# New nonasymptotic convergence rates of stochastic proximal pointalgorithm for convex optimization problems

## TL;DR

This paper establishes new nonasymptotic convergence rates for the stochastic proximal point algorithm in convex optimization, under weak regularity conditions, including linear convergence in certain settings.

## Contribution

It introduces a weak linear regularity condition that enhances convergence rate analysis of stochastic proximal point methods for convex problems.

## Key findings

- Achieves an $rac{1}{k}$ convergence rate under weak regularity assumptions.
- Demonstrates linear convergence in the interpolation setting.
- Applicable to many non-strongly convex functions in machine learning.

## Abstract

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to noisy stochastic optimization, with a resulting iteration related to stochastic alternating projections. Inspired by the scalability of alternating projection methods, we start from the (linear) regularity assumption, typically used in convex feasiblity problems to guarantee the linear convergence of stochastic alternating projection methods, and analyze a general weak linear regularity condition which facilitates convergence rate boosts in stochastic proximal point schemes. Our applications include many non-strongly convex functions classes often used in machine learning and statistics. Moreover, under weak linear regularity assumption we guarantee $\mathcal{O}\left(\frac{1}{k}\right)$ convergence rate for SPP, in terms of the distance to the optimal set, using only projections onto a simple component set. Linear convergence is obtained for interpolation setting, when the optimal set of the expected cost is included into the optimal sets of each functional component.