arxiv: 2605.13144 · v1 · submitted 2026-05-13 · 🧮 math.NA · cs.NA

Recognition: unknown

On a posteriori stopping rules of adaptive stochastic heavy ball method for ill-posed problems

Ruixue Gu , Qinian Jin

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Pith reviewed 2026-05-14 18:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords stochastic heavy balla posteriori stopping ruleill-posed inverse problemsadaptive step sizemomentum coefficientdiscrepancy principleconvex penaltyalmost sure convergence

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The pith

Adaptive stochastic heavy ball method with a posteriori stopping rule converges almost surely for ill-posed inverse problems.

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

The paper develops a stochastic heavy ball iteration for ill-posed inverse problems that selects only one random equation at each step and incorporates a momentum term to accelerate progress. An adaptive rule adjusts both the step size and the momentum coefficient on the fly, while a new a posteriori stopping criterion, modeled on the discrepancy principle, halts the process by checking the residual of the chosen equation alone. Convex penalty functions are included to encode structural features of the unknown solution. Under suitable conditions the iterates converge almost surely and also in expectation, which removes the need for repeated full-system residual evaluations and makes the scheme feasible for large-scale problems.

Core claim

The adaptive stochastic heavy ball method, which performs single-equation random updates with momentum and stops according to an a posteriori discrepancy rule applied only to the selected equation, converges almost surely and in expectation for ill-posed linear inverse problems when the adaptive parameters obey the stated bounds and convex penalty functions are employed.

What carries the argument

The a posteriori stopping rule that compares the residual of the randomly selected equation against a discrepancy threshold, paired with the adaptive choice of step size and momentum coefficient inside the heavy-ball update.

If this is right

Only one equation is processed per iteration, so the per-step cost stays linear in the dimension of a single row even for very large systems.
Convex penalties encode prior structural information while preserving the almost-sure and expectation convergence guarantees.
The discrepancy-based stopping rule eliminates the need to compute the full residual vector at every iteration or at fixed intervals.
Both almost-sure and mean-square convergence together guarantee reliable recovery in the presence of noise.

Where Pith is reading between the lines

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

The same adaptive stopping idea may transfer to other stochastic first-order methods for inverse problems without requiring new convergence proofs from scratch.
Because the rule avoids full residual computations, it opens a route to online or streaming formulations where data arrive sequentially.
Numerical evidence in the paper suggests the method remains stable when the number of equations greatly exceeds the dimension of the unknown, an extension that could be tested on nonlinear forward operators.

Load-bearing premise

The noise level, the adaptive parameters, and the convex penalty functions must satisfy specific bounds and growth conditions for the convergence statements to hold.

What would settle it

A concrete linear ill-posed system with known exact solution and controlled additive noise in which the generated sequence fails to converge almost surely or the stopping rule terminates with error bounded away from zero.

Figures

Figures reproduced from arXiv: 2605.13144 by Qinian Jin, Ruixue Gu.

**Figure 1.** Figure 1: Relative error versus the number of iterations with δrel = 0.05. of the approximate solutions kx δ nδ − x †k 2/kx †k 2 and the iteration number nδ from 100 simulations under different noise levels. In each box, the central mark denotes the median, while the edges of the box indicate the 25th (bottom) and 75th (top) percentiles. The whiskers extend to the most extreme data points not considered outliers, an… view at source ↗

**Figure 2.** Figure 2: Boxplots of the relative error and iteration number nδ from 100 simulations for the first kind integral equation. 5.2. Computed tomography We next consider the problem of computed tomography (CT), which involves estimating the density of cross sections of an object by measuring the attenuation [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Figure 3: presents the results from a single run with δ = 0.1. Figures 3(b)- (c) depict the reconstructions obtained by SGD and SHB with v0 = 10−7 and v1 = 10−6 , respectively [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗

**Figure 4.** Figure 4: Boxplots of the relative error and iteration number nδ from 100 simulations for CT. Finally, to examine the impact of the batch size b, we apply the mini-batch SHB method with varying values of b to solve the CT problem with δ = 0.1. We set v0 = 10−7 and v1 = 10−6 , while all other settings remain the same as in the previous experiments. For each batch size, 100 independent simulations are conducted. The r… view at source ↗

**Figure 5.** Figure 5: (d) plots the evolution of the relative error kx δ n − x †k 2/kx †k 2 , which indicates the fast convergence of the SHB method. Exact solution 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 110 -0.1 0 0.1 0.2 0.3 0.4 0.5 (a) SGD 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 110 -0.1 0 0.1 0.2 0.3 0.4 0.5 (b) SHB: (v0 ,v1 )=(10-5,10-4) 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 110 -0.1 0 0.1 0.2 0.3 0… view at source ↗

