We present tools for the analysis of Follow-The-Regularized- Leader (FTRL), Dual Averaging, and Mirror Descent algorithms when the regularizer (equivalently, prox-function or learning rate schedule) is chosen adaptively based on the data. Adaptivity can be used to prove regret bounds that hold on every round, and also allows for data-dependent regret bounds as in AdaGrad-style algorithms (e.g., Online Gradient Descent with adaptive per-coordinate learning rates). We present results from a large number of prior works in a unified manner, using a modular and tight analysis that isolates the key arguments in easily re-usable lemmas. This approach strengthens previously known FTRL analysis techniques to produce bounds as tight as those achieved by potential functions or primal-dual analysis. Further, we prove a general and exact equivalence between adaptive Mirror Descent algorithms and a corresponding FTRL update, which allows us to analyze Mirror Descent algorithms in the same framework. The key to bridging the gap between Dural Averaging and Mirror Descent algorithms lies in an analysis of the FTRL-Proximal algorithm family. Our regret bounds are proved in the most general form, holding for arbitrary norms and non- smooth regularizers with time-varying weight.
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