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We consider a stochastic version of the proximal point algorithm foroptimization problems posed on a Hilbert space. A typical application of thisis supervised learning. While the method is not new, it has not beenextensively analyzed in this form. Indeed, most related results are confined tothe finite-dimensional setting, where error bounds could depend on thedimension of the space. On the other hand, the few existing results in theinfinite-dimensional setting only prove very weak types of convergence, owingto weak assumptions on the problem. In particular, there are no results thatshow convergence with a rate. In this article, we bridge these two worlds byassuming more regularity of the optimization problem, which allows us to proveconvergence with an (optimal) sub-linear rate also in an infinite-dimensionalsetting.  In particular, we assume that the objective function is the expected value ofa family of convex differentiable functions. While we require that the fullobjective function is strongly convex, we do not assume that its constituentparts are so. Further, we require that the gradient satisfies a weak localLipschitz continuity property, where the Lipschitz constant may growpolynomially given certain guarantees on the variance and higher moments nearthe minimum.  We illustrate these results by discretizing a concrete infinite-dimensionalclassification problem with varying degrees of accuracy.

Sub-linear convergence of a stochastic proximal iteration method in Hilbert space