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

Nicholas Pischke

TL;DR

This work extends stochastic proximal point methods to nonlinear, nonpositively curved spaces by formulating and analyzing a stochastic proximal point algorithm on separable Hadamard (CAT(0)) spaces. It establishes strong convergence to the unique zero of the mean monotone vector field, under a strong monotonicity assumption together with mild independence and tangent-space separability, and provides explicit, uniform nonasymptotic rates in both expectation and almost surely. The contributions generalize Hilbert-space results to a broad geometric setting, yielding fast convergence guarantees under additional second-moment conditions and delivering corollaries for stochastic convex minimization. The results bridge stochastic approximation with metric geometry, offering quantitative convergence certificates that are robust to the underlying space, distribution, and data geometry.

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.

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

TL;DR

This work extends stochastic proximal point methods to nonlinear, nonpositively curved spaces by formulating and analyzing a stochastic proximal point algorithm on separable Hadamard (CAT(0)) spaces. It establishes strong convergence to the unique zero of the mean monotone vector field, under a strong monotonicity assumption together with mild independence and tangent-space separability, and provides explicit, uniform nonasymptotic rates in both expectation and almost surely. The contributions generalize Hilbert-space results to a broad geometric setting, yielding fast convergence guarantees under additional second-moment conditions and delivering corollaries for stochastic convex minimization. The results bridge stochastic approximation with metric geometry, offering quantitative convergence certificates that are robust to the underlying space, distribution, and data geometry.

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.

Paper Structure

This paper contains 9 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The contributions of the present paper and related work
  4. Preliminaries
  5. Geodesics, ${\rm{CAT}(0)}$ spaces and tangent spaces
  6. Monotone vector fields, maximality and resolvents
  7. Measurability and integration on ${\rm{CAT}(0)}$ spaces
  8. Random monotone vector fields on nonlinear spaces
  9. A stochastic proximal point algorithm

Key Result

Lemma 2.1

For any $x,a,b\in X$:

Theorems & Definitions (32)

  • Lemma 2.2
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8: p. 371 in Sturm2003
  • ...and 22 more