Bayesian Learning in Episodic Zero-Sum Games
arXiv:2603.20604v1 Announce Type: new Abstract: We study Bayesian learning in episodic, finite-horizon zero-sum Markov games with unknown transition and reward models. We investigate a posterior algorithm in which each player maintains a Bayesian posterior over the game model, independently samples a game model at the beginning of each episode, and computes an equilibrium policy for the sampled model. We analyze two settings: (i) Both players use the posterior sampling algorithm, and (ii) Only one player uses posterior sampling while the opponent follows an arbitrary learning algorithm. In each setting, we provide guarantees on the expected regret of the posterior sampling agent. Our notion of regret compares the expected total reward of the learning agent against the expected total reward under equilibrium policies of the true game. Our main theoretical result is an expected regret bound for the posterior sampling agent of order $O(HS\sqrt{ABHK\log(SABHK)})$ where $K$ is the number o
arXiv:2603.20604v1 Announce Type: new Abstract: We study Bayesian learning in episodic, finite-horizon zero-sum Markov games with unknown transition and reward models. We investigate a posterior algorithm in which each player maintains a Bayesian posterior over the game model, independently samples a game model at the beginning of each episode, and computes an equilibrium policy for the sampled model. We analyze two settings: (i) Both players use the posterior sampling algorithm, and (ii) Only one player uses posterior sampling while the opponent follows an arbitrary learning algorithm. In each setting, we provide guarantees on the expected regret of the posterior sampling agent. Our notion of regret compares the expected total reward of the learning agent against the expected total reward under equilibrium policies of the true game. Our main theoretical result is an expected regret bound for the posterior sampling agent of order $O(HS\sqrt{ABHK\log(SABHK)})$ where $K$ is the number of episodes, $H$ is the episode length, $S$ is the number of states, and $A,B$ are the action space sizes of the two players. Experiments in a grid-world predator--prey domain illustrate the sublinear regret scaling and show that posterior sampling competes favorably with a fictitious-play baseline.
Executive Summary
This article presents a theoretical framework for Bayesian learning in episodic zero-sum Markov games, where two players with unknown transition and reward models engage in a series of finite-horizon games. The authors develop a posterior sampling algorithm, where each player maintains a Bayesian posterior over the game model, independently samples a game model at the beginning of each episode, and computes an equilibrium policy for the sampled model. The authors provide expected regret bounds for the posterior sampling agent and demonstrate the efficacy of this approach through experiments in a grid-world predator-prey domain. Overall, this article makes significant contributions to the field of game theory and machine learning, with potential applications in areas such as autonomous systems and multi-agent decision-making.
Key Points
- ▸ The authors develop a posterior sampling algorithm for Bayesian learning in episodic zero-sum Markov games.
- ▸ The algorithm involves maintaining a Bayesian posterior over the game model and independently sampling a game model at the beginning of each episode.
- ▸ The authors provide expected regret bounds for the posterior sampling agent and demonstrate the efficacy of this approach through experiments.
Merits
Strength of Theoretical Contributions
The article presents a rigorous and well-reasoned theoretical framework for Bayesian learning in episodic zero-sum Markov games, with expected regret bounds that provide a clear understanding of the algorithm's performance.
Methodological Advancements
The posterior sampling algorithm developed in this article offers a novel approach to Bayesian learning in games, leveraging the power of Bayesian inference to improve decision-making under uncertainty.
Empirical Validity
The experiments in a grid-world predator-prey domain demonstrate the efficacy of the posterior sampling algorithm, providing a clear illustration of its potential applications in real-world scenarios.
Demerits
Limitation in Handling Complex Environments
The article assumes a relatively simple game environment, with a finite number of states and actions. Further research is needed to extend this framework to more complex environments with continuous state and action spaces.
Assumptions on Opponent's Behavior
The article assumes that the opponent follows an arbitrary learning algorithm, which may not always be the case in real-world scenarios. Further research is needed to investigate the robustness of the posterior sampling algorithm to different opponent behaviors.
Expert Commentary
This article makes significant contributions to the field of game theory and machine learning, with potential applications in areas such as autonomous systems and multi-agent decision-making. The posterior sampling algorithm developed in this article offers a novel approach to Bayesian learning in games, leveraging the power of Bayesian inference to improve decision-making under uncertainty. While the article assumes a relatively simple game environment, further research is needed to extend this framework to more complex environments with continuous state and action spaces. Additionally, the article highlights the importance of Bayesian learning in games, where players must adapt to changing environments and opponents. This has implications for policy-making in areas such as economics and international relations.
Recommendations
- ✓ Further research is needed to extend the posterior sampling algorithm to more complex environments with continuous state and action spaces.
- ✓ The article highlights the importance of Bayesian learning in games, where players must adapt to changing environments and opponents. This has implications for policy-making in areas such as economics and international relations.
Sources
Original: arXiv - cs.LG
