarithmic factors.

Results
The proposed algorithm demonstrates a policy regret of $\tilde{O}(\sqrt{T})$, which is a significant improvement over existing methods in the literature. The results are benchmarked against standard regret measures in POMGs, although specific baseline algorithms are not named in the abstract. The authors also extend their findings to provide horizon-adaptive guarantees and consider adversaries with geometric fading memory, further enhancing the applicability of their approach.

Limitations
The authors acknowledge that their results are contingent on fixed problem parameters and do not account for dynamic changes in the environment or adversary strategies beyond the specified memory model. Additionally, the algorithm’s performance may degrade in highly complex environments where the assumptions of the observable-operator class do not hold. The paper does not explore the computational complexity of implementing the proposed algorithm, which could be a practical limitation in real-world applications.

Why it matters
This research contributes to the understanding of strategic interactions in partially observable environments, providing a robust framework for policy learning against adaptive adversaries. The minimax-optimality of the proposed algorithm has significant implications for fields such as reinforcement learning, game theory, and multi-agent systems, where decision-making under uncertainty is critical. The findings can inform future work on developing more sophisticated algorithms that can handle a broader range of adversarial behaviors and environmental dynamics, as published in arXiv.