Inertial Methods with Viscous and Hessian driven Damping for Non-Convex Optimization

TL;DR

This paper investigates second-order non-convex optimization dynamics with viscous and Hessian-driven damping, providing convergence conditions, rates, and algorithms, including novel strategies for locally Lipschitz smooth functions.

Contribution

It introduces new convergence analysis for second-order dynamics with Hessian-driven damping, including cases with only local Lipschitz smoothness and backtracking strategies.

Findings

Convergence of gradient to zero under specific damping conditions.

Global convergence to critical points for definable functions.

Almost sure convergence to local minima for Morse functions.

Abstract

In this paper, we aim to study non-convex minimization problems via second-order (in-time) dynamics, including a non-vanishing viscous damping and a geometric Hessian-driven damping. Second-order systems that only rely on a viscous damping may suffer from oscillation problems towards the minima, while the inclusion of a Hessian-driven damping term is known to reduce this effect without explicit construction of the Hessian in practice. There are essentially two ways to introduce the Hessian-driven damping term: explicitly or implicitly. For each setting, we provide conditions on the damping coefficients to ensure convergence of the gradient towards zero. Moreover, if the objective function is definable, we show global convergence of the trajectory towards a critical point as well as convergence rates. Besides, in the autonomous case, if the objective function is Morse, we conclude that the trajectory converges to a local minimum of the objective for almost all initializations. We also study algorithmic schemes for both dynamics and prove discrete analogues of the previous properties under appropriate stepsize conditions. In particular, we consider the case where the objective is only locally Lipschitz smooth and propose a backtracking strategy for which we establish convergence guarantees. Our work is the first one that handles this situation.