🤖 AI Summary
This work addresses the absence of principled value-based reinforcement learning algorithms for optimizing exponential utility under fixed risk aversion. It establishes, for the first time, a value-based framework tailored to discounted Markov decision processes in this setting. Building upon a Bellman-type equation for exponential utility, the paper introduces two Q-value-style operators and rigorously proves their contractivity and almost sure convergence under both the $L^\infty$ and sup-log (Thompson) metrics—without requiring a global contraction assumption for the single-timescale algorithm. Furthermore, it provides finite-time convergence rates and demonstrates that the induced greedy stationary policy is optimal in the sense of exponential utility.
📝 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.