Learning Chance-Constrained MDPs with Bellman Distributional Certificates

Safety is a fundamental challenge in reinforcement learning (RL), particularly in real-world applications such as autonomous driving, robotics, and healthcare. To address this, Constrained Markov Decision Processes (CMDPs) are commonly used to enforce safety constraints while optimizing performance. However, existing methods often suffer from significant safety violations or require a high sample complexity to generate near-optimal policies. We address two settings: relaxed feasibility, where small violations are allowed, and strict feasibility, where no violation is allowed. We propose a model-based primal-dual algorithm that balances regret and bounded constraint violations, drawing on techniques from online RL and constrained optimization. For relaxed feasibility, we prove that our algorithm returns an $\varepsilon$-optimal policy with $\varepsilon$-bounded violation with arbitrarily high probability, requiring $\tilde{O}\left(\frac{SAH^3}{\varepsilon^2}\right)$ learning episodes, matching the lower bound for unconstrained MDPs. For strict feasibility, we prove that our algorithm returns an $\varepsilon$-optimal policy with zero violation with arbitrarily high probability, requiring $\tilde{O}\left(\frac{SAH^5}{\varepsilon^2ζ^2}\right)$ learning episodes, where $ζ$ is the problem-dependent Slater constant characterizing the size of the feasible region. This result matches the lower bound for learning CMDPs with access to a generative model. Our results demonstrate that learning CMDPs in an online setting is as easy as learning with a generative model and is no more challenging than learning unconstrained MDPs when small violations are allowed.

Constrained MDPs (CMDPs) are a widely adopted framework for incorporating safety into RL agents; however, the framework does not support risk-sensitive constraints. This can be problematic: For example, CMDPs allow for optimal solutions that, in order to satisfy the risk-neutral constraints, mix infrequent catastrophic behaviors and frequent, overly conservative ones. Moreover, prior empirical results suggest that enforcing stricter, risk-sensitive constraints can improve performance even under risk-neutral evaluation. The natural framework to incorporate risk-sensitive constraints is utility-constrained MDPs (UCMDPs), but no practical solutions for this problem existed. In this work, we introduce a simple yet powerful methodology for UCMDPs and constrained RL. Besides allowing for risk-sensitive constraints, our framework does not require us to fix constraint limits in advance of training the agent, provided that a sensible range is known. This increases policy flexibility and, in practice, allows for adjustments to these limits at no extra training cost. Besides benefiting from the generality of the framework, our agent shows strong performance in practice, consistently matching or outperforming existing baselines in several Safety Gymnasium benchmark tasks.

{\em Specification-Guided Reinforcement Learning (RL)} has largely concerned itself with learning an optimal policy against a formal specification in an unknown {\em Markov Decision Process (MDP)}. We take up the dual question. Given a policy already in hand, can one verify that it is correct against a formal specification in an unknown MDP, using sampling access alone? Such a question arises naturally in safety-critical systems. We investigate it for the {\em almost-sure reachability property}, whose correctness is witnessed by a {\em ranking supermartingale (RSM)} certificate. Our first finding is negative. In the absence of any further information about the MDP, one can neither learn such a certificate nor verify a candidate one with probabilistic guarantees. On the positive side, we show that, under a mild restriction, a known lower bound $p_{\min}$ on nonzero transition probabilities, a correct certificate can be learned with PAC-style guarantees. To the best of our knowledge, this is the first study of learning and verifying certificates for almost-sure reachability in unknown MDPs from sampling access alone. Finally, our experiments show that the $p_{\min}$-based sufficient condition, while sound, is conservative: the learned certificate often remains valid on the true MDP even when the condition fails.