In what follows, we review the Decision-Estimation Coefficient and Estimation-to-Decisions meta-algorithm (Section 1.2), highlighting opportunities for improvement.

A central problem in online learning and decision making---from bandits to reinforcement learning---is to understand what modeling assumptions lead to sample-efficient learning guarantees. We consider a general adversarial decision making framework that encompasses (structured) bandit problems with adversarial rewards and reinforcement learning problems with adversarial dynamics. Our main result is to show---via new upper and lower bounds---that the Decision-Estimation Coefficient, a complexity measure introduced by Foster et al. in the stochastic counterpart to our setting, is necessary and sufficient to obtain low regret for adversarial decision making. However, compared to the stochastic setting, one must apply the Decision-Estimation Coefficient to the convex hull of the class of models (or, hypotheses) under consideration. This establishes that the price of accommodating adversarial rewards or dynamics is governed by the behavior of the model class under convexification, and recovers a number of existing results --both positive and negative. En route to obtaining these guarantees, we provide new structural results that connect the Decision-Estimation Coefficient to variants of other well-known complexity measures, including the Information Ratio of Russo and Van Roy and the Exploration-by-Optimization objective of Lattimore and György.

We consider the problem of interactive decision making, encompassing structured bandits and reinforcementlearning with general function approximation. Recently, Foster et al. (2021) introduced theDecision-Estimation Coefficient, a measure of statistical complexity that lower bounds the optimal regret for interactive decisionmaking, as well as a meta-algorithm, Estimation-to-Decisions, which achieves upperbounds in terms of the same quantity. Estimation-to-Decisions is a reduction, which liftsalgorithms for (supervised) online estimation into algorithms fordecision making. In this paper, we show that by combining Estimation-to-Decisions witha specialized form of optimistic estimation introduced byZhang (2022), it is possible to obtain guaranteesthat improve upon those of Foster et al. (2021) byaccommodating more lenient notions of estimation error. We use this approach to derive regret bounds formodel-free reinforcement learning with value function approximation, and give structural results showing when it can and cannot help more generally.

A long line of works characterizes the sample complexity of regret minimization in sequential decision-making by min-max programs. In the corresponding saddle-point game, the min-player optimizes the sampling distribution against an adversarial max-player that chooses confusing models leading to large regret.