The Statistical Complexity of Interactive Decision Making (2112.13487v3)

Abstract: A fundamental challenge in interactive learning and decision making, ranging from bandit problems to reinforcement learning, is to provide sample-efficient, adaptive learning algorithms that achieve near-optimal regret. This question is analogous to the classical problem of optimal (supervised) statistical learning, where there are well-known complexity measures (e.g., VC dimension and Rademacher complexity) that govern the statistical complexity of learning. However, characterizing the statistical complexity of interactive learning is substantially more challenging due to the adaptive nature of the problem. The main result of this work provides a complexity measure, the Decision-Estimation Coefficient, that is proven to be both necessary and sufficient for sample-efficient interactive learning. In particular, we provide: 1. a lower bound on the optimal regret for any interactive decision making problem, establishing the Decision-Estimation Coefficient as a fundamental limit. 2. a unified algorithm design principle, Estimation-to-Decisions (E2D), which transforms any algorithm for supervised estimation into an online algorithm for decision making. E2D attains a regret bound that matches our lower bound up to dependence on a notion of estimation performance, thereby achieving optimal sample-efficient learning as characterized by the Decision-Estimation Coefficient. Taken together, these results constitute a theory of learnability for interactive decision making. When applied to reinforcement learning settings, the Decision-Estimation Coefficient recovers essentially all existing hardness results and lower bounds. More broadly, the approach can be viewed as a decision-theoretic analogue of the classical Le Cam theory of statistical estimation; it also unifies a number of existing approaches -- both Bayesian and frequentist.

• The paper introduces the DEC, establishing it as a necessary and sufficient condition for sample-efficient interactive learning.
• A localized DEC framework integrates regret and Hellinger distance estimation to derive universal lower bounds in decision-making.
• The DEC framework extends to reinforcement learning, enabling reduction-based algorithms that achieve near-optimal regret.

Essay on the Statistical Complexity of Interactive Decision Making

The complexity of interactive decision-making tasks, ranging from bandit problems to reinforcement learning (RL), hinges upon developing sample-efficient algorithms achieving near-optimal regret. This paper introduces the Decision-Estimation Coefficient (DEC), a new complexity measure capturing the statistical intricacies of interactive learning. The paper establishes the DEC's role as a necessary and sufficient condition for sample-efficient learning, thus forging a unified framework for understanding the learnability of interactive decision-making tasks.

A salient contribution of this work is the theoretical development and analysis of the DEC. This coefficient quantifies the trade-off between a decision's informativeness and its associated regret, thereby reflecting the statistical bounds of interactive learning tasks. The DEC hinges upon two critical components: the regret associated with a decision and the Hellinger distance's estimation error. It avoids the prevalent pitfall of relying solely on regret-centric measures, a significant advancement for understanding decision-making complexity.

Lower Bounds and the Decision-Estimation Coefficient

A key theoretical result establishes a universal lower bound using the DEC for any interactive decision-making problem. The lower bound strengthens earlier approaches by integrating the decision-making dynamics with statistical estimation theory. Specifically, the paper employs a likelihood-based localization methodology, developing a "localized" DEC that incorporates both hard and easy instances in its evaluation—highlighting that sample complexity is not universally uniform across instances.

Beyond Bandits: Extending to Reinforcement Learning

Notably, the DEC recovers and extends several existing results in reinforcement learning. It renders a systematic approach to capturing sample efficiency across various RL problems, including those thought to be intractable in the past. The introduction of the Estimation-to-Decisions (E) meta-algorithm, a powerful reduction transforming supervised estimation algorithms into interactive decision-making strategies, is particularly noteworthy. By leveraging DEC, E achieves optimal regret consistent with the information-theoretic limits set by the local DEC.

Implications and Future Directions

The theoretical innovations in this paper significantly impact the broader AI and machine learning disciplines. By clarifying the limits of decision-making under sample constraints, the DEC informs algorithm design, particularly for complex, high-dimensional spaces. Practically, bounded DECs mark domains where algorithms can achieve sublinear regret—a critical insight for applications like robotics and dialogue systems requiring sophisticated decision-making under uncertainty.

Moreover, this work sets a fertile ground for future research on improving computational efficiency and developing DEC-based algorithms applicable to POMDPs, contextual decision problems, and RL paradigms with multi-agent interactions. Understanding how DEC might be employed in real-world function approximation scenarios and deep learning presents further avenues for exploration.

This paper's framework opens a new chapter in our theoretical understanding of interactive decision-making, proving indispensable for practitioners and theorists aiming to harness AI's full potential in decision-centric applications. The DEC's innovative integration of regret and estimation error into a singular robust measure paves the way not only for theoretical advancement but for tangible improvement in AI systems' decision-making capabilities.