Convergence Rates for Stochastic Proximal and Projection Estimators

In this paper, we establish explicit convergence rates for the stochastic smooth approximations of infimal convolutions introduced and developed in citeMR4581306,MR4923371. In particular, we quantify the convergence of the associated barycentric estimators toward proximal mappings and metric projections. We prove a dimension-explicit √δ bound, with explicit constants for the proximal mapping, in the ρ-weakly convex (possibly nonsmooth) setting, and we also obtain a dimension-explicit √δ rate for the metric projection onto an arbitrary convex set with nonempty interior. Under additional regularity, namely C2 smoothness with globally Lipschitz Hessian, we derive an improved linear O(δ) rate with explicit constants, and we obtain refined projection estimates for convex sets with local C2,1 boundary. Examples demonstrate that these rates are optimal.