The statistical complexity of interactive decision making: A minimax optimization perspective

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. In this talk, we will introduce a new complexity measure, the Decision-Estimation Coefficient (DEC), whose value is given by a certain zero-sum game which represents the tradeoff between minimizing regret and acquiring information in the face of an unknown environment. We show that the DEC is necessary and sufficient for sample-efficient interactive learning, and in particular 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, which attains a regret bound matching our lower bound, thereby achieving optimal sample-efficient learning as characterized by the Decision-Estimation Coefficient.

Taken together, these results give 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.