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In this paper, we investigate a novel safe reinforcement learning problem with step-wise violation constraints. Our problem differs from existing works in that we consider stricter step-wise violation constraints and do not assume the existence of safe actions, making our formulation more suitable for safety-critical applications which need to ensure safety in all decision steps and may not always possess safe actions, e.g., robot control and autonomous driving. We propose a novel algorithm SUCBVI, which guarantees O( √ ST ) step-wise violation and O( √ H 3 SAT ) regret. Lower bounds are provided to validate the optimality in both violation and regret performance with respect to S and T . Moreover, we further study a novel safe reward-free exploration problem with step-wise violation constraints. For this problem, we design an (ε, δ)-PAC algorithm SRF-UCRL, which achieves nearly state-of-the-art sample complexity O(( S 2 AH 2 ε + H 4 SA ε 2 )(log( 1 δ ) + S)), and guarantees O( √ ST ) violation during the exploration. The experimental results demonstrate the superiority of our algorithms in safety performance, and corroborate our theoretical results.

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