Convergence Rates for Stochastic Proximal and Projection Estimators
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910027653054464 |
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| author | Morales, Diego Pérez-Aros, Pedro Vilches, Emilio |
| author_facet | Morales, Diego Pérez-Aros, Pedro Vilches, Emilio |
| contents | 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 $\sqrtδ$ bound, with explicit constants for the proximal mapping, in the $ρ$-weakly convex (possibly nonsmooth) setting, and we also obtain a dimension-explicit $\sqrtδ$ 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 $O(δ)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06750 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence Rates for Stochastic Proximal and Projection Estimators Morales, Diego Pérez-Aros, Pedro Vilches, Emilio Optimization and Control 49J53 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 $\sqrtδ$ bound, with explicit constants for the proximal mapping, in the $ρ$-weakly convex (possibly nonsmooth) setting, and we also obtain a dimension-explicit $\sqrtδ$ 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 $O(δ)$ 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. |
| title | Convergence Rates for Stochastic Proximal and Projection Estimators |
| topic | Optimization and Control 49J53 |
| url | https://arxiv.org/abs/2602.06750 |