Distributed Algorithms for $α$-Potential Functions in General-Sum Games

We study decentralized learning of Nash equilibria (NE) in infinite-horizon discounted Markov games under bandit feedback, focusing on Markov $α$-potential games. We develop KL-projected natural policy gradient (NPG) algorithms in two settings: an episodic setting with frozen policies during sampling and a fully online setting in which players receive a single realized cost sample per time step and update their policies asynchronously along a continuing trajectory. We establish finite-time high-probability NE regret bounds of order $\widetilde O(T^{-1/4})$ and $\widetilde O(T^{-2/15})$ for the episodic and fully online settings, respectively, up to fixed approximation terms. Crucially, our bounds eliminate the distribution-mismatch coefficient, which can scale prohibitively with the size of the state space, while accommodating potential approximation, estimation-oracle bias, and transition sensitivity. We further identify a state-wise potential structure that yields sharper guarantees with additive dependence on the potential approximation error $α$. We specialize the framework to independent-resource Markov congestion games (IMCGs), establish their approximate-potential and transition-sensitivity properties, and construct decentralized estimation oracles from realized costs. As an application, we introduce strategic online job scheduling on stochastic machines and obtain a scalable decentralized algorithm for learning stable dispatching policies. Overall, our results provide the first finite-time high-probability NE regret guarantees for fully online asynchronous decentralized learning in Markov $α$-potential games, remove distribution-mismatch coefficients from the regret bounds, accommodate fixed estimation-oracle bias, and provide scalable decentralized learning with finite-time guarantees for IMCGs.

We study last-iterate convergence in unknown two-player zero-sum matrix games with bandit payoff feedback and observed opponent actions. For games with $d$ actions per player, we develop an algorithm achieving a duality gap of $\widetilde{\mathcal{O}}(\sqrt{d/t})$ with high probability, simultaneously at every round $t$. This improves the dimension dependence of the best previously known guarantee by a factor of $d^{3/2}$. The rate matches a standard bandit lower bound, establishing minimax optimality in both the number of actions and the number of rounds, up to logarithmic factors. The algorithm is computationally efficient, requiring only $\mathcal{O}(d)$ time and memory per round. Our technical contribution is a joint design of adaptive averaging and corrected exponential weights that absorbs estimation variance, together with a potential argument that bounds phase durations.

The distributed setting for Saddle Problems (SPs) has recently emerged as a framework for various modern applications in machine learning and multiagent systems. Despite its relevance, the theoretical foundations of this setting have not yet been thoroughly established. In this paper, we advance this research direction by formalizing the distributed setup for SPs and providing rigorous definitions of communication and computational costs. Our main result is a novel decoupled method that achieves optimal communication cost within the zero-respecting framework. Our method is based on a multi-stage reduction to the decoupled minimization of residual norms, which yields strict improvements over the best known communication cost for the class and the long-standing oracle cost of the Extragradient method. Further, we show by a matching lower bound that our method is communication-optimal within the family of gradient-span algorithms. Finally, we study the extension of distributed SP into Variational Inequality Problem (VIP), which generalizes two-player zero-sum games to multiplayer general-sum games. We show that our decoupled method achieves a new state-of-the-art communication complexity for this broader class.