Zusammenfassung
We propose a family of optimization methods that achieve linear convergence
using first-order gradient information and constant step sizes on a class of
convex functions much larger than the smooth and strongly convex ones. This
larger class includes functions whose second derivatives may be singular or
unbounded at their minima. Our methods are discretizations of conformal
Hamiltonian dynamics, which generalize the classical momentum method to model
the motion of a particle with non-standard kinetic energy exposed to a
dissipative force and the gradient field of the function of interest. They are
first-order in the sense that they require only gradient computation. Yet,
crucially the kinetic gradient map can be designed to incorporate information
about the convex conjugate in a fashion that allows for linear convergence on
convex functions that may be non-smooth or non-strongly convex. We study in
detail one implicit and two explicit methods. For one explicit method, we
provide conditions under which it converges to stationary points of non-convex
functions. For all, we provide conditions on the convex function and kinetic
energy pair that guarantee linear convergence, and show that these conditions
can be satisfied by functions with power growth. In sum, these methods expand
the class of convex functions on which linear convergence is possible with
first-order computation.
Nutzer