To help address these issues, a natural approach is to consider games with special structures. In this paper, we focus on congestion games.

Regret in stochastic multi-armed bandits traditionally measures the difference between the highest reward and either the arithmetic mean of accumulated rewards or the final reward. These conventional metrics often fail to address fairness among agents receiving rewards, particularly in settings where rewards are distributed across a population, such as patients in clinical trials. To address this, a recent body of work has introduced Nash regret, which evaluates performance via the geometric mean of accumulated rewards, aligning with the Nash social welfare function known for satisfying fairness axioms. To minimize Nash regret, existing approaches require specialized algorithm designs and strong assumptions, such as multiplicative concentration inequalities and bounded, non-negative rewards, making them unsuitable for even Gaussian reward distributions. We demonstrate that an initial uniform exploration phase followed by a standard Upper Confidence Bound (UCB) algorithm achieves near-optimal Nash regret, while relying only on additive Hoeffding bounds, and naturally extending to sub-Gaussian rewards. Furthermore, we generalize the algorithm to a broad class of fairness metrics called the $p$-mean regret, proving (nearly) optimal regret bounds uniformly across all $p$ values. This is in contrast to prior work, which made extremely restrictive assumptions on the bandit instances and even then achieved suboptimal regret bounds.

To help address these issues, a natural approach is to consider games with special structures. In this paper, we focus on congestion games.

To help address these issues, a natural approach is to consider games with special structures. In this paper, we focus on congestion games.

In this work, we extend the concept of the $p$-mean welfare objective from social choice theory (Moulin 2004) to study $p$-mean regret in stochastic multi-armed bandit problems. The $p$-mean regret, defined as the difference between the optimal mean among the arms and the $p$-mean of the expected rewards, offers a flexible framework for evaluating bandit algorithms, enabling algorithm designers to balance fairness and efficiency by adjusting the parameter $p$. Our framework encompasses both average cumulative regret and Nash regret as special cases. We introduce a simple, unified UCB-based algorithm (Explore-Then-UCB) that achieves novel $p$-mean regret bounds. Our algorithm consists of two phases: a carefully calibrated uniform exploration phase to initialize sample means, followed by the UCB1 algorithm of Auer, Cesa-Bianchi, and Fischer (2002). Under mild assumptions, we prove that our algorithm achieves a $p$-mean regret bound of $\tilde{O}\left(\sqrt{\frac{k}{T^{\frac{1}{2|p|}}}}\right)$ for all $p \leq -1$, where $k$ represents the number of arms and $T$ the time horizon. When $-1

Markov Potential Games (MPGs) form an important sub-class of Markov games, which are a common framework to model multi-agent reinforcement learning problems. In particular, MPGs include as a special case the identical-interest setting where all the agents share the same reward function. Scaling the performance of Nash equilibrium learning algorithms to a large number of agents is crucial for multi-agent systems. To address this important challenge, we focus on the independent learning setting where agents can only have access to their local information to update their own policy. In prior work on MPGs, the iteration complexity for obtaining $\epsilon$-Nash regret scales linearly with the number of agents $N$. In this work, we investigate the iteration complexity of an independent policy mirror descent (PMD) algorithm for MPGs. We show that PMD with KL regularization, also known as natural policy gradient, enjoys a better $\sqrt{N}$ dependence on the number of agents, improving over PMD with Euclidean regularization and prior work. Furthermore, the iteration complexity is also independent of the sizes of the agents' action spaces.

In this work, we study potential games and Markov potential games under stochastic cost and bandit feedback. We propose a variant of the Frank-Wolfe algorithm with sufficient exploration and recursive gradient estimation, which provably converges to the Nash equilibrium while attaining sublinear regret for each individual player. Our algorithm simultaneously achieves a Nash regret and a regret bound of $O(T^{4/5})$ for potential games, which matches the best available result, without using additional projection steps. Through carefully balancing the reuse of past samples and exploration of new samples, we then extend the results to Markov potential games and improve the best available Nash regret from $O(T^{5/6})$ to $O(T^{4/5})$. Moreover, our algorithm requires no knowledge of the game, such as the distribution mismatch coefficient, which provides more flexibility in its practical implementation. Experimental results corroborate our theoretical findings and underscore the practical effectiveness of our method.