  Much of the work in online learning focuses on the study of sublinear upper
bounds on the regret. In this work, we initiate the study of best-case lower
bounds in online convex optimization, wherein we bound the largest improvement
an algorithm can obtain relative to the single best action in hindsight. This
problem is motivated by the goal of better understanding the adaptivity of a
learning algorithm. Another motivation comes from fairness: it is known that
best-case lower bounds are instrumental in obtaining algorithms for
decision-theoretic online learning (DTOL) that satisfy a notion of group
fairness. Our contributions are a general method to provide best-case lower
bounds in Follow The Regularized Leader (FTRL) algorithms with time-varying
regularizers, which we use to show that best-case lower bounds are of the same
order as existing upper regret bounds: this includes situations with a fixed
learning rate, decreasing learning rates, timeless methods, and adaptive
gradient methods. In stark contrast, we show that the linearized version of
FTRL can attain negative linear regret. Finally, in DTOL with two experts and
binary predictions, we fully characterize the best-case sequences, which
provides a finer understanding of the best-case lower bounds.
