Data-Parallel Neural Network Training via Nonlinearly Preconditioned Trust-Region Method

Alena Kopani\v{c}\'akov\'a; Ken Trotti; Rolf Krause; Samuel A. Cruz Alegr\'ia

REVIEW 4 major objections 5 minor 33 references

Data-Parallel Neural Network Training via Nonlinearly Preconditioned Trust-Region Method

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A data-parallel trust-region method trains neural networks without learning-rate tuning and matches fine-tuned Adam on MNIST and CIFAR-10.

desk verdict A plausible APTS variant for data-parallel NN training, but the accuracy and no-tuning claims outrun the evidence: no error bars, no SGD baseline, and no convergence proof for the minibatch variant. read the letter →

arxiv 2502.05133 v1 pith:2H7JLOSC submitted 2025-02-07 cs.LG cs.NAmath.NA

Samuel A. Cruz Alegría , Ken Trotti , Alena Kopaničáková , Rolf Krause This is my paper · ORCID

classification cs.LGcs.NAmath.NA MSC 90C5365K0568T0768W10

keywords data-paralleltrainingtrust-regionmethodsnonlinearpreconditioningdomaindecompositionneuralnetworkhyperparameter-freeoptimizationL-SR1paralleldeeplearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a data-parallel trust-region method, APTS, for training deep neural networks without learning-rate tuning. It claims APTS reaches validation accuracy comparable to fine-tuned Adam and SGD on MNIST and CIFAR-10, while reducing communication compared with synchronized parallel SGD. The training data is split into overlapping minibatches, each processed by a local trust-region solver; a consistency term makes each local gradient agree with the global gradient, and step sizes are adapted automatically via trust-region radii. The authors argue this makes the method globally convergent and inherently parallelizable, unlike gradient-descent variants whose convergence depends on delicate hyperparameters.

What carries the argument

The key object is the modified local loss (3) with its first-order consistency term (4), together with the acceptance ratio (6). The consistency term makes each local gradient equal to the full global gradient at the start of the outer iteration, so local trust-region steps approximate a global step; the ratio then measures whether the summed local reductions predict the actual global decrease, triggering trust-region radius updates (7). The global step bound (5) keeps the sum of local steps within the global trust-region radius. Local subproblems are solved with an L-SR1 Hessian approximation and the OBS trust-region solver.

What would settle it

Run APTS with the paper's fixed trust-region settings on a standard image-classification task with many subdomains (e.g., a 16-way split of CIFAR-10) and check whether the full training loss decreases monotonically across accepted iterations and whether final accuracy stays within a couple of points of a well-tuned Adam baseline.

Watch

Extended reading notes

Core claim

The central claim is that a trust-region method can be made data-parallel through additive nonlinear preconditioning: each worker solves a trust-region subproblem on a modified local loss $f_i(\theta)=f(\theta, D_i)+\langle r_i, \theta-\theta_{i,0}\rangle$, where $r_i$ is the difference between global and local gradients at the current iterate, so the local objective is first-order consistent with the global one. The local steps are summed, and the combined step is accepted only if the ratio $\rho$ of actual global decrease to summed local predicted decrease exceeds $\eta_1$. This mechanism transfers the global convergence properties of trust-region methods to the parallel setting and makes explicit step-size selection unnecessary. On MNIST and CIFAR-10 the method matches fine-tuned Adam accuracy with a single fixed set of trust-region parameters.

Load-bearing premise

The method's convergence depends on the assumption that the steps computed locally on each data subset, when added together, still reduce the full training loss reliably; the paper notes this is not proven for the stochastic version.

