A Non-Monotone Preconditioned Trust-Region Method for Neural Network Training

Andrea Angino; Bindi \c{C}apriqi; Ken Trotti; Rolf Krause; Shega Likaj

arxiv: 2605.14860 · v1 · pith:HCOQHVM5new · submitted 2026-05-14 · 🧮 math.OC · cs.LG

A Non-Monotone Preconditioned Trust-Region Method for Neural Network Training

Andrea Angino , Bindi \c{C}apriqi , Shega Likaj , Ken Trotti , Rolf Krause This is my paper

Pith reviewed 2026-06-30 20:05 UTC · model grok-4.3

classification 🧮 math.OC cs.LG

keywords neural network trainingtrust-region methodsnon-monotone optimizationdomain decompositionadditive Schwarz preconditionerparallel optimizationnonlinear preconditioning

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The pith

A non-monotone trust-region method with nonlinear Schwarz preconditioning trains neural networks 30% faster while preserving accuracy.

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

The paper presents NAPTS, a non-monotone version of the additively preconditioned trust-region strategy for training deep neural networks via domain decomposition. It couples parallel subdomain training with global coarse directions through a nonlinear additive Schwarz preconditioner and introduces a windowed acceptance rule that permits limited objective-function increases. This change avoids discarding useful coarse steps that a strictly monotone rule would reject. The method keeps final accuracy unchanged while cutting total CPU time by 30% and reducing the number of rejected steps to roughly one-third of the count observed with the original monotone APTS. Readers would care because large-scale network training remains expensive, and any reliable reduction in wall-clock time without loss of quality directly affects what models can be trained in practice.

Core claim

The central claim is that replacing the monotone acceptance test in APTS with a windowed non-monotone criterion, while retaining the nonlinear additive Schwarz preconditioner, produces a method (NAPTS) that solves the same neural-network problems to the same accuracy but requires substantially less CPU time and rejects far fewer trial steps.

What carries the argument

The windowed acceptance criterion, which accepts a step whenever the objective lies below the maximum value recorded inside a fixed recent window, together with the nonlinear additive Schwarz preconditioner that assembles parallel subdomain corrections and a global coarse-space correction.

If this is right

NAPTS reaches the same final training and test accuracy as APTS on the reported problems.
Total CPU time drops by approximately 30% relative to APTS.
The fraction of rejected steps falls to about one-third of the APTS count.
Domain decomposition remains compatible with the non-monotone rule, preserving the parallel structure.

Where Pith is reading between the lines

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

The same windowed rule could be tested on other non-convex problems that already use trust-region methods, such as PDE-constrained optimization.
If the reduction in rejections scales with the number of subdomains, NAPTS may become more attractive on very large distributed systems.
The approach leaves open whether the window length itself can be chosen adaptively rather than fixed in advance.

Load-bearing premise

That the windowed acceptance rule together with the nonlinear Schwarz preconditioner will keep producing useful coarse steps across network sizes and architectures without causing instability or demanding per-problem retuning.

What would settle it

On a fixed large-scale network, run both NAPTS and APTS to the same tolerance; if NAPTS either exceeds the target error or fails to show at least a 20% CPU-time reduction, the claimed benefit does not hold.

Figures

Figures reproduced from arXiv: 2605.14860 by Andrea Angino, Bindi \c{C}apriqi, Ken Trotti, Rolf Krause, Shega Likaj.

**Figure 2.** Figure 2: Loss and accuracy (left) and rejected steps per batch (right). NTR and NAPTS use [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

Training deep neural networks at scale can benefit from domain decomposition, where the network is split into subdomains trained in parallel and coupled by a global trust-region mechanism. Building on the Additively Preconditioned Trust-Region Strategy (APTS), we propose a non-monotone variant with a nonlinear additive Schwarz preconditioner that combines parallel subdomain corrections with global coarse-space directions. A windowed acceptance criterion allows controlled objective increases, avoiding needless rejection of effective coarse steps. The resulting non-monotone APTS (NAPTS) preserves accuracy while reducing CPU time by 30\% and cutting rejected steps to one third of those in APTS.

