# Nesterov's acceleration and Polyak's heavy ball method in continuous time: convergence rate analysis under geometric conditions and perturbations

## TL;DR

This paper analyzes the convergence rates of a family of second order ODEs related to inertial gradient methods, considering geometric conditions and perturbations, to understand their behavior in optimization tasks.

## Contribution

It provides the first joint analysis of geometrical properties and perturbations for these ODEs, offering new asymptotic bounds on convergence rates.

## Key findings

- Derived new asymptotic bounds for convergence rates.
- Analyzed effects of damping and perturbations on convergence.
- Insights applicable to stochastic optimization algorithms.

## Abstract

In this article a family of second order ODEs associated to inertial gradient descend is studied. These ODEs are widely used to build trajectories converging to a minimizer $x^*$ of a function $F$, possibly convex. This family includes the continuous version of the Nesterov inertial scheme and the continuous heavy ball method. Several damping parameters, not necessarily vanishing, and a perturbation term $g$ are thus considered. The damping parameter is linked to the inertia of the associated inertial scheme and the perturbation term $g$ is linked to the error that can be done on the gradient of the function $F$. This article presents new asymptotic bounds on $F(x(t))-F(x^*)$ where $x$ is a solution of the ODE, when $F$ is convex and satisfies local geometrical properties such as {\L}ojasiewicz properties and under integrability conditions on $g$. Even if geometrical properties and perturbations were already studied for most ODEs of these families, it is the first time they are jointly studied. All these results give an insight on the behavior of these inertial and perturbed algorithms if $F$ satisfies some {\L}ojasiewicz properties especially in the setting of stochastic algorithms.