Asymptotic behaviour of stochastic inertial dynamics incorporating a Tikhonov regularization term

TL;DR

This paper analyzes the asymptotic behavior of stochastic inertial dynamics with Tikhonov regularization in Hilbert spaces, establishing convergence rates and solution properties under noise and regularization decay conditions.

Contribution

It introduces a stochastic inertial system with Tikhonov regularization for convex optimization, proving existence, uniqueness, and convergence rates of solutions in noisy settings.

Findings

Established existence and uniqueness of solution trajectories.

Derived convergence rates in expectation and almost surely.

Demonstrated asymptotic convergence of trajectories to optimal solutions.

Abstract

In a separable Hilbert space, we study the minimization problem of a convex smooth function with Lipschitz continuous gradient whose evaluations are corrupted by random noise. To this end, we associate a stochastic inertial system that incorporates Tikhonov regularization with the optimization problem. We establish existence and uniqueness of a solution trajectory for this system. Then, we derive an upper bound on the expected value of an appropriate associated energy function given square-integrability of the diffusion $\sigma_X$ before focusing on the particular case where the parameter function multiplied by the Tikhonov term is given by $\frac{1}{t^r}$ for $0<r<2$. For this setting, we show a.s. convergence rates as well as convergence rates in expectation for the function values along the trajectory to an infimal value, the trajectory process to an optimal solution and its time derivative to zero under a stronger integrability condition on $\sigma_X$.