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Recently, much work has been done on extending the scope of online learningand incremental stochastic optimization algorithms. In this paper we contributeto this effort in two ways: First, based on a new regret decomposition and ageneralization of Bregman divergences, we provide a self-contained, modularanalysis of the two workhorses of online learning: (general) adaptive versionsof Mirror Descent (MD) and the Follow-the-Regularized-Leader (FTRL) algorithms.The analysis is done with extra care so as not to introduce assumptions notneeded in the proofs and allows to combine, in a straightforward way, differentalgorithmic ideas (e.g., adaptivity, optimism, implicit updates) and learningsettings (e.g., strongly convex or composite objectives). This way we are ableto reprove, extend and refine a large body of the literature, while keeping theproofs concise. The second contribution is a byproduct of this carefulanalysis: We present algorithms with improved variational bounds for smooth,composite objectives, including a new family of optimistic MD algorithms withonly one projection step per round. Furthermore, we provide a simple extensionof adaptive regret bounds to practically relevant non-convex problem settingswith essentially no extra effort.

A Modular Analysis of Adaptive (Non-)Convex Optimization: Optimism, Composite Objectives, and Variational Bounds