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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.08934 |
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| _version_ | 1866908279468195840 |
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| author | Deng, Zilong Khan, Simon Zou, Shaofeng |
| author_facet | Deng, Zilong Khan, Simon Zou, Shaofeng |
| contents | 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 $τ$ 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 $ε$-optimal policy with at most $\tilde{O} \left(\frac{SA}{(1-γ)^4τ^2ε^2} \right)$ samples, where $γ$ is the discount factor, and $S, A$ are the sizes of the state and action spaces. Furthermore, when $τ\geq γ$, the sample complexity improves to $\tilde{O} \left( \frac{SA}{(1-γ)^3ε^2} \right)$. We further show a minimax lower bound of $\tilde{O} \left(\frac{(1-γτ)SA}{(1-γ)^4τε^2} \right)$. For a fixed risk level $τ\in (0,1]$, 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 $τ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08934 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-Optimal Sample Complexity for Iterated CVaR Reinforcement Learning with a Generative Model Deng, Zilong Khan, Simon Zou, Shaofeng Machine Learning 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 $τ$ 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 $ε$-optimal policy with at most $\tilde{O} \left(\frac{SA}{(1-γ)^4τ^2ε^2} \right)$ samples, where $γ$ is the discount factor, and $S, A$ are the sizes of the state and action spaces. Furthermore, when $τ\geq γ$, the sample complexity improves to $\tilde{O} \left( \frac{SA}{(1-γ)^3ε^2} \right)$. We further show a minimax lower bound of $\tilde{O} \left(\frac{(1-γτ)SA}{(1-γ)^4τε^2} \right)$. For a fixed risk level $τ\in (0,1]$, 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 $τ$, 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. |
| title | Near-Optimal Sample Complexity for Iterated CVaR Reinforcement Learning with a Generative Model |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2503.08934 |