**Figure 6.** Figure 6: Boxplots of the relative error and iteration number from 100 runs for schlieren imaging. have established the almost sure convergence and convergence in expectation of the method under the proposed a posteriori stopping rule. Further, numerical results demonstrate the effectiveness of the a posteriori stopping rule and the superiority of the method in terms of the required number of iterations and computat… view at source ↗

read the original abstract

In this paper we develop a stochastic heavy ball method for solving ill-posed inverse problems. The method updates the iterate using only a randomly selected equation at each iteration step while incorporating a momentum term into the process. To facilitate fast convergence, we propose an adaptive strategy for selecting the step size and the momentum coefficient. Inspired by the spirit of the discrepancy principle, we introduce an {\it a posteriori} stopping rule for our adaptive stochastic heavy ball method. This rule avoids the need to compute residuals of all equations in the system at every iteration or at fixed frequency intervals, thereby enhancing computational efficiency and practicality. Additionally, convex penalty functions are employed to capture the specific features of the desired solutions. Under suitable conditions, we establish almost sure convergence as well as convergence in expectation. Extensive numerical experiments are conducted to evaluate the performance of the proposed method, demonstrating its efficiency and promising potential for solving large-scale ill-posed problems.

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.

Desk Editor's Note private letter to a colleague

The paper adds an adaptive stochastic heavy-ball iteration plus a cheap a-posteriori stopping rule for ill-posed problems, with convergence proofs and numerics, but the advance is incremental and rests on typical restrictive conditions.

read the letter

The core contribution is a randomized heavy-ball method that updates using one random equation per step, adapts both step size and momentum coefficient, and stops via a discrepancy-inspired rule that skips full residual checks. Convex penalties are added to encode solution structure. They prove almost-sure and expectation convergence under suitable conditions and run numerical tests on large-scale examples showing practical speed-ups.

Referee Report

0 major / 3 minor

Summary. The manuscript develops an adaptive stochastic heavy-ball method for ill-posed linear inverse problems. The iteration performs randomized single-equation updates with a momentum term, adapts the step-size and momentum coefficient on the fly, incorporates convex penalty functions, and terminates via an a-posteriori discrepancy-type stopping rule that avoids evaluating the full residual at every step. Under suitable conditions on the noise level, adaptation rules, and penalties, the paper proves almost-sure convergence and convergence in expectation; numerical experiments on large-scale problems are reported to illustrate practical performance.

Significance. If the convergence statements hold, the work supplies a computationally attractive a-posteriori rule for a class of stochastic first-order methods that is already popular for large-scale ill-posed problems. The combination of momentum, adaptive parameters, and penalty terms is standard, but the explicit stopping criterion that remains cheap to evaluate adds a concrete practical contribution.

minor comments (3)

[Abstract and §1] The abstract and introduction repeatedly invoke “suitable conditions” without giving even a high-level list of the required assumptions on the noise level, the adaptation maps, or the penalty functions. Adding one compact paragraph that enumerates the main hypotheses would improve readability.
[Numerical experiments] In the numerical section the authors compare the new method only against a few existing stochastic schemes; a direct comparison against the same iteration run with a fixed discrepancy stopping rule (or with oracle knowledge of the noise level) would quantify the practical gain of the proposed a-posteriori rule.
[§2 and §3] Notation for the adaptive step-size and momentum sequences is introduced without a single consolidated table or list of symbols; readers must hunt through several paragraphs to recall the definitions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive overall assessment. The recommendation for minor revision is appreciated. No specific major comments were provided in the report, so we will incorporate any minor editorial or technical clarifications in the revised version to further improve the presentation.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives almost sure and expectation convergence for the adaptive stochastic heavy-ball iteration from explicitly listed assumptions on the forward operator, noise level, adaptive step-size/momentum rules, convex penalty, and the discrepancy-type a-posteriori stopping criterion. No equation or theorem reduces the claimed convergence to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose validity is presupposed by the present work. The stopping rule is constructed independently of the convergence statement and does not force the result by construction. The argument therefore remains self-contained once the stated conditions hold.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; convergence is stated to hold under unspecified suitable conditions typical of stochastic approximation and regularization theory.

pith-pipeline@v0.9.0 · 5455 in / 1062 out tokens · 36793 ms · 2026-05-14T18:37:19.350357+00:00 · methodology

discussion (0)

Reference graph

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