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First-order methods play a central role in large-scale machine learning. Eventhough many variations exist, each suited to a particular problem, almost allsuch methods fundamentally rely on two types of algorithmic steps: gradientdescent, which yields primal progress, and mirror descent, which yields dualprogress.  We observe that the performances of gradient and mirror descent arecomplementary, so that faster algorithms can be designed by LINEARLY COUPLINGthe two. We show how to reconstruct Nesterov's accelerated gradient methodsusing linear coupling, which gives a cleaner interpretation than Nesterov'soriginal proofs. We also discuss the power of linear coupling by extending itto many other settings that Nesterov's methods cannot apply to.

Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent