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

arXiv: https://arxiv.org/abs/2510.10697 · HTML: https://mlsanity.com/paper/2510.10697/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 69 references that are in the map.
- First posted on arXiv 12 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Stochastic (Approximate) Proximal Point Methods: Convergence, Optimality, and Adaptivity](https://mlsanity.com/paper/1810.05633/) (2018)
- [Incremental Gradient, Subgradient, and Proximal Methods for Convex Optimization: A Survey](https://mlsanity.com/paper/1507.01030/) (2015)
- [Ergodic convergence of a stochastic proximal point algorithm](https://mlsanity.com/paper/1504.05400/) (2015)
- [Riemannian stochastic optimization methods avoid strict saddle points](https://mlsanity.com/paper/2311.02374/) (2023)
- [Asymptotic regularity of a generalised stochastic Halpern scheme](https://mlsanity.com/paper/2411.04845/) (2024)
- [A quantitative Robbins-Siegmund theorem](https://mlsanity.com/paper/2410.15986/) (2024)
- [Stochastic Quasi-Fej'er Block-Coordinate Fixed Point Iterations With Random Sweeping II: Mean-Square and Linear Convergence](https://mlsanity.com/paper/1704.08083/) (2017)
- [Stochastic Proximal Point Methods for Monotone Inclusions under Expected Similarity](https://mlsanity.com/paper/2405.14255/) (2024)
- [A variational approach to stochastic minimization of convex functionals](https://mlsanity.com/paper/1605.03289/) (2016)
- [Effective strong convergence of the proximal point algorithm in CAT(0) spaces](https://mlsanity.com/paper/1801.02179/) (2018)
- [Monotone vector fields and generation of nonexpansive semigroups in complete CAT(0) spaces](https://mlsanity.com/paper/1906.05984/) (2019)
- [Sub-linear convergence of a stochastic proximal iteration method in Hilbert space](https://mlsanity.com/paper/2010.12348/) (2020)
- [Basic convex analysis in metric spaces with bounded curvature](https://mlsanity.com/paper/2302.03588/) (2023)
- [An abstract effective convergence theorem for stochastic processes, with applications to stochastic approximation](https://mlsanity.com/paper/2504.12922/) (2025)

## Abstract (arXiv)

We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field. Generalizing previous work by P. Bianchi, we prove the convergence of this method under a suitable strong monotonicity assumption in (separable) Hilbert-Hadamard spaces, that is assuming that all tangent spaces isometrically embed into Hilbert spaces (covering, but not being limited to, the setting of 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. Sublinear nonasymptotic guarantees under additional second-moment conditions on the Yosida approximates and special cases of stochastic convex minimization are discussed.