Risk-sensitive Markov Decision Process and Learning under General Utility Functions

Reinforcement Learning (RL) has gained substantial attention across diverse application domains and theoretical investigations. Existing literature on RL theory largely focuses on risk-neutral settings where the decision-maker learns to maximize the expected cumulative reward. However, in practical scenarios such as portfolio management and e-commerce recommendations, decision-makers often persist in heterogeneous risk preferences subject to outcome uncertainties, which can not be well-captured by the risk-neural framework. Incorporating these preferences can be approached through utility theory, yet the development of risk-sensitive RL under general utility functions remains an open question for theoretical exploration. In this paper, we consider a scenario where the decision-maker seeks to optimize a general utility function of the cumulative reward in the framework of a Markov decision process (MDP). To facilitate the Dynamic Programming Principle and Bellman equation, we enlarge the state space with an additional dimension that accounts for the cumulative reward. We propose a discretized approximation scheme to the MDP under enlarged state space, which is tractable and key for algorithmic design. We then propose a modified value iteration algorithm that employs an epsilon-covering over the space of cumulative reward. When a simulator is accessible, our algorithm efficiently learns a near-optimal policy with guaranteed sample complexity. In the absence of a simulator, our algorithm, designed with an upper-confidence-bound exploration approach, identifies a near-optimal policy while ensuring a guaranteed regret bound. Finally, we establish a novel theoretical regret lower bound for the risk-sensitive setting, and show that the regret of our algorithm matches this lower bound up to a small polynomial factor