Settling the Sample Complexity of Model-Based Offline Reinforcement Learning (2204.05275v4)

Abstract: This paper is concerned with offline reinforcement learning (RL), which learns using pre-collected data without further exploration. Effective offline RL would be able to accommodate distribution shift and limited data coverage. However, prior algorithms or analyses either suffer from suboptimal sample complexities or incur high burn-in cost to reach sample optimality, thus posing an impediment to efficient offline RL in sample-starved applications. We demonstrate that the model-based (or "plug-in") approach achieves minimax-optimal sample complexity without burn-in cost for tabular Markov decision processes (MDPs). Concretely, consider a finite-horizon (resp. $\gamma$-discounted infinite-horizon) MDP with $S$ states and horizon $H$ (resp. effective horizon $\frac{1}{1-\gamma}$), and suppose the distribution shift of data is reflected by some single-policy clipped concentrability coefficient $C^{\star}_{\text{clipped}}$. We prove that model-based offline RL yields $\varepsilon$-accuracy with a sample complexity of [ \begin{cases} \frac{H^{4}SC_{\text{clipped}}^{\star}}{\varepsilon^{2}} & (\text{finite-horizon MDPs}) \frac{SC_{\text{clipped}}^{\star}}{(1-\gamma)^{3}\varepsilon^{2}} & (\text{infinite-horizon MDPs}) \end{cases} ] up to log factor, which is minimax optimal for the entire $\varepsilon$-range. The proposed algorithms are "pessimistic" variants of value iteration with Bernstein-style penalties, and do not require sophisticated variance reduction. Our analysis framework is established upon delicate leave-one-out decoupling arguments in conjunction with careful self-bounding techniques tailored to MDPs.

• The paper presents a model-based offline RL approach that attains minimax-optimal sample complexity for both infinite-horizon and finite-horizon MDPs without requiring a burn-in phase.
• It demonstrates ε-accuracy with sample complexities of Õ(S/((1-γ)^3ε²)) for discounted MDPs and Õ(H⁴S/ε²) for episodic settings using a novel subsampling technique.
• A Bernstein-style penalty and leave-one-out analysis are employed to enhance variance estimation and decouple statistical dependencies in sample-sparse environments.

Overview of "Settling the Sample Complexity of Model-Based Offline Reinforcement Learning"

The paper "Settling the Sample Complexity of Model-Based Offline Reinforcement Learning" by Gen Li, Laixi Shi, Yuxin Chen, Yuejie Chi, and Yuting Wei presents a comprehensive examination of the sample complexity involved in model-based offline reinforcement learning (RL), focusing on minimizing the number of samples necessary for achieving $\varepsilon$-accuracy. This work aims to address the limitations of previous analyses that either yielded suboptimal sample complexities or imposed significant sample requirements to reach optimal efficiency, particularly in sample-sparse environments.

Key Contributions

1. Model-Based Approach and Minimax Optimality:
   • The authors establish that a model-based RL approach, implementing value iteration with pessimistic variants, achieves a minimax-optimal sample complexity for both infinite-horizon and finite-horizon Markov decision processes (MDPs). Notably, this approach operates without incurring a burn-in cost, which contrasts sharply with earlier methods that demanded large initial sample sizes to achieve sample optimality.
2. Discounted Infinite-Horizon MDPs:
   • The paper explores $\gamma$-discounted infinite-horizon MDPs, demonstrating that the model-based offline RL can achieve $\varepsilon$-accuracy with a sample complexity of $$\widetilde{O} \left( \frac{S}{(1-\gamma)^3 \varepsilon^2} \right)$$, covering the entire $\varepsilon$-range effectively. This result is independent of whether the clipped concentrability coefficient or its standard counterpart is used, marking a significant improvement over previous algorithms which lacked the ability to achieve minimax optimality over the full range of $\varepsilon$-values.
3. Finite-Horizon MDPs:
   • For episodic finite-horizon MDPs, the authors provide an equally significant result: a sample complexity of $$\widetilde{O} \left( \frac{H^4 S}{\varepsilon^2} \right)$$, again achieving minimax-optimal sample efficiency across the full $\varepsilon$-range. This is achieved through a novel subsampling technique that effectively decouples statistical dependencies.
4. Bernstein-Style Penalty and Leave-One-Out Analysis:
   • The paper emphasizes the adoption of a Bernstein-style penalty for variance estimation, enhancing the learned model's stability in uncertain environments. The use of a leave-one-out analysis framework allows for finer control over the statistical dependencies between samples, ensuring that the analysis retains its sharpness even in the most constrained data regimes.
5. Theoretical Implications and Future Directions:
   • The research suggests that model-based approaches can achieve sample efficiency without needing variance reduction, a technique previously considered necessary for optimality. This holds potential for developing simpler, more efficient offline RL algorithms. Moreover, the findings pave the way for extending these analyses and methods to scenarios involving function approximation.

Implications for Theory and Practice

The implications of this paper reach both theoretical and practical dimensions within the RL community. Theoretically, the results settle long-standing questions about the sample efficiency of model-based approaches in settings where sample collection is expensive or infeasible. Practically, the algorithms proposed are straightforward to implement, without the need for sophisticated variance reduction methods, and are directly applicable to real-world RL applications such as robotics and autonomous systems, where offline learning from limited data is crucial.

In conclusion, this paper makes significant strides in advancing our understanding of the sample complexity requirements in offline RL, providing a clear path toward more efficient, data-conservative RL algorithm design. The elimination of the need for large burn-in sample sizes and the demonstrated efficacy of Bernstein-style penalties offer practical avenues for enhancing RL applications across various domains.