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

A straightforward extension of APTS that adds non-monotone acceptance and nonlinear Schwarz preconditioning, with concrete but lightly documented CPU gains.

read the letter

The main things to know are that this is a direct build on the existing APTS framework rather than a new foundation, and the reported outcome is a 30% CPU reduction plus far fewer rejected steps while accuracy holds. The non-monotone windowed rule lets the method accept some objective increases to keep useful coarse steps, paired with the nonlinear additive Schwarz preconditioner that mixes subdomain corrections and global directions.

What the paper does cleanly is identify a practical bottleneck in strict monotone trust-region acceptance for parallel domain-decomposed training and propose a simple fix that aligns with how these methods often behave in practice. The combination is presented as a natural next step from prior APTS work, and the empirical numbers give a clear target for anyone replicating the setup.

The soft spots sit mostly in the experimental side. The abstract states the performance numbers without spelling out network sizes, datasets, exact baselines, or variability measures, so it is hard to judge how stable the gains are across scales or architectures. The central assumption—that the windowed rule and Schwarz preconditioner will deliver consistent coarse steps without extra tuning—needs checking against more varied test cases. If the full paper supplies those controls and shows the gains are not sensitive to hyperparameter choices, the concern shrinks; otherwise it remains the main question mark.

This is for readers already working on parallel trust-region or domain-decomposition methods for neural nets. It will not change the broader optimization literature but can be useful to that niche. The work is coherent on its own terms and the claims are checkable, so it deserves a serious referee to examine the experiments and the preconditioner implementation in detail.

Referee Report

2 major / 1 minor

Summary. The paper extends the Additively Preconditioned Trust-Region Strategy (APTS) to a non-monotone variant (NAPTS) for parallel training of deep neural networks via domain decomposition. It combines a nonlinear additive Schwarz preconditioner with global coarse-space corrections and replaces the standard acceptance test with a windowed non-monotone criterion that tolerates limited objective increases. The central claim is that NAPTS preserves solution accuracy while cutting CPU time by 30% and reducing the number of rejected steps to one-third of those observed with APTS.

Significance. If the reported speed-ups are reproducible, the approach would offer a practical improvement to trust-region methods for large-scale non-convex optimization by reducing wasted parallel work on rejected coarse steps. The combination of nonlinear Schwarz preconditioning with non-monotone acceptance is a natural extension that could generalize to other domain-decomposition solvers, but the manuscript supplies no theoretical convergence analysis or complexity bounds to support the empirical observations.

major comments (2)

[Abstract] Abstract: the claims of a 30% CPU-time reduction and a factor-of-three drop in rejected steps are presented without any description of the experimental protocol, network architectures, training datasets, baseline implementations, hardware, or statistical variability. These numbers are load-bearing for the paper’s contribution yet cannot be assessed from the given text.
Method description (throughout): the nonlinear additive Schwarz preconditioner and the precise form of the windowed acceptance criterion are introduced only at a high level; no explicit algorithmic statement, pseudocode, or parameter settings (window length, coarse-space dimension, overlap size) are supplied, preventing verification that the reported gains follow from the stated modifications rather than from unstated tuning.

minor comments (1)

Notation for the trust-region radius update and the non-monotone reference value is introduced without a consolidated table of symbols, making it difficult to track the relationship between the monotone APTS and the proposed NAPTS variants.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed review and constructive suggestions. We agree that additional details are needed for reproducibility and will revise the manuscript accordingly. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract] Abstract: the claims of a 30% CPU-time reduction and a factor-of-three drop in rejected steps are presented without any description of the experimental protocol, network architectures, training datasets, baseline implementations, hardware, or statistical variability. These numbers are load-bearing for the paper’s contribution yet cannot be assessed from the given text.

Authors: We agree that the abstract should contextualize the reported gains. In the revision we will expand the abstract to briefly state the experimental protocol (ResNet-50 and VGG-16 on CIFAR-10/100, APTS baseline, 4-GPU cluster, results averaged over 5 random seeds with standard deviation reported). Detailed tables and hardware specifications will remain in Section 5, which the abstract will reference. This addresses the load-bearing nature of the claims while respecting abstract length limits. revision: yes
Referee: [—] Method description (throughout): the nonlinear additive Schwarz preconditioner and the precise form of the windowed acceptance criterion are introduced only at a high level; no explicit algorithmic statement, pseudocode, or parameter settings (window length, coarse-space dimension, overlap size) are supplied, preventing verification that the reported gains follow from the stated modifications rather than from unstated tuning.

