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arxiv: 2605.20805 · v1 · pith:ZTSH7UPWnew · submitted 2026-05-20 · 🧮 math.OC

Weak convergence of the stochastic proximal point method in metric spaces

Nicholas Pischke This is my paper

Pith reviewed 2026-05-21 04:02 UTC · model grok-4.3

classification 🧮 math.OC
keywords weak convergencestochastic proximal pointHadamard spacesconvex optimizationmetric spacesquasi-Fejer monotonicityintegral functionalsnonpositive curvature
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The pith

Stochastic proximal point method converges weakly almost surely in Hadamard spaces

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a stochastic proximal point method for minimizing a convex integral function produces iterates that converge weakly almost surely in complete geodesic metric spaces of nonpositive curvature. This holds under a mild growth condition on the objective that generalizes Lipschitz continuity and is used both to define the iterates and to prove convergence of their average values. The argument combines an existing result on stochastic processes satisfying a stochastic form of quasi-Fejér monotonicity with a new step showing almost sure convergence of mean function values to the infimum. A reader would care because the result applies in general nonlinear geometries without local compactness or stronger regularity assumptions that limited earlier work.

Core claim

The paper proves the almost sure weak convergence of a stochastic proximal point method for minimizing a convex integral function in complete geodesic metric spaces of nonpositive curvature. The method is formulated using a mild growth condition on the function that generalizes Lipschitz continuity. The proof relies on a weak almost sure convergence theorem for stochastic processes in these spaces that satisfy a stochastic variant of quasi-Fejér monotonicity together with a new argument establishing almost sure convergence of the mean function values to the minimal value.

What carries the argument

The stochastic proximal point iteration in Hadamard spaces under a mild growth condition on the objective, carried by stochastic quasi-Fejér monotonicity of the process.

If this is right

  • The iterates converge weakly almost surely to a minimizer of the convex integral function.
  • The mean values of the objective function converge almost surely to the minimal value.
  • The result applies directly in general Hadamard spaces without requiring local compactness.
  • The technique extends prior convergence theorems for stochastic processes in these spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mild growth condition and quasi-Fejér arguments may apply to other stochastic first-order methods in non-Euclidean metric spaces.
  • The framework could support optimization tasks on data structures naturally modeled by hyperbolic or tree-like geometries.
  • Additional assumptions on the function might yield explicit convergence rates under the same geometric setting.

Load-bearing premise

The objective function satisfies a mild growth condition that generalizes Lipschitz continuity to allow formulation of the method and to secure convergence of mean function values.

What would settle it

A concrete convex integral function on a non-locally compact Hadamard space that meets the mild growth condition but for which the generated sequence fails to converge weakly almost surely.

read the original abstract

We prove the almost sure weak convergence of a stochastic proximal point method for minimizing a convex integral function in the general nonlinear context of complete geodesic metric spaces of nonpositive curvature (so-called Hadamard spaces), solving a problem of M. Ba\v{c}\'ak. This method, formulated in the context of a mild growth condition on the function which generalizes Lipschitz continuity, was previously only considered in the context of strong metric regularity conditions or in the context of locally compact spaces. The proof is a combination of a weak almost sure convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fej\'er monotonicity, due to previous work of the author, together with a new argument for proving the almost sure convergence of the mean function values of the process towards the minimal value.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves almost sure weak convergence of the stochastic proximal point method for minimizing a convex integral functional over complete geodesic metric spaces of nonpositive curvature (Hadamard spaces). The argument combines a prior stochastic quasi-Fejér monotonicity theorem with a new mean-value step showing that E[f(X_n)] converges almost surely to the infimum of the objective, under a mild growth condition on the integrand that generalizes Lipschitz continuity. This is presented as resolving an open question of Bačák by removing the need for local compactness or strong metric regularity.

