CONVERGENCE RATES OF DAMPED INERTIAL DYNAMICS UNDER GEOMETRIC CONDITIONS AND PERTURBATIONS

In this article a family of second-order ODEs associated with the inertial gradient descent 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 made 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 Lojasiewicz 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 Lojasiewicz properties especially in the setting of stochastic algorithms.