Cascaded Gaps: Towards Gap-Dependent Regret for Risk-Sensitive Reinforcement Learning
This work addresses a theoretical gap in risk-sensitive RL for researchers, providing near-optimal regret guarantees that are incremental but offer specific improvements.
The paper tackles the problem of deriving gap-dependent regret bounds for risk-sensitive reinforcement learning using the entropic risk measure, proposing cascaded gaps to achieve non-asymptotic and logarithmic regret bounds with exponential improvement over gap-independent ones in certain settings.
In this paper, we study gap-dependent regret guarantees for risk-sensitive reinforcement learning based on the entropic risk measure. We propose a novel definition of sub-optimality gaps, which we call cascaded gaps, and we discuss their key components that adapt to the underlying structures of the problem. Based on the cascaded gaps, we derive non-asymptotic and logarithmic regret bounds for two model-free algorithms under episodic Markov decision processes. We show that, in appropriate settings, these bounds feature exponential improvement over existing ones that are independent of gaps. We also prove gap-dependent lower bounds, which certify the near optimality of the upper bounds.