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Provable Risk-Sensitive Distributional Reinforcement Learning with General Function Approximation

Yu Chen, Xiangcheng Zhang, Siwei Wang, Longbo Huang

TL;DR

The paper addresses risk-aware sequential decision-making by marrying risk-sensitive objectives with distributional RL in a general function-approximation setting. It develops a unifying framework (RS-DisRL) and two meta-algorithms, RS-DisRL-M for model-based and RS-DisRL-V for model-free scenarios, both achieving sublinear regret with respect to the number of episodes via LSR and MLE estimation within augmented MDPs. The key theoretical contributions include a first ${\widetilde{\mathcal{O}}}(\sqrt{K})$ regret bound for static Lipschitz risk measures, novel augmented-simulation lemmas, and extensive analyses leveraging eluder and Bellman-eluder dimensions. These results yield statistically efficient, scalable methods for risk-sensitive distributional RL, including special cases like CVaR with linear function approximation, with broad implications for safety-critical applications. Overall, the work provides a foundational, rigorously analyzed path to practical, risk-aware RL in high-dimensional or continuous-state settings.

Abstract

In the realm of reinforcement learning (RL), accounting for risk is crucial for making decisions under uncertainty, particularly in applications where safety and reliability are paramount. In this paper, we introduce a general framework on Risk-Sensitive Distributional Reinforcement Learning (RS-DisRL), with static Lipschitz Risk Measures (LRM) and general function approximation. Our framework covers a broad class of risk-sensitive RL, and facilitates analysis of the impact of estimation functions on the effectiveness of RSRL strategies and evaluation of their sample complexity. We design two innovative meta-algorithms: \texttt{RS-DisRL-M}, a model-based strategy for model-based function approximation, and \texttt{RS-DisRL-V}, a model-free approach for general value function approximation. With our novel estimation techniques via Least Squares Regression (LSR) and Maximum Likelihood Estimation (MLE) in distributional RL with augmented Markov Decision Process (MDP), we derive the first $\widetilde{\mathcal{O}}(\sqrt{K})$ dependency of the regret upper bound for RSRL with static LRM, marking a pioneering contribution towards statistically efficient algorithms in this domain.

Provable Risk-Sensitive Distributional Reinforcement Learning with General Function Approximation

TL;DR

The paper addresses risk-aware sequential decision-making by marrying risk-sensitive objectives with distributional RL in a general function-approximation setting. It develops a unifying framework (RS-DisRL) and two meta-algorithms, RS-DisRL-M for model-based and RS-DisRL-V for model-free scenarios, both achieving sublinear regret with respect to the number of episodes via LSR and MLE estimation within augmented MDPs. The key theoretical contributions include a first regret bound for static Lipschitz risk measures, novel augmented-simulation lemmas, and extensive analyses leveraging eluder and Bellman-eluder dimensions. These results yield statistically efficient, scalable methods for risk-sensitive distributional RL, including special cases like CVaR with linear function approximation, with broad implications for safety-critical applications. Overall, the work provides a foundational, rigorously analyzed path to practical, risk-aware RL in high-dimensional or continuous-state settings.

Abstract

In the realm of reinforcement learning (RL), accounting for risk is crucial for making decisions under uncertainty, particularly in applications where safety and reliability are paramount. In this paper, we introduce a general framework on Risk-Sensitive Distributional Reinforcement Learning (RS-DisRL), with static Lipschitz Risk Measures (LRM) and general function approximation. Our framework covers a broad class of risk-sensitive RL, and facilitates analysis of the impact of estimation functions on the effectiveness of RSRL strategies and evaluation of their sample complexity. We design two innovative meta-algorithms: \texttt{RS-DisRL-M}, a model-based strategy for model-based function approximation, and \texttt{RS-DisRL-V}, a model-free approach for general value function approximation. With our novel estimation techniques via Least Squares Regression (LSR) and Maximum Likelihood Estimation (MLE) in distributional RL with augmented Markov Decision Process (MDP), we derive the first dependency of the regret upper bound for RSRL with static LRM, marking a pioneering contribution towards statistically efficient algorithms in this domain.
Paper Structure (56 sections, 35 theorems, 173 equations, 1 figure, 1 table, 10 algorithms)

This paper contains 56 sections, 35 theorems, 173 equations, 1 figure, 1 table, 10 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. RSRL
  4. DisRL
  5. Notations
  6. Problem Formulation
  7. Episodic Markov Decision Process
  8. Lipschitz Risk Measures
  9. RSRL with Static LRM
  10. Augmented MDPs and Distributional Bellman Equation
  11. Augmented MDPs
  12. Distributional Bellman Equation
  13. Model-Based Meta-Algorithm RS-DisRL-M
  14. Theoretical Guarantees
  15. Estimation by Model-Based LSR Approach
  16. ...and 41 more sections

Key Result

Theorem 6.4

Under Assumption ass:mbreal, if the estimation function M-Est satisfies Conditions con:mbconcentration and con:mbelliptical, then the regret of RS-DisRL-M (Algorithm alg:mbframe) can be bounded by $\operatorname{Regret}(K) \leq L_\infty(\rho)\xi(K, H, {\bm{\Theta}}, \beta, \delta)$.

Figures (1)

  • Figure 1: Comparison for different algorithms for the CVaR objective $\operatorname{CVaR}_{\tau}$ under different risk parameter $\tau$.

Theorems & Definitions (67)

  • Definition 5.1: Distributional Bellman Equation wang2023benefitsbastani2022regret
  • Theorem 6.4
  • Theorem 6.5
  • Theorem 6.6: Estimation by Model-Based MLE Approach
  • Theorem 7.4
  • Theorem 7.5
  • Theorem 7.6
  • Definition A.1: Covering Number
  • Definition A.2: Bracketing Number
  • Definition A.3: $\varepsilon$-dependence russo2013eluder
  • ...and 57 more