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

• URL: http://arxiv.org/abs/2605.08053v1
• Date: Fri, 08 May 2026 17:41:48 GMT
• Title: Reinforcement Learning for Exponential Utility: Algorithms and Convergence in Discounted MDPs
• Abstract summary: Reinforcement learning (RL) for exponential-utility optimization in discounted Markov decision processes lacks principled value-based algorithms.<n>We derive two Q-value-style extensions and show that the associated operators are contractions in the $L_infty$ and sup-log/Thompson metrics.<n>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.
• Score: 2.574071344130061
• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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.

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