Near-Optimal Sample Complexity for Iterated CVaR Reinforcement Learning with a Generative Model

Abstract

In this work, we study the sample complexity problem of risk-sensitive Reinforcement Learning (RL) with a generative model, where we aim to maximize the Conditional Value at Risk (CVaR) with risk tolerance level $\tau$ at each step, a criterion we refer to as Iterated CVaR. We first build a connection between Iterated CVaR RL and $(s, a)$-rectangular distributional robust RL with a specific uncertainty set for CVaR. We establish nearly matching upper and lower bounds on the sample complexity of this problem. Specifically, we first prove that a value iteration-based algorithm, ICVaR-VI, achieves an $\epsilon$-optimal policy with at most $\tilde{O} \left(\frac{SA}{(1-\gamma)^4\tau^2\epsilon^2} \right)$ samples, where $\gamma$ is the discount factor, and $S, A$ are the sizes of the state and action spaces. Furthermore, when $\tau \geq \gamma$, the sample complexity improves to $\tilde{O} \left( \frac{SA}{(1-\gamma)^3\epsilon^2} \right)$. We further show a minimax lower bound of $\tilde{O} \left(\frac{(1-\gamma \tau)SA}{(1-\gamma)^4\tau\epsilon^2} \right)$. For a fixed risk level $\tau \in (0,1]$, our upper and lower bounds match, demonstrating the tightness and optimality of our analysis. We also investigate a limiting case with a small risk level $\tau$, called Worst-Path RL, where the objective is to maximize the minimum possible cumulative reward. We develop matching upper and lower bounds of $\tilde{O} \left(\frac{SA}{p_{\min}} \right)$, where $p_{\min}$ denotes the minimum non-zero reaching probability of the transition kernel.