Efficient No-regret Learning Algorithms for Convergence to Nash Equilibrium in Potential and Markov Potential Games

Core Concepts

The authors 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 in potential games and Markov potential games.

Abstract

The paper studies potential games and Markov potential games under stochastic cost and bandit feedback. The authors propose a variant of the Frank-Wolfe algorithm that simultaneously achieves a Nash regret and a regret bound of O(T^4/5) for potential games, matching the best available result without using additional projection steps. The key highlights are: The algorithm uses a recursive gradient estimator to reduce the gradient estimation error, enabling fast convergence to the Nash equilibrium and sublinear regret for each player. For Markov potential games, the authors' algorithm improves the previous best Nash regret from O(T^5/6) to O(T^4/5), and also guarantees sublinear regret for individual players. The algorithm does not require any knowledge of the game, such as the distribution mismatch coefficient, providing more flexibility in practical implementation. Experimental results on a Markov congestion game validate the theoretical findings and demonstrate the practical effectiveness of the method.

Stats

The maximum size of the action space among all players is m. The number of players is n. The size of the state space in Markov potential games is S. The minimum stopping probability of the game is κ. The distribution mismatch coefficient is D_∞. The time horizon is T.

Quotes

"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."

Key Insights Distilled From

Convergence to Nash Equilibrium and No-regret Guarantee in (Markov) Potential Games

by Jing Dong,Ba... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.06516.pdf

Deeper Inquiries

What are the potential applications of the proposed algorithms beyond the game theory domain, such as in multi-agent reinforcement learning or distributed optimization

The proposed algorithms for potential games and Markov potential games have potential applications beyond the game theory domain. One application is in multi-agent reinforcement learning, where agents need to learn to cooperate or compete in complex environments. The algorithms can be used to help agents converge to Nash equilibria efficiently, leading to more stable and optimal strategies. Another application is in distributed optimization, where multiple agents need to coordinate their actions to optimize a global objective function. By applying the algorithms in a distributed setting, agents can learn to make decisions that collectively lead to better outcomes without the need for centralized control.

How can the algorithms be further extended to handle more general game settings, such as non-convex cost functions or incomplete information

To handle more general game settings, such as non-convex cost functions or incomplete information, the algorithms can be extended in several ways. For non-convex cost functions, the optimization techniques used in the algorithms can be adapted to handle non-convex optimization problems. This may involve using more advanced optimization methods or incorporating regularization techniques to prevent getting stuck in local minima. For incomplete information settings, techniques from Bayesian optimization or reinforcement learning with partial observability can be integrated to account for uncertainty and make decisions based on limited information. Additionally, game-theoretic concepts like Bayesian games can be utilized to model strategic interactions in settings with incomplete information.

Can the techniques used in the recursive gradient estimation be applied to other online learning problems to improve the sample efficiency and convergence rate

The techniques used in recursive gradient estimation can be applied to other online learning problems to improve sample efficiency and convergence rates. One potential application is in online convex optimization, where the goal is to optimize a convex function over a sequence of decision points. By using recursive gradient estimation, algorithms can adaptively estimate gradients and update decision variables, leading to faster convergence and better performance. Additionally, the techniques can be applied to online learning problems in reinforcement learning, where agents need to learn optimal policies in dynamic environments. By incorporating recursive gradient estimation, agents can learn more efficiently from limited feedback and adapt their strategies in real-time.