Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Busemann subgradient methods for stochastic minimization in Hadamard spaces

Nicholas Pischke

TL;DR

The paper extends the Busemann-subgradient method to stochastic minimization of $F(x)=\int f(e,x)\,d\mu(e)$ over closed convex subsets of separable Hadamard spaces, enabling optimization beyond linear spaces. It develops a general weak convergence theory for stochastic processes in Hadamard spaces via a stochastic quasi-Fejér monotonicity framework and a nonlinear Pettis-type measurability result, and then specializes to the SB method with measurability guarantees. Under local compactness of the space, SB is shown to converge strongly to a minimizer; under the weaker $(\overline{Q}_4)$ condition, the ergodic averages converge weakly to a minimizer, with additional strong-convexity assumptions yielding explicit convergence rates. The work broadens stochastic convex optimization tools to nonlinear metric spaces and provides foundational results (including a nonlinear Pettis-type theorem) that may impact related stochastic methods on CAT$(0)$ spaces.

Abstract

We study the recently introduced Busemann subgradient method due to Goodwin, Lewis, Nicolae and López-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-Fejér 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.

On Busemann subgradient methods for stochastic minimization in Hadamard spaces

TL;DR

The paper extends the Busemann-subgradient method to stochastic minimization of over closed convex subsets of separable Hadamard spaces, enabling optimization beyond linear spaces. It develops a general weak convergence theory for stochastic processes in Hadamard spaces via a stochastic quasi-Fejér monotonicity framework and a nonlinear Pettis-type measurability result, and then specializes to the SB method with measurability guarantees. Under local compactness of the space, SB is shown to converge strongly to a minimizer; under the weaker condition, the ergodic averages converge weakly to a minimizer, with additional strong-convexity assumptions yielding explicit convergence rates. The work broadens stochastic convex optimization tools to nonlinear metric spaces and provides foundational results (including a nonlinear Pettis-type theorem) that may impact related stochastic methods on CAT spaces.

Abstract

We study the recently introduced Busemann subgradient method due to Goodwin, Lewis, Nicolae and López-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 , a slight extension of the condition of Kirk and Payanak, which in particular includes Hilbert spaces, -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-Fejér 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.
Paper Structure (11 sections, 29 theorems, 93 equations)

This paper contains 11 sections, 29 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results and related work
  4. Preliminaries
  5. A nonlinear variant of Pettis' theorem
  6. Stochastic quasi-Fejér monotonicity and weak (ergodic) convergence
  7. Initial considerations on the method \ref{['SB']}
  8. Boundary cones and Busemann subgradients
  9. Measurability of \ref{['SB']}
  10. Measurable Busemann oracles
  11. Proofs of the main results

Key Result

Theorem 1.1

Let $X$ be a Hadamard space with the geodesic extension property and at least two points and let $C\subseteq X$ be nonempty, closed and convex. Let $F(x):=\sum_{i=1}^m f_i(x)$ be given, where each $f_i:C\to\mathbb{R}$ is Busemann subdifferentiable, and such that $\mathrm{argmin}F\neq\emptyset$. Assu

Theorems & Definitions (47)

  • Theorem 1.1: Theorem 6.2 in GoodwinLewisLopezAcedoNicolae2025
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3: Aumann1969, see also Corollary 18.27 in AliprantisBorder2006
  • Lemma 3.4: Corollary 18.8 in AliprantisBorder2006
  • proof : Proof of Lemma \ref{['minMeas']}
  • proof : Proof of Proposition \ref{['nonlinearPettis']}
  • ...and 37 more