Mean-square and linear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature

2025-10-12

AI Summary

This paper addresses the approximation of mean zero points of randomly perturbed monotone vector fields in nonlinear Hadamard spaces. To tackle this problem in a nonlinear, non-separable metric setting, we generalize the stochastic proximal point algorithm—originally developed for Hilbert spaces—to Hadamard manifolds for the first time. Under a tangent-space separation assumption and strong monotonicity, convergence is rigorously established via Yosida regularization and second-moment analysis. Our main contributions are: (1) explicit convergence rates in both mean-square and almost-sure senses; (2) non-asymptotic linear convergence guarantees; and (3) novel, unifying results even in the Hilbert space special case. Collectively, these advances provide a foundational theoretical framework for stochastic optimization on nonpositively curved geometries.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence under a suitable strong monotonicity assumption, together with a probabilistic independence assumption and a separability assumption on the tangent spaces. As a particular case, our results transfer previous work by P. Bianchi on that method in Hilbert spaces for the first time to Hadamard manifolds. Moreover, our convergence proof is fully effective and allows for the construction of explicit rates of convergence for the iteration towards the (unique) solution both in mean and almost surely. These rates are moreover highly uniform, being independent of most data surrounding the iteration, space or distribution. In that generality, these rates are novel already in the context of Hilbert spaces. Linear nonasymptotic guarantees under additional second-moment conditions on the Yosida approximates and special cases of stochastic convex minimization are discussed.

Problem

• Extends stochastic proximal point algorithm to nonlinear Hadamard spaces
• Proves convergence rates for monotone vector field zeros approximation
• Provides uniform nonasymptotic guarantees for stochastic optimization problems

Innovation

• Stochastic proximal point algorithm in Hadamard spaces
• Convergence with strong monotonicity and independence assumptions
• Effective construction of uniform explicit convergence rates