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Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces

Ariel Goodwin, Adrian S. Lewis, Genaro López-Acedo, Adriana Nicolae

TL;DR

This work extends convex optimization to Hadamard spaces by introducing Busemann subgradients and Busemann envelopes to enable global, primal subgradient methods without relying on linear structure or curvature lower bounds. It develops stochastic and incremental splitting algorithms for minimizing sums of geodesically convex components, proving complexity bounds parallel to Euclidean theory (e.g., an $O(\varepsilon^{-2})$ iteration rate in appropriate settings) and providing a concrete analysis for the median/p-mean problems. The authors connect this framework to horospherical convexity, compare with Alexandrov-space subgradients, and show how a Busemann oracle can yield tractable subgradient steps on nonlinear spaces. They validate the approach computationally in BHV tree space, illustrating practical viability for problems like computing medians in complex non-Euclidean geometries. Overall, the paper broadens the toolkit for non-Euclidean convex optimization, enabling scalable, provably convergent methods for important geometric data analysis tasks.

Abstract

As a foundation for optimization, convexity is useful beyond the classical settings of Euclidean and Hilbert space. The broader arena of nonpositively curved metric spaces, which includes manifolds like hyperbolic space, as well as metric trees and more general CAT(0) cubical complexes, supports primal tools like proximal operations for geodesically convex functions. However, the lack of linear structure in such spaces complicates dual constructions like subgradients. To address this hurdle, we introduce a new type of subgradient for functions on Hadamard spaces, based on Busemann functions. Our notion supports generalizations of classical stochastic and incremental subgradient methods, with guaranteed complexity bounds. We illustrate with subgradient algorithms for $p$-mean problems in general Hadamard spaces, in particular computing medians in BHV tree space.

Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces

TL;DR

This work extends convex optimization to Hadamard spaces by introducing Busemann subgradients and Busemann envelopes to enable global, primal subgradient methods without relying on linear structure or curvature lower bounds. It develops stochastic and incremental splitting algorithms for minimizing sums of geodesically convex components, proving complexity bounds parallel to Euclidean theory (e.g., an iteration rate in appropriate settings) and providing a concrete analysis for the median/p-mean problems. The authors connect this framework to horospherical convexity, compare with Alexandrov-space subgradients, and show how a Busemann oracle can yield tractable subgradient steps on nonlinear spaces. They validate the approach computationally in BHV tree space, illustrating practical viability for problems like computing medians in complex non-Euclidean geometries. Overall, the paper broadens the toolkit for non-Euclidean convex optimization, enabling scalable, provably convergent methods for important geometric data analysis tasks.

Abstract

As a foundation for optimization, convexity is useful beyond the classical settings of Euclidean and Hilbert space. The broader arena of nonpositively curved metric spaces, which includes manifolds like hyperbolic space, as well as metric trees and more general CAT(0) cubical complexes, supports primal tools like proximal operations for geodesically convex functions. However, the lack of linear structure in such spaces complicates dual constructions like subgradients. To address this hurdle, we introduce a new type of subgradient for functions on Hadamard spaces, based on Busemann functions. Our notion supports generalizations of classical stochastic and incremental subgradient methods, with guaranteed complexity bounds. We illustrate with subgradient algorithms for -mean problems in general Hadamard spaces, in particular computing medians in BHV tree space.

Paper Structure

This paper contains 18 sections, 18 theorems, 152 equations, 5 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Geodesic geometry and convexity
  3. Busemann subgradients and envelopes
  4. The geometry of Busemann subdifferentiability
  5. Subgradients in Alexandrov space
  6. Subgradient-based minimization of sums
  7. Subgradient methods for the median problem
  8. Computational experiments
  9. Proof of Proposition \ref{['prop:boundconv']}
  10. Proof of Lemma \ref{['lem:convergence']}
  11. Proof of Example \ref{['ex:euclid']}
  12. Proof of Example \ref{['ex:convcont']}
  13. Proof of Example \ref{['ex:dist']}
  14. Proof of Example \ref{['ex:distballs']}
  15. Proof of Proposition \ref{['biconjugate']}
  16. ...and 3 more sections

Key Result

Proposition 2.1

The Busemann function corresponding to any fixed direction depends on the reference point only through an additive constant.

Figures (5)

  • Figure 4.1: Part of a level set that fails to be horospherically convex.
  • Figure 8.1: Three trees in $\mathcal{T}_4$ with neighboring respective orthants, embedded isometrically in $\mathbb{R}^2$ (recreated from sturmmean).
  • Figure 8.2: Convergence of $f(x^k)$ to $f_{\text{opt}}$ in Example \ref{['ex:ex1']} using different stepsizes and choices of initial tree.
  • Figure 8.3: The tree topology defining three neighboring quadrants and their common spine.
  • Figure 8.4: Convergence of $f(x^k)$ to $f_{\text{opt}}$ in Example \ref{['ex:ex2']} using different stepsizes and choices of initial tree.

Theorems & Definitions (54)

  • Proposition 2.1
  • Proposition 2.2: Convergence of directions
  • Example 2.1: Boundary of Euclidean space
  • Example 2.2: Boundary of the tripod
  • Example 2.3: Boundary of hyperbolic space
  • Lemma 2.4: Convergence in the boundary cone
  • Proposition 2.3: Properties of the pairing
  • proof
  • Definition 3.1
  • Proposition 3.1
  • ...and 44 more