Optimal convergence rate of inertial gradient system with flat geometries and perturbations

In this paper, we study the decay estimate of the trajectories of the second order differential equation:$ \ddot{x}(t)+\gamma (t) \dot{x}(t)+\nabla F(x(t)) = g(t), $where $ \gamma (t) = \frac{\alpha }{t} $ is a viscous damping coefficient with $ \alpha >0 $, $ F $ is a convex differentiable function, and $ g(t) $ is a small perturbation term. We are concerned in the optimal convergence rates for damped inertial gradient dynamics with flat geometries, which is proposed in Sebbouh and Dossal (SIAM J Optim 30: 1850-1877, 2020). Making use of some Lyapunov functions and iteration method, we get some results. More specifically, we obtained the convergence rate $ t^{-{\frac{2\gamma \alpha }{\gamma +2}} } $ of $ F(x(t))-F^* $ when $ 1+\frac{2}{\gamma } < \alpha <\frac{\gamma +2}{\gamma -2} $. In addition, for the case $ \alpha<1+\frac{2}{\gamma} $, we proved that the convergence rate cannot be better than $ t^{-\frac{2\bar{\gamma } \alpha }{\gamma +2} } $ for $ \bar{ \gamma }\le\gamma+2 $.