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In this paper, we investigate a novel safe reinforcement learning problemwith step-wise violation constraints. Our problem differs from existing worksin that we consider stricter step-wise violation constraints and do not assumethe existence of safe actions, making our formulation more suitable forsafety-critical applications which need to ensure safety in all decision stepsand may not always possess safe actions, e.g., robot control and autonomousdriving. We propose a novel algorithm SUCBVI, which guarantees$\widetilde{O}(\sqrt{ST})$ step-wise violation and$\widetilde{O}(\sqrt{H^3SAT})$ regret. Lower bounds are provided to validatethe optimality in both violation and regret performance with respect to $S$ and$T$. Moreover, we further study a novel safe reward-free exploration problemwith step-wise violation constraints. For this problem, we design an$(\varepsilon,\delta)$-PAC algorithm SRF-UCRL, which achieves nearlystate-of-the-art sample complexity$\widetilde{O}((\frac{S^2AH^2}{\varepsilon}+\frac{H^4SA}{\varepsilon^2})(\log(\frac{1}{\delta})+S))$,and guarantees $\widetilde{O}(\sqrt{ST})$ violation during the exploration. Theexperimental results demonstrate the superiority of our algorithms in safetyperformance, and corroborate our theoretical results.

Provably Safe Reinforcement Learning with Step-wise Violation Constraints