  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
Gy\"{o}rgy.
