Weak convergence of the stochastic proximal point method in metric spaces

TL;DR

This paper proves the almost sure weak convergence of a stochastic proximal point method for convex functions in Hadamard spaces, extending previous results to a more general nonlinear metric space setting.

Contribution

It establishes weak convergence of the stochastic proximal point method in Hadamard spaces under mild growth conditions, broadening applicability beyond prior regularity assumptions.

Findings

Proves almost sure weak convergence in Hadamard spaces.

Extends convergence results to functions with mild growth conditions.

Combines stochastic process convergence with new mean value convergence argument.

Abstract

We prove the almost sure weak convergence of a stochastic proximal point method for minimizing a convex integral function in the general nonlinear context of complete geodesic metric spaces of nonpositive curvature (so-called Hadamard spaces), solving a problem of M. Ba\v{c}\'ak. This method, formulated in the context of a mild growth condition on the function which generalizes Lipschitz continuity, was previously only considered in the context of strong metric regularity conditions or in the context of locally compact spaces. The proof is a combination of a weak almost sure convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fej\'er monotonicity, due to previous work of the author, together with a new argument for proving the almost sure convergence of the mean function values of the process towards the minimal value.