On Busemann subgradient methods for stochastic minimization in Hadamard spaces

TL;DR

This paper extends Busemann subgradient methods to stochastic minimization in Hadamard spaces, proving convergence under various conditions and providing explicit rates under strong convexity.

Contribution

It introduces a convergence analysis for stochastic Busemann subgradient methods in Hadamard spaces, including new weak and strong convergence results.

Findings

Proved strong convergence under local compactness.

Established weak ergodic convergence under condition $(ar{Q}_4)$.

Derived explicit convergence rates under strong convexity.

Abstract

We study the recently introduced Busemann subgradient method due to Goodwin, Lewis, Nicolae and L\'opez-Acedo, extending it to minimize the mean of a stochastic function over general Hadamard spaces. We prove a strong convergence theorem under a local compactness assumption and further prove weak ergodic convergence of the method over Hadamard spaces satisfying condition $(\overline{Q}_4)$, a slight extension of the $(Q_4)$ condition of Kirk and Payanak, which in particular includes Hilbert spaces, $\mathbb{R}$-trees and spaces of constant curvature. The proof is based on a general (weak) convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fej\'er monotonicity, together with a nonlinear variant of Pettis' theorem, which are of independent interest. Lastly, we provide a strong convergence result under a strong convexity assumption, and in that case in particular derive explicit rates of convergence.