Tight Sample Complexity Bounds for Entropic Best Policy Identification

TL;DR

This paper improves the sample complexity bounds for entropic risk-sensitive policy identification in reinforcement learning, closing the exponential gap between existing lower and upper bounds by leveraging sharper concentration bounds and a new stopping rule.

Contribution

It introduces a forward-model based algorithm with KL-based exploration bonuses and novel technical innovations to match the lower bound on sample complexity.

Findings

Achieves a sample complexity bound that matches the lower bound.

Uses sharper concentration bounds derived from the exponential utility's smoothness.

Proposes a new stopping rule to exploit the tightness of the bounds.

Abstract

We study best-policy identification for finite-horizon risk-sensitive reinforcement learning under the entropic risk measure. Recent work established a constant gap in the exponential horizon dependence between lower and upper bounds on the number of samples required to identify an approximately optimal policy. Precisely, known lower bounds scale in $\Omega(e^{|\beta| H})$ where $H$ is the horizon of the MDP, while the state-of-the-art upper bound achieves at best $O(e^{2|\beta| H})$ (arXiv:2506.00286v2) using a generative model. We show that this extra exponential factor can be traced to overly loose concentration control for exponential utilities. To close this open gap, we revisit the analysis of this problem through a forward-model based algorithm building on KL-based exploration bonuses that we adapt to the entropic criterion. The improvement we get is due to two main novel technical innovations. We leverage the smoothness properties of the exponential utility to derive sharper concentration bounds, and we propose a new stopping rule that exploits further this tightness to obtain a sample complexity that matches the lower bound.