On Busemann subgradient methods for stochastic minimization in Hadamard spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910015679365120 |
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| author | Pischke, Nicholas |
| author_facet | Pischke, Nicholas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Busemann subgradient methods for stochastic minimization in Hadamard spaces Pischke, Nicholas Optimization and Control 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. |
| title | On Busemann subgradient methods for stochastic minimization in Hadamard spaces |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2602.08127 |