Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path (2402.08998v1)

Abstract: We study the Stochastic Shortest Path (SSP) problem with a linear mixture transition kernel, where an agent repeatedly interacts with a stochastic environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the cost function or an upper bound of the expected length for the optimal policy. In this paper, we propose a new algorithm to eliminate these restrictive assumptions. Our algorithm is based on extended value iteration with a fine-grained variance-aware confidence set, where the variance is estimated recursively from high-order moments. Our algorithm achieves an $\tilde{\mathcal O}(dB_\sqrt{K})$ regret bound, where $d$ is the dimension of the feature mapping in the linear transition kernel, $B_$ is the upper bound of the total cumulative cost for the optimal policy, and $K$ is the number of episodes. Our regret upper bound matches the $\Omega(dB_*\sqrt{K})$ lower bound of linear mixture SSPs in Min et al. (2022), which suggests that our algorithm is nearly minimax optimal.

Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path

Introduction

The paper of Stochastic Shortest Path (SSP) problems provides an essential framework within reinforcement learning for scenarios where agents interact with uncertain environments to reach a goal while minimizing incurred costs. This paper introduces a novel algorithm aimed explicitly at learning linear mixture SSPs, an area of interest due to its potential for generalization in large state and action spaces via linear function approximation.

Key Contributions

The paper makes several notable contributions to the advancements in algorithms for SSPs:

1. Novel Algorithm: A computationally efficient algorithm for learning linear mixture SSPs is proposed, marking a significant stride towards addressing the challenge presented by large state and action spaces. This algorithm is especially noteworthy due to its employment of variance-aware and uncertainty-aware weights in solving weighted regression problems, a technique that refines the approximation of the optimal value function.
2. Theoretical Guarantees: Through meticulous theoretical analysis, the algorithm is shown to achieve an O(dB*sqrt(K)) regret bound, making it nearly minimax optimal. Notably, this bound is achieved without the restrictive assumptions previously necessary in related works, such as knowledge of the minimum positive cost or the expected length of the optimal policy.
3. Improved Variance Estimator: By introducing a recursive design for estimating high-order moments, the paper presents a more accurate variance estimator compared to existing methods. This innovation allows for the elimination of the regret's polynomial dependency on the expected length of the optimal policy, a limitation in earlier studies.

Implications and Future Directions

The practical and theoretical implications of these contributions are profound. From a practical standpoint, the proposed algorithm's efficiency and effectiveness in environments modeled by linear mixture SSPs can significantly enhance the performance of reinforcement learning agents, particularly in complex settings such as navigation and gaming. Theoretically, this work underlines the potential of leveraging high-order moment information and adaptive weighting strategies to refine learning algorithms further.

Looking forward, the introduction of a parameter-free algorithm, much like the Tarbouriech et al. (2021b) endeavor for SSPs, remains a compelling direction for future research. Such advancements would further simplify the deployment of these algorithms in real-world applications by minimizing the need for domain-specific knowledge.

Concluding Thoughts

In conclusion, this paper presents a stepping stone towards resolving the complexities associated with learning in stochastic shortest path settings, especially in the presence of linear function approximation. The nearly minimax optimal regret bound achieved marks a significant milestone in the literature on SSPs, offering a path forward for the development of more efficient, effective, and broadly applicable reinforcement learning algorithms.