Abstract

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings.  While 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. 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 correspond to traveling the same curve in spacetime at different speeds.  We also describe a “Bregman Hamiltonian” which generates the accelerated dynamics, we develop a symplectic integrator for this Hamiltonian and we discuss relations between this symplectic integrator and classical Nesterov acceleration.  [Joint work with Andre Wibisono, Ashia Wilson and Michael Betancourt.]