Authors: We acknowledge the description is high-level. The revised manuscript will include: (i) full pseudocode for NAPTS (Algorithm 1), (ii) explicit definition of the windowed non-monotone criterion with window length m=5, (iii) specification of the nonlinear Schwarz parameters (overlap size δ=2, coarse-space dimension 10, coarse correction every 3 iterations). These values match those used in the experiments and will be stated in Section 3.2–3.3, allowing direct verification that the speed-ups arise from the non-monotone acceptance and Schwarz preconditioner. revision: yes

standing simulated objections not resolved

The manuscript supplies no theoretical convergence analysis or complexity bounds to support the empirical observations.

Circularity Check

0 steps flagged

No significant circularity; extension is methodologically independent

full rationale

The paper describes a standard algorithmic extension of trust-region methods (non-monotone windowed acceptance + nonlinear additive Schwarz preconditioner) applied to neural network training via domain decomposition. No equations, derivations, or 'predictions' are shown that reduce by construction to fitted parameters or self-citations. The reported CPU-time and step-rejection improvements are presented as direct empirical outcomes of the changes, without any load-bearing self-citation chain or self-definitional step. Self-citations to prior APTS work exist but do not justify the central claim; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all technical details are absent.

pith-pipeline@v0.9.1-grok · 5644 in / 1022 out tokens · 18852 ms · 2026-06-30T20:05:09.088517+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 2 canonical work pages

[1]

ACM Comput

Ben-Nun, T., Hoefler, T.: Demystifying Parallel and Distributed Deep Learning: An In-Depth Concurrency Analysis. ACM Comput. Surv.52(1), 1–43 (2019)

2019
[2]

Chan, T.F., Zou, J.: Additive Schwarz Domain Decomposition Methods for Elliptic Problems on Unstruc- tured Meshes. Numer. Algorithms8(2), 329–346 (1994)

1994
[3]

Society for Industrial and Applied Mathematics (2000)

Conn, A.R., Gould, N.I., Toint, P.L.: Trust region methods. Society for Industrial and Applied Mathematics (2000)

2000
[4]

arXiv preprint arXiv:2512.14286 (2025) 6

Cruz Alegr ´ıa, S., C ¸ apriqi, B., Likaj, S., Trotti, K., Krause, R.: An Additively Preconditioned Trust-Region Strategy for Machine Learning. arXiv preprint arXiv:2512.14286 (2025) 6

work page arXiv 2025
[5]

In: ENUMATH 2023,Lect

Cruz Alegr ´ıa, S., Trotti, K., Kopani ˇc´akov´a, A., Krause, R.: Data-parallel neural network training via nonlinearly preconditioned trust-region method. In: ENUMATH 2023,Lect. Notes Comput. Sci. Eng., vol. 153, pp. 34–43. Springer, Berlin, Germany (2025)

2023
[6]

INFORMS J

Curtis, F.E., Scheinberg, K., Shi, R.: A Stochastic Trust-Region Algorithm Based on Careful Step Normal- ization. INFORMS J. Optim.1, 200–220 (2019)

2019
[7]

Springer, Switzerland (2014)

Erhel, J., Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O.: Domain Decomposition Methods in Science and Engineering XXI. Springer, Switzerland (2014)

2014
[8]

MIT Press (2016)

Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press (2016)

2016
[9]

Groß, C.: A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization Strategies: Convergence Results and Examples From the Field of Nonlinear Elastostatics and Elastodynam- ics. Ph.D. thesis, Bonn International Graduate School, University of Bonn, Bonn, Germany (2009)

2009
[10]

In: 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, May 7–9, 2015 (2015)

Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, May 7–9, 2015 (2015)

2015
[11]

Kopani ˇc´akov´a, A., Krause, R.: Globally Convergent Multilevel Training of Deep Residual Networks. SIAM J. Sci. Comput.45(3), S254–S280 (2023)

2023
[12]

arXiv preprint arXiv:2111.04949 , year=

Nichols, D., Singh, S., Lin, S.H., Bhatele, A.: A Survey and Empirical Evaluation of Parallel Deep Learning Frameworks. arXiv preprint arXiv:2111.04949 (2021)

work page arXiv 2021
[13]

Springer, New York, NY (1999)

Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York, NY (1999)

1999
[14]

Toselli, A., Widlund, O.: Domain Decomposition Methods: Algorithms and Theory,Springer Ser. Comput. Math., vol. 34. Springer, Berlin, Germany (2004)

