An SDE Perspective on Stochastic Inertial Gradient Dynamics with Time-Dependent Viscosity and Geometric Damping

TL;DR

This paper develops a stochastic differential equation framework with time-dependent viscosity and geometric damping to analyze and improve convex optimization algorithms, demonstrating fast convergence and new complexity results.

Contribution

It introduces second-order stochastic differential equations with viscous and Hessian-driven damping for convex optimization, extending previous first-order analyses.

Findings

Almost sure convergence of values with time-dependent damping

Fast convergence rates in expectation for convex and strongly convex cases

Almost sure weak convergence of trajectories when Hessian damping is zero

Abstract

Our approach is part of the close link between continuous dissipative dynamical systems and optimization algorithms. We aim to solve convex minimization problems by means of stochastic inertial differential equations which are driven by the gradient of the objective function. This will provide a general mathematical framework for analyzing fast optimization algorithms with stochastic gradient input. Our study is a natural extension of our previous work devoted to the first-order in time stochastic steepest descent. Our goal is to develop these results further by considering second-order stochastic differential equations in time, incorporating a viscous time-dependent damping and a Hessian-driven damping. To develop this program, we rely on stochastic Lyapunov analysis. Assuming a square-integrability condition on the diffusion term times a function dependant on the viscous damping, and that the Hessian-driven damping is a positive constant, our first main result shows that almost surely, there is convergence of the values, and states fast convergence of the values in expectation. Besides, in the case where the Hessian-driven damping is zero, we conclude with the fast convergence of the values in expectation and in almost sure sense, we also managed to prove almost sure weak convergence of the trajectory. We provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex and strongly convex case.