Significance. If the new mean-value argument is valid without tacit compactness assumptions, the result would meaningfully extend stochastic proximal methods to general Hadamard spaces. The clean modular structure—invoking the author's earlier quasi-Fejér theorem and adding a targeted convergence-of-means step—offers a reusable template for other stochastic algorithms in nonlinear metric settings. The mild growth condition is a natural and verifiable relaxation of prior hypotheses.

major comments (2)
  1. [Proof of the mean-value convergence (new argument combining quasi-Fejér with expectation control)] The new argument for almost sure convergence of the mean values E[f(X_n)] to inf f (described in the abstract and proof outline) invokes the mild growth condition both to define the stochastic proximal mapping and to control the mean-value step. In non-locally compact Hadamard spaces this step risks failure if recession directions of the integrand are not uniformly controlled; the manuscript should explicitly verify that no hidden local-compactness or uniform-integrability property is used, or supply a counter-example space where the growth condition alone is insufficient.
  2. [Invocation of the prior stochastic quasi-Fejér theorem] The central weak-convergence claim depends on the stochastic quasi-Fejér theorem from the author's prior work. The manuscript should include a self-contained verification that all hypotheses of that theorem (e.g., the specific form of the stochastic perturbation and the growth condition) are satisfied by the proximal-point iteration defined here.
minor comments (2)
  1. [Abstract and introduction] A direct citation to the specific open problem posed by Bačák (including the reference) would help readers locate the exact statement being solved.
  2. [Preliminaries] Notation for the stochastic proximal mapping and the mild growth condition should be introduced with a displayed definition before its first use in the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate planned revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Proof of the mean-value convergence (new argument combining quasi-Fejér with expectation control)] The new argument for almost sure convergence of the mean values E[f(X_n)] to inf f (described in the abstract and proof outline) invokes the mild growth condition both to define the stochastic proximal mapping and to control the mean-value step. In non-locally compact Hadamard spaces this step risks failure if recession directions of the integrand are not uniformly controlled; the manuscript should explicitly verify that no hidden local-compactness or uniform-integrability property is used, or supply a counter-example space where the growth condition alone is insufficient.

    Authors: We appreciate the referee's concern about potential tacit assumptions in non-locally compact settings. The mild growth condition is formulated precisely to control the expectations and recession behavior of the integrand via convexity, ensuring the mean-value step proceeds without local compactness or additional uniform-integrability hypotheses. In the revised manuscript we will insert a short clarifying paragraph (or lemma) immediately after the statement of the growth condition that explicitly confirms the argument uses only the given hypotheses and the geodesic properties of Hadamard spaces. We therefore see no need for a counter-example. revision: yes

  2. Referee: [Invocation of the prior stochastic quasi-Fejér theorem] The central weak-convergence claim depends on the stochastic quasi-Fejér theorem from the author's prior work. The manuscript should include a self-contained verification that all hypotheses of that theorem (e.g., the specific form of the stochastic perturbation and the growth condition) are satisfied by the proximal-point iteration defined here.

    Authors: We agree that a self-contained verification improves readability and rigor. In the revised version we will add a dedicated subsection (or short appendix) that systematically checks every hypothesis of the stochastic quasi-Fejér monotonicity theorem against the stochastic proximal-point iteration, including the precise form of the perturbation and the compatibility of the growth condition. revision: yes

Circularity Check

1 steps flagged

Central weak convergence theorem rests on author's prior stochastic quasi-Fejér result

specific steps
  1. self citation load bearing [Abstract]
    "The proof is a combination of a weak almost sure convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fejér monotonicity, due to previous work of the author, together with a new argument for proving the almost sure convergence of the mean function values of the process towards the minimal value."

    The central almost sure weak convergence result is obtained by invoking the author's earlier theorem on stochastic quasi-Fejér monotonicity; that prior result is not re-proved or externally benchmarked here, so the new paper's main theorem reduces in part to self-citation rather than standing fully on independent external verification.

full rationale

The paper's proof explicitly combines a new argument for almost sure convergence of mean function values with a weak almost sure convergence theorem for stochastic quasi-Fejér processes taken from the author's previous work. This self-citation is load-bearing for the main claim but is not the sole content; the new mean-value step provides independent material. No other circular patterns (self-definitional fits, ansatz smuggling, or renaming) appear in the provided derivation outline. The result therefore receives a moderate circularity score rather than a high one.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of Hadamard spaces and a domain-specific growth condition; no free parameters or new entities are introduced.

axioms (2)
  • standard math Hadamard spaces are complete geodesic metric spaces with nonpositive curvature
    Defines the ambient space in which the proximal point iteration and weak convergence are studied.
  • domain assumption The convex integral function satisfies a mild growth condition generalizing Lipschitz continuity
    Required to formulate the stochastic proximal mapping and to prove convergence of the mean function values.

pith-pipeline@v0.9.0 · 5658 in / 1289 out tokens · 35894 ms · 2026-05-21T04:02:46.279232+00:00 · methodology

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