Editorial extensions

If this is right

Training deep networks would no longer require per-problem learning-rate schedules; a fixed trust-region configuration suffices across datasets and architectures.
Data-parallel training would communicate only after several local trust-region iterations, reducing synchronization overhead relative to gradient-synchronous SGD and Adam.
The deterministic APTS variant inherits a global convergence guarantee from trust-region and nonlinear-preconditioning theory, something SGD and Adam lack in the nonconvex regime.
The method should scale with the number of subdomains up to the point where local gradients stop being good approximations of the full gradient.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The consistency term is conceptually a variance-reduction device, and one could test whether replacing the full-gradient consistency term with a stochastic estimate would make the stochastic SAPTS variant theoretically tractable.
The observed accuracy degradation with more subdomains suggests a trade-off between parallelism and approximation quality; increasing overlap between minibatches might delay that degradation, a testable prediction.
The method currently applies trust region locally first and then once globally per outer loop; a natural extension would be to make the global trust-region step itself a distributed operation, potentially improving scaling further.
The claim of no hyperparameter tuning is limited to the tested architectures; verifying it on transformers or recurrent networks would be a direct next test.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper is a reasonable incremental contribution—applying APTS, a nonlinear preconditioned trust-region method from domain decomposition, to data-parallel NN training—but its headline claims (comparable to Adam, no hyperparameter tuning) are not supported by the evidence as presented. The consistency term (3)-(4) is a clean idea: it makes each local loss gradient match the global gradient at the start of each outer iteration, which should reduce minibatch variance. The combined local/global acceptance ratio (6) is also sensible. The method is clearly described, the algorithm is explicit, and the authors point to a code repository, which is good practice.

The soft spots are material. First, the empirical comparison: Figures 1 and 2 show APTS curves against Adam, but there are no error bars for APTS and no hyperparameter values for the 'constant parameters.' The abstract promises comparison with SGD and Adam, but the experiments only show Adam; SGD is dismissed in one sentence without showing data. Second, the convergence claim: the paper says APTS is 'globally convergent' and cites Gross's thesis, but no theorem is stated for the nonconvex DNN setting with L-SR1 Hessian approximations. More importantly, Section 2.2 explicitly concedes that the existing convergence analysis does not extend to the stochastic SAPTS variant, yet the minibatch experiments (which are SAPTS runs) are presented without that caveat in the results. Third, the 'no tuning' claim is weakened by the fact that the method has at least six TR hyperparameters (Δ0, η1, η2, α, β, ν, overlap). Not tuning them is nice, but they still need to be chosen; the paper doesn't report what they were. The scaling claims are based on a single node type and a small number of subdomains, so the communication advantage is plausible but not measured carefully.

These are fixable problems. The core idea has merit and the paper is honest about the SAPTS limitation. What it needs before the claims can be accepted is a proper empirical study: error bars, a real SGD baseline, a sensitivity table for the TR parameters, and either a convergence proof for the deterministic case or a clear statement that only empirical convergence is claimed.

Who this is for: researchers working on parallel nonlinear optimization for ML, and people interested in trust-region methods as an alternative to SGD/Adam. It deserves a serious referee, but the referee should push for the missing comparisons and a more careful claims-rewrite. My recommendation: send it to review, but condition acceptance on addressing the empirical gaps and softening the 'no tuning' claim.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a data-parallel variant of the Additively Preconditioned Trust-Region Strategy (APTS) for training deep neural networks. The dataset is partitioned into minibatches or microbatches, and each subproblem trains a copy of the network with a local trust-region method on a modified local loss (3) that includes a first-order consistency term (4) matching the global gradient at the first local iterate. The local steps are summed, their quality is assessed via the ratio (6), and the global parameters are updated with a combined rule (7) that also updates the trust-region radius; one additional global trust-region step is then performed (Algorithm 1, line 9). A stochastic variant, SAPTS, is also described. Experiments on MNIST and CIFAR-10 compare APTS/SAPTS with Adam using fixed, unreported TR parameters, with first- and second-order local models. The paper claims comparable accuracy to SGD and Adam, global convergence, parallelizability, and no hyperparameter tuning.

Significance. The algorithmic idea is genuinely interesting: combining additive Schwarz-style nonlinear preconditioning with trust-region globalization for DNN training is a plausible way to reduce communication frequency and avoid learning-rate schedules. The paper makes a useful connection to the APTS/ASPIN literature and gives a concrete algorithm (Algorithm 1) with a software reference, which is a positive reproducibility practice. However, the strength of the claims is far ahead of the evidence: only two small datasets are considered, no error bars or hyperparameter values are reported for the new method, the SGD baseline named in the abstract is missing, and the convergence argument is cited rather than proved for the stochastic and nonconvex setting. With these gaps closed, the contribution could be valuable; as it stands, the significance is potential rather than demonstrated.

major comments (4)

