Provably Efficient RL under Episode-Wise Safety in Constrained MDPs with Linear Function Approximation
(2502.10138v2)
Published 14 Feb 2025 in cs.LG
Abstract: We study the reinforcement learning (RL) problem in a constrained Markov decision process (CMDP), where an agent explores the environment to maximize the expected cumulative reward while satisfying a single constraint on the expected total utility value in every episode. While this problem is well understood in the tabular setting, theoretical results for function approximation remain scarce. This paper closes the gap by proposing an RL algorithm for linear CMDPs that achieves $\tilde{\mathcal{O}}(\sqrt{K})$ regret with an episode-wise zero-violation guarantee. Furthermore, our method is computationally efficient, scaling polynomially with problem-dependent parameters while remaining independent of the state space size. Our results significantly improve upon recent linear CMDP algorithms, which either violate the constraint or incur exponential computational costs.
Provably Efficient RL under Episode-Wise Safety in Linear CMDPs
The paper addresses the challenge of reinforcement learning (RL) in constrained Markov decision processes (CMDPs) employing linear function approximation. It targets scenarios where an RL agent must explore and learn while satisfying specific safety constraints at each episode. Traditionally, such safety constraints are well-studied in tabular CMDPs, but this research extends these ideas to CMDPs featuring linear function approximation—where state spaces can be large or even infinite.
Problem Statement and Contributions
The main objective of the paper is to develop an RL algorithm, termed OPSE-LCMDP (Optimistic-Pessimistic Softmax Exploration for Linear CMDP), that achieves sublinear regret and zero episode-wise constraint violations while remaining computationally efficient. The authors assert that their approach significantly improves over existing methods that either fail to ensure constraint satisfaction or suffer from exponential computational complexity.
The research introduces innovative techniques for integrating optimism and pessimism in policy updates, allowing for efficient and safe exploration without compromising the computational tractability. The OPSE-LCMDP operates under the following assumptions and methodological decisions:
Access to a strictly safe policy, which serves as a fallback strategy when the algorithm is uncertain of the environment.
Utilization of a softmax policy framework to balance optimistic exploration against strict constraint satisfaction.
Application of a novel technique to determine an optimal degree of exploration (via the parameter λ), ensuring that the pessimistic constraints hold.
Theoretical Foundations
The theory underlying this work hinges on a fine balance between exploration (seeking new information) and exploitation (making use of existing knowledge), with the added challenge of maintaining safety constraints. More specifically:
The regret bound for the proposed algorithm is O(H4ξ−1d5K), where applicable parameters are meticulously bounded.
The introduction of clipped optimistic and pessimistic value functions is a significant methodological advancement; it restrains overly optimistic or pessimistic estimates, thus avoiding overcommitment to uncertain policies.
Algorithmic Insights
Central to the approach is a bisection search to dynamically adapt λ, which controls the trade-off between exploration and constraint satisfaction. This method:
Identifies small feasible λ values through iterative refinement, ensuring that the algorithm does not exceed state space-dependent computational costs.
Systematically and efficiently incorporates uncertainty margins into the policy decision-making process, essential for handling large-scale CMDPs.
Results and Future Directions
The paper's results are quantitatively validated through regret analysis, showcasing the statistical robustness of the OPSE-LCMDP algorithm. Despite the algorithm's complexities, the authors successfully demonstrate its capacity to operate independently of state space size—overcoming a critical limitation in RL research.
This research opens new avenues for further exploration in artificial intelligence, particularly in adapting RL models to contexts where safety cannot be compromised, such as autonomous driving or industrial automation. Future work might involve scaling these approaches to multi-constraint CMDPs or developing enhanced softmax policies that reduce the dependency on sophisticated bounding techniques.
In conclusion, the paper substantially advances the understanding of safe reinforcement learning with linear function approximation, establishing new benchmarks for efficient and safe decision-making algorithms in CMDPs.