Convergence Rates for Stochastic Proximal and Projection Estimators
Diego Morales, Pedro Pérez-Aros, Emilio Vilches
TL;DR
This work provides non-asymptotic, dimension-explicit convergence rates for stochastic barycentric estimators that approximate proximal mappings and metric projections via Gaussian perturbations and Laplace-type weighting. It proves a sharp \mathcal{O}(\sqrt{\delta}) bias in the nonsmooth, ρ-weakly convex setting, and, under C^2 smoothness with globally Lipschitz Hessian, an improved \mathcal{O}(\delta) bias with explicit constants, underscoring the role of curvature. For projections, it yields a quantitative bound ||p_\delta(x)-proj_C(x)|| ≤ \sqrt{n\delta} for convex C, and establishes boundary-sensitive refinements giving an O(δ) expansion when ∂C has a local C^{2,1} chart. The results are sharp and provide practical guidance on choosing the smoothing parameter δ and understanding dimension- and geometry-dependent behavior in black-box proximal/projection estimation.
Abstract
In this paper, we establish explicit, non-asymptotic 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 $\sqrtδ$ bound in the $ρ$-weakly convex (possibly nonsmooth) setting and show, by examples, that this order is sharp. Under additional regularity, namely $C^{2}$ 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 $C^{2,1}$ boundary.