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Large sectors of the recent optimization literature focused in the lastdecade on the development of optimal stochastic first order schemes forconstrained convex models under progressively relaxed assumptions. Stochasticproximal point is an iterative scheme born from the adaptation of proximalpoint algorithm to noisy stochastic optimization, with a resulting iterationrelated to stochastic alternating projections. Inspired by the scalability ofalternating projection methods, we start from the (linear) regularityassumption, typically used in convex feasiblity problems to guarantee thelinear convergence of stochastic alternating projection methods, and analyze ageneral weak linear regularity condition which facilitates convergence rateboosts in stochastic proximal point schemes. Our applications include manynon-strongly convex functions classes often used in machine learning andstatistics. Moreover, under weak linear regularity assumption we guarantee$\mathcal{O}\left(\frac{1}{k}\right)$ convergence rate for SPP, in terms of thedistance to the optimal set, using only projections onto a simple componentset. Linear convergence is obtained for interpolation setting, when the optimalset of the expected cost is included into the optimal sets of each functionalcomponent.

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