Convergence Rates for Stochastic Proximal and Projection Estimators

TL;DR

This paper derives explicit convergence rates for stochastic approximations of infimal convolutions, including proximal mappings and metric projections, with optimal bounds in various regularity settings.

Contribution

It provides the first explicit, dimension-dependent convergence rates for stochastic proximal and projection estimators, including optimal bounds under smoothness assumptions.

Findings

Dimension-explicit bcb7b4^{1/2} rate for weakly convex functions

Dimension-explicit bcb7b4 rate for metric projections onto convex sets

Improved linear bcb7b4 rate under smoothness assumptions

Abstract

In this paper, we establish explicit convergence rates for the stochastic smooth approximations of infimal convolutions introduced and developed in \cite{MR4581306,MR4923371}. In particular, we quantify the convergence of the associated barycentric estimators toward proximal mappings and metric projections. We prove a dimension-explicit $\sqrt{\delta}$ bound, with explicit constants for the proximal mapping, in the $\rho$-weakly convex (possibly nonsmooth) setting, and we also obtain a dimension-explicit $\sqrt{\delta}$ rate for the metric projection onto an arbitrary convex set with nonempty interior. Under additional regularity, namely $C^{2}$ smoothness with globally Lipschitz Hessian, we derive an improved linear $O(\delta)$ rate with explicit constants, and we obtain refined projection estimates for convex sets with local $C^{2,1}$ boundary. Examples demonstrate that these rates are optimal.