Exponential Bellman Equation and Improved Regret Bounds for Risk-Sensitive Reinforcement Learning

Part of Advances in Neural Information Processing Systems 34 pre-proceedings (NeurIPS 2021)

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Yingjie Fei, Zhuoran Yang, Yudong Chen, Zhaoran Wang


We study risk-sensitive reinforcement learning (RL) based on the entropic risk measure. Although existing works have established non-asymptotic regret guarantees for this problem, they leave open an exponential gap between the upper and lower bounds. We identify the deficiencies in existing algorithms and their analysis that result in such a gap. To remedy these deficiencies, we investigate a simple transformation of the risk-sensitive Bellman equations, which we call the exponential Bellman equation. The exponential Bellman equation inspires us to develop a novel analysis of Bellman backup procedures in risk-sensitive RL algorithms, and further motivates the design of a novel exploration mechanism. We show that these analytic and algorithmic innovations together lead to improved regret upper bounds over existing ones.