Robust Bayesian Dynamic Programming for On-policy Risk-sensitive Reinforcement Learning

TL;DR

This paper introduces a robust Bayesian framework for risk-sensitive reinforcement learning that handles transition uncertainty, unifies existing methods, and demonstrates strong theoretical and empirical performance including in option hedging.

Contribution

It develops a novel risk-sensitive Bayesian dynamic programming algorithm with convergence guarantees and robustness analysis, extending RL to handle transition uncertainties comprehensively.

Findings

Algorithm converges to near-optimal policies in training environments.

Strong consistency guarantees for the risk-based Bellman operator estimator.

Validated effectiveness in risk-sensitivity and robustness through numerical experiments.

Abstract

We propose a novel framework for risk-sensitive reinforcement learning (RSRL) that incorporates robustness against transition uncertainty. We define two distinct yet coupled risk measures: an inner risk measure addressing state and cost randomness and an outer risk measure capturing transition dynamics uncertainty. Our framework unifies and generalizes most existing RL frameworks by permitting general coherent risk measures for both inner and outer risk measures. Within this framework, we construct a risk-sensitive robust Markov decision process (RSRMDP), derive its Bellman equation, and provide error analysis under a given posterior distribution. We further develop a Bayesian Dynamic Programming (Bayesian DP) algorithm that alternates between posterior updates and value iteration. The approach employs an estimator for the risk-based Bellman operator that combines Monte Carlo sampling with convex optimization, for which we prove strong consistency guarantees. Furthermore, we demonstrate that the algorithm converges to a near-optimal policy in the training environment and analyze both the sample complexity and the computational complexity under the Dirichlet posterior and CVaR. Finally, we validate our approach through two numerical experiments. The results exhibit excellent convergence properties while providing intuitive demonstrations of its advantages in both risk-sensitivity and robustness. Empirically, we further demonstrate the advantages of the proposed algorithm through an application on option hedging.