[Abstract; §3] The abstract claims that the proposed APTS variant achieves validation accuracy comparable to SGD and Adam, but Section 3 states that APTS is compared only with the sequential Adam optimizer and that 'SGD and TR performed less optimally' without reporting any SGD experiments. The claim of parity with SGD is therefore not supported by the presented evidence.
[§3] The experimental section reports no error bars, confidence intervals, or repeated-trial statistics for APTS, and it does not list the values of the supposedly constant TR parameters (Δ0, η1, η2, α, β, ν, overlap, and L-SR1 memory m). Adam, by contrast, is reported as the best of 13 learning rates over 10 trials. Without the APTS parameter values and variance information, the claims 'comparable validation accuracy' and 'no hyperparameter tuning' cannot be verified or reproduced.
[§1; §2.2] The paper asserts in Section 1 that the method is globally convergent and inherently parallelizable, but the convergence statement is only a citation to [11]. No theorem or verification of the assumptions of that theory (nonconvex DNN loss, L-SR1 Hessian approximations, the local models fi in (3), the acceptance ratio (6), and the global TR step in Algorithm 1 line 9) is provided. Section 2.2 explicitly disclaims the extension of the convergence analysis to SAPTS, yet the reported minibatch experiments (minibatches of size 10,000 with 5% overlap) are SAPTS runs. Thus the evaluated stochastic variant has no convergence guarantee, and the deterministic variant's guarantee is not established in this setting.
[§2.2; §3] The claim of reduced communication cost is not substantiated. In the deterministic APTS variant, the consistency term (4) requires the full gradient ∇f(θ_k, D), and Algorithm 1 line 9 performs a global TR iteration on f with the full dataset D; both incur synchronization and full-data evaluation costs every outer iteration. The paper provides no wall-clock communication comparison with data-parallel SGD/Adam and states that scalability studies are planned. The statement that 'communication overhead depends only on the connection speed' therefore overstates what is demonstrated.

minor comments (5)

[Throughout] The dataset name is inconsistently typeset as 'CIF AR-10' in several places; it should be CIFAR-10.
[§3] The phrase 'communication overheard' should be 'communication overhead'.
[§2.2; Eq. (6)–(7)] The text says 'If ρk > η' but equation (7) uses η1 and η2; the acceptance threshold η is never defined. Please clarify the notation and align the text with the update rule.
[Algorithm 1, line 3] The notation 'D ∈ [D]' for APTS and 'D ∈ [D1, D2, ...]' for SAPTS is unclear; please specify how the full dataset, minibatches, and microbatches are enumerated and ordered.
[Figures 1 and 2] The figures lack explicit captions and legends identifying which curves correspond to Adam versus APTS and to which number of subdomains; the text references 'first-order model' and 'second-order model' but the reader cannot infer all settings from the figures alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the APTS construction is a consistency design, not a fitted prediction.

full rationale

The paper's core construction defines local losses fi in (3) with the consistency term (4) chosen so that the local gradient matches the global gradient at the first local TR iteration; this is an algorithmic consistency condition, not an input that is later repackaged as a prediction. The acceptance ratio (6) compares the global decrease to the sum of local decreases, and the update rule (7) is a standard trust-region globalization mechanism. Convergence is attributed to the external APTS theory of Groß [11] rather than derived here, and the paper explicitly disclaims that this analysis extends to SAPTS; this is a limitation, not circularity. The empirical claims compare APTS against fine-tuned Adam, and the absence of SGD results or error bars weakens the evidence but does not make the derivation circular. Self-citations ([2], [23], [26], [28], [30], [33]) appear but none is load-bearing for the main construction: [23] supports a variance-reduction remark, and [2] is only mentioned as a possible future combination. The 'no hyperparameter tuning' claim is a free-parameter or experimental-support concern, not a circularity concern, because the constant TR parameters are not fitted to the reported validation accuracies.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical or mathematical entities. The central claim depends on the choice of several TR hyperparameters (Δ0, η1, η2, α, β, ν, overlap) that are fixed but undisclosed, and on the transfer of APTS convergence theory to the DNN setting without proof.

free parameters (6)

initial trust-region radius Δ0
Not specified numerically, fixed across runs; required by Algorithm 1.
acceptance thresholds η1, η2
