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On The Statistical Complexity of Offline Decision-Making

Thanh Nguyen-Tang, Raman Arora

TL;DR

This work develops a unified minimax theory for offline decision-making with function approximation, identifying the pseudo-dimension of the value-function class and a new policy transfer coefficient framework as the core drivers of learnability. By introducing policy transfer coefficients that subsume prior data-coverage notions, the authors derive near-optimal lower and upper bounds for offline contextual bandits and MDPs, and extend the analysis to a hybrid offline-online setting with adaptive, Hedge-based procedures. The results show when and how offline data can accelerate online decision-making, and provide stable, distribution-shift-aware algorithms such as OfDM-Hedge and OfDM-Hedge-MDP that adapt to unknown transfer regimes. The findings illuminate the fundamental role of data quality and function-class complexity in offline RL with function approximation, while also outlining key gaps and directions for future work in nonparametric settings and fully adaptive hybrids.

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.

On The Statistical Complexity of Offline Decision-Making

TL;DR

This work develops a unified minimax theory for offline decision-making with function approximation, identifying the pseudo-dimension of the value-function class and a new policy transfer coefficient framework as the core drivers of learnability. By introducing policy transfer coefficients that subsume prior data-coverage notions, the authors derive near-optimal lower and upper bounds for offline contextual bandits and MDPs, and extend the analysis to a hybrid offline-online setting with adaptive, Hedge-based procedures. The results show when and how offline data can accelerate online decision-making, and provide stable, distribution-shift-aware algorithms such as OfDM-Hedge and OfDM-Hedge-MDP that adapt to unknown transfer regimes. The findings illuminate the fundamental role of data quality and function-class complexity in offline RL with function approximation, while also outlining key gaps and directions for future work in nonparametric settings and fully adaptive hybrids.

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.
Paper Structure (39 sections, 24 theorems, 105 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 39 sections, 24 theorems, 105 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Problem Formulation
  3. Stochastic contextual bandits
  4. Offline data
  5. Function approximation
  6. Notation.
  7. Offline Decision-Making as Transfer Learning
  8. Relations with other notions of data coverage
  9. Compared with concentrability coefficients.
  10. Compared with the data diversity of nguyen-tang2023on.
  11. Lower Bounds
  12. Upper Bounds
  13. Offline Data-assisted Online Decision-Making
  14. Lower bounds
  15. Upper bounds
  16. ...and 24 more sections

Key Result

Theorem 4.1

For any $C > 0, \rho \geq 1, n \geq d \cdot \max\{2^{2 \rho - 4} C, C^{\frac{1}{\rho-1}}/32 \}$, we have 0 1where the infimum is taken over all offline algorithm $\hat{\pi}(\cdot)$ (a randomized mapping from the offline data to a policy). 0where the infimum is taken over all offline algorithm $\hat{\pi}(\cdot)$ (a randomized mapping from the offline data to a policy).

Figures (2)

  • Figure 1: Hard MDPs
  • Figure :

Theorems & Definitions (57)

  • Definition 2.2: Pseudo-dimension
  • Definition 2.4: Covering number
  • Definition 3.1: Policy transfer coefficients
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 4.1
  • Theorem 5.1
  • Remark 5.2: Adaptive to policy transfer coefficients
  • Remark 5.3: No cost blowup of the Hedge algorithm
  • ...and 47 more