Reinforcement Learning for Exponential Utility: Algorithms and Convergence in Discounted MDPs

TL;DR

This paper develops and analyzes new reinforcement learning algorithms for exponential utility optimization in discounted MDPs, establishing their convergence and optimality properties.

Contribution

It introduces two novel Q-value-style algorithms with convergence guarantees for exponential utility in discounted MDPs, filling a key gap in principled value-based RL methods.

Findings

Proved contraction properties of the operators in specific metrics.

Established almost-sure convergence of the two-timescale Q-learning algorithm.

Provided finite-time convergence rates and analyzed challenges for the sublinear operator.

Abstract

Reinforcement learning (RL) for exponential-utility optimization in discounted Markov decision processes (MDPs) lacks principled value-based algorithms. We address this gap in the fixed risk-aversion setting. Building on the Bellman-type equation for exponential utility studied in \cite{porteus1975optimality}, we derive two Q-value-style extensions and show that the associated operators are contractions in the $L_\infty$ and sup-log/Thompson metrics, respectively. We characterize their fixed points and prove that the induced greedy stationary policy is optimal for the exponential-utility objective among stationary policies. These structural results lead to two model-free algorithms: a two-timescale Q-learning--style algorithm, for which we establish almost-sure convergence and provide finite-time convergence rates via timescale separation, and a one-timescale algorithm governed by a sublinear power-law operator. Since the latter does not admit a global contraction in standard metrics, we prove its convergence using delicate arguments based on local Lipschitzness, monotonicity, homogeneity, and Dini derivatives, and provide a scalar finite-time analysis that highlights the challenges in obtaining convergence rates in the vector case. Our work provides a foundation for value-based RL under exponential-utility objectives.