Sample Complexity of Episodic Fixed-Horizon Reinforcement Learning (1510.08906v3)

Abstract: Recently, there has been significant progress in understanding reinforcement learning in discounted infinite-horizon Markov decision processes (MDPs) by deriving tight sample complexity bounds. However, in many real-world applications, an interactive learning agent operates for a fixed or bounded period of time, for example tutoring students for exams or handling customer service requests. Such scenarios can often be better treated as episodic fixed-horizon MDPs, for which only looser bounds on the sample complexity exist. A natural notion of sample complexity in this setting is the number of episodes required to guarantee a certain performance with high probability (PAC guarantee). In this paper, we derive an upper PAC bound $\tilde O(\frac{|\mathcal S|^2 |\mathcal A| H^2}{\epsilon^2} \ln\frac 1 \delta)$ and a lower PAC bound $\tilde \Omega(\frac{|\mathcal S| |\mathcal A| H^2}{\epsilon^2} \ln \frac 1 {\delta + c})$ that match up to log-terms and an additional linear dependency on the number of states $|\mathcal S|$. The lower bound is the first of its kind for this setting. Our upper bound leverages Bernstein's inequality to improve on previous bounds for episodic finite-horizon MDPs which have a time-horizon dependency of at least $H^3$.

• The paper introduces the UCFH algorithm, leveraging Bernstein’s inequality to reduce the horizon dependency in sample complexity from H³ to H².
• It establishes matching lower bounds that set a theoretical performance threshold for any RL algorithm in fixed-horizon MDPs.
• The authors construct robust confidence sets using variance-based criteria to ensure accurate MDP modeling, enhancing algorithm applicability.

Sample Complexity of Episodic Fixed-Horizon Reinforcement Learning

The paper "Sample Complexity of Episodic Fixed-Horizon Reinforcement Learning" by Christoph Dann and Emma Brunskill extends the rigorous analysis of sample complexity within reinforcement learning (RL) to episodic fixed-horizon Markov Decision Processes (MDPs). It presents both lower and upper bounds, providing a comprehensive understanding of the sample complexity for this specific setting.

Summary of Contributions

1. Upper Bound Derivation: The authors propose a new model-based algorithm, UCFH (Upper-Confidence Fixed-Horizon), which demonstrates improved performance in terms of sample complexity for episodic fixed-horizon RL when compared to prior art. The key innovation involves leveraging Bernstein's inequality for tighter confidence bounds, yielding an improved horizon dependency from $H^3$ to $H^2$. The presented PAC (Probably Approximately Correct) sample complexity upper bound is $$\tilde O(\frac{|^2| H^2}{\epsilon^2} \ln \frac 1 \delta)$$, where $|S|$ and $|A|$ denote the cardinalities of state and action spaces, respectively.
2. Lower Bound Derivation: In parallel, the paper establishes a companion lower bound on the sample complexity as $$\tilde \Omega(\frac{|S||A| H^2}{\epsilon^2} \ln \frac{1}{\delta + c})$$. These bounds match up to logarithmic terms, and significantly, the lower bound provides a fundamental performance threshold for any RL algorithm in fixed-horizon settings.
3. Confidence Set Construction: A critical aspect investigated in the paper is the construction of confidence sets that capture the true MDP with high probability. Importantly, the inclusion of variance-based conditions in addition to optimism ensures that the estimated transition models remain within these confidence bounds, validating use in arbitrary MDPs with linear overhead related to the possible number of successor states.

Key Results and Implications

• Improved Sample Complexity: The reduction in horizon dependency reflects a critical improvement, potentially impacting how RL applications in constrained episodic environments, such as pedagogical agents or customer service automation, could be approached with more efficiency.
• Algorithm Compatibility: The UCFH algorithm can be applied directly to a wide range of MDPs without requiring additional structures like a generative model or sparse transitions, broadening its applicability significantly across domains with diverse characteristics.
• Foundational Bounds: The introduction of lower bounds provides a theoretical benchmark against which RL methods can be evaluated. This recognition of a lower limit highlights the potential gaps and inefficiencies in existing algorithms, motivating further research into novel techniques that could approach these bounds.

Future Directions

The theoretical advancements presented in this paper offer several avenues for further research and development. One potential direction is an empirical evaluation of the UCFH algorithm across various environments to validate its theoretical promise in practical settings. Moreover, exploring whether the dependency on the state space size can be minimized further, similar to results achieved by methods like Mormax for infinite-horizon MDPs, remains an open challenge.

In essence, this work contributes to a deeper understanding of sample efficiency in episodic fixed-horizon RL, helping to narrow the gap between theoretical insights and practical implementations in AI systems requiring adaptable decision-making frameworks.