On The Statistical Complexity of Offline Decision-Making (2501.06339v1)

Abstract: We study the statistical complexity of offline decision-making with function approximation, establishing (near) minimax-optimal rates for stochastic contextual bandits and Markov decision processes. The performance limits are captured by the pseudo-dimension of the (value) function class and a new characterization of the behavior policy that \emph{strictly} subsumes all the previous notions of data coverage in the offline decision-making literature. In addition, we seek to understand the benefits of using offline data in online decision-making and show nearly minimax-optimal rates in a wide range of regimes.

• The paper introduces novel policy transfer coefficients to precisely measure data coverage in offline reinforcement learning, refining learnability characterizations.
• It establishes near minimax-optimal learning rates for offline decision-making in various settings by combining pseudo-dimension with these new coefficients.
• The findings have practical implications for domains like autonomous driving and healthcare, offering theoretical insights into data coverage assumptions and future research directions.

On The Statistical Complexity of Offline Decision-Making

The paper, "On The Statistical Complexity of Offline Decision-Making," focuses on the theoretical underpinnings of offline decision-making processes within machine learning, particularly using reinforcement learning (RL) frameworks like stochastic contextual bandits and Markov decision processes (MDPs).

Key Contributions

1. Policy Transfer Coefficients: The authors propose a novel notion called "policy transfer coefficients" to measure data coverage in offline RL. This new concept provides a refined characterization of learnability from offline data, subsuming existing notions like single-policy concentrability coefficients and data diversity.
2. Lower and Upper Bounds: The paper establishes (near) minimax-optimal rates for offline decision-making by leveraging the pseudo-dimension of function classes combined with the newly introduced policy transfer coefficients. By doing so, it provides comprehensive lower and upper bounds for learning with offline data in various settings, including multi-armed bandits, contextual bandits, and MDPs.
3. Function Approximation: The research explores various function approximation classes, ensuring that these approximations align with the complexities involved in offline learning. The coverage includes linear and neural network-based classes and generally, any class with bounded $L_1$ covering numbers.
4. Hybrid Offline-Online Learning: Extending beyond purely offline contexts, the paper characterizes the value of offline data when used alongside online decision-making data. Through this insight, the authors explore the possible gains in efficiency and efficacy when mixing pre-collected offline datasets with new interactions.
5. Technical Innovations: The authors solve various technical challenges, such as providing a uniform Bernstein's inequality for BeLLMan-like loss under empirical $L_1$ covering numbers, addressing issues like the absence of cost blowup in the number of iterations of the Hedge algorithm, and utilizing properties like the pseudo-dimension for providing strong theoretical bounds on learning complexities.

Numerical Results and Claims

The paper demonstrates its theoretical findings through examples comparing offline decision-making paradigms with known notions like HK transfer exponents, showing concrete cases where policy transfer coefficients yield tighter bounds on the efficiency of learning processes.

Implications and Future Directions

Practical Implications: The insights from this research can be instrumental in domains where offline data collection is the norm, such as autonomous driving, healthcare, and financial services. Practitioners can apply the bounds and characterizations developed in this paper to evaluate and improve the expected performance of offline learning models in these areas.

Theoretical Implications: The paper paves the way for a reevaluation of classical assumptions about data coverage in offline learning. The introduction of policy transfer coefficients can spur further theoretical inquiries into the nuanced interactions between offline data and decision-making algorithms.

Future Developments: This endeavor leaves open areas for exploration, such as extending these analyses to nonparametric function classes or devising adaptive algorithms that efficiently utilize compound datasets of varying quality. There is also potential in exploring if these principles can be applied to broader AI applications involving complex sequential decision-making tasks.

In conclusion, this paper offers a deep dive into the statistical complexities of offline decision-making, furnishing both rigorous characterizations and practical guidance for researchers and professionals aiming to leverage offline data towards optimal decision-making policies.