Hamiltonian Descent Methods

TL;DR

This paper introduces Hamiltonian descent methods that achieve linear convergence on a broad class of convex functions, including non-smooth and unbounded cases, using first-order gradient information and conformal Hamiltonian dynamics.

Contribution

It presents a new family of optimization algorithms based on conformal Hamiltonian dynamics that extend linear convergence guarantees to larger classes of convex functions.

Findings

Achieves linear convergence on convex functions with singular or unbounded second derivatives.

Provides conditions for convergence of explicit and implicit Hamiltonian methods.

Expands the class of functions where first-order methods can guarantee linear convergence.

Abstract

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.