A variational perspective on accelerated methods in optimization.
Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. Although many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the Bregman Lagrangian, which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods corresponds to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720.
Andre
A
Wibisono
Department of Statistics, University of California, Berkeley, CA 94720.
Ashia C
AC
Wilson
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720; jordan@cs.berkeley.edu.|Department of Statistics, University of California, Berkeley, CA 94720.
Michael I
MI
Jordan
Bregman divergence; Lagrangian framework; accelerated methods; convex optimization; mirror descent