Finite-Time Analysis of Discounted Exponential-Utility Reinforcement Learning

arXiv:2608.01917v1 Announce Type: new Abstract: Discounted exponential utility provides a principled criterion for risk-sensitive sequential decision-making, but its nonlinear structure complicates reinforcement learning. A recent work itep{thoppe2026reinforcement} addressed this difficulty by introducing a Bellman-compatible surrogate and two model-free fixed-point algorithms for optimizing it over stationary policies. However, their main convergence results are asymptotic. In this work, we establish finite-time rates of ilde{O} (1/sqrt{n}) for the aforementioned two algorithms under asynchronous Markovian sampling, where n is the iteration index and ilde{O} hides logarithmic expressions. Importantly, we employ parameter-free choices for the stepsize parameter to derive these rate results. For the algorithmically simpler one-timescale method, the main challenge is that its update equation is not directly aligned with the contraction geometry of its underlying power-law operator. We overcome this mismatch by exploiting the boundedness, monotonicity, and homogeneity of the operator to obtain a local pseudo-contraction property for the relative-error dynamics. We then use a Moreau-envelope-based Lyapunov function and Polyak--Ruppert averaging to obtain the stated convergence rate with parameter-free stepsizes. For the two-timescale method, the main challenge is to control a tracking error on the faster timescale. These results provide the first finite-time guarantees for model-free discounted exponential-utility reinforcement learning.

Source: arXiv cs.LG | 2026-08-04