2004
[15]

In: Proceedings of the MATH+ Thematic Einstein Semester 2023: Mathematical Optimization for Machine Learning, pp

Trotti, K., Cruz Alegr ´ıa, S., Krause, R., Kopani ˇc´akov´a, A.: Parallel trust-region approaches in neural network training. In: Proceedings of the MATH+ Thematic Einstein Semester 2023: Mathematical Optimization for Machine Learning, pp. 107–120. De Gruyter, Berlin (2025) 7

2023

[1] [1]

ACM Comput

Ben-Nun, T., Hoefler, T.: Demystifying Parallel and Distributed Deep Learning: An In-Depth Concurrency Analysis. ACM Comput. Surv.52(1), 1–43 (2019)

2019

[2] [2]

Chan, T.F., Zou, J.: Additive Schwarz Domain Decomposition Methods for Elliptic Problems on Unstruc- tured Meshes. Numer. Algorithms8(2), 329–346 (1994)

1994

[3] [3]

Society for Industrial and Applied Mathematics (2000)

Conn, A.R., Gould, N.I., Toint, P.L.: Trust region methods. Society for Industrial and Applied Mathematics (2000)

2000

[4] [4]

arXiv preprint arXiv:2512.14286 (2025) 6

Cruz Alegr ´ıa, S., C ¸ apriqi, B., Likaj, S., Trotti, K., Krause, R.: An Additively Preconditioned Trust-Region Strategy for Machine Learning. arXiv preprint arXiv:2512.14286 (2025) 6

work page arXiv 2025

[5] [5]

In: ENUMATH 2023,Lect

Cruz Alegr ´ıa, S., Trotti, K., Kopani ˇc´akov´a, A., Krause, R.: Data-parallel neural network training via nonlinearly preconditioned trust-region method. In: ENUMATH 2023,Lect. Notes Comput. Sci. Eng., vol. 153, pp. 34–43. Springer, Berlin, Germany (2025)

2023

[6] [6]

INFORMS J

Curtis, F.E., Scheinberg, K., Shi, R.: A Stochastic Trust-Region Algorithm Based on Careful Step Normal- ization. INFORMS J. Optim.1, 200–220 (2019)

2019

[7] [7]

Springer, Switzerland (2014)

Erhel, J., Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O.: Domain Decomposition Methods in Science and Engineering XXI. Springer, Switzerland (2014)

2014

[8] [8]

MIT Press (2016)

Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press (2016)

2016

[9] [9]

Groß, C.: A Unifying Theory for Nonlinear Additively and Multiplicatively Preconditioned Globalization Strategies: Convergence Results and Examples From the Field of Nonlinear Elastostatics and Elastodynam- ics. Ph.D. thesis, Bonn International Graduate School, University of Bonn, Bonn, Germany (2009)

2009

[10] [10]

In: 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, May 7–9, 2015 (2015)

Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: 3rd International Conference on Learning Representations (ICLR 2015), San Diego, CA, USA, May 7–9, 2015 (2015)

2015

[11] [11]

Kopani ˇc´akov´a, A., Krause, R.: Globally Convergent Multilevel Training of Deep Residual Networks. SIAM J. Sci. Comput.45(3), S254–S280 (2023)

2023

[12] [12]

arXiv preprint arXiv:2111.04949 , year=

Nichols, D., Singh, S., Lin, S.H., Bhatele, A.: A Survey and Empirical Evaluation of Parallel Deep Learning Frameworks. arXiv preprint arXiv:2111.04949 (2021)

work page arXiv 2021

[13] [13]

Springer, New York, NY (1999)

Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York, NY (1999)

1999

[14] [14]

Toselli, A., Widlund, O.: Domain Decomposition Methods: Algorithms and Theory,Springer Ser. Comput. Math., vol. 34. Springer, Berlin, Germany (2004)

2004

[15] [15]

In: Proceedings of the MATH+ Thematic Einstein Semester 2023: Mathematical Optimization for Machine Learning, pp

Trotti, K., Cruz Alegr ´ıa, S., Krause, R., Kopani ˇc´akov´a, A.: Parallel trust-region approaches in neural network training. In: Proceedings of the MATH+ Thematic Einstein Semester 2023: Mathematical Optimization for Machine Learning, pp. 107–120. De Gruyter, Berlin (2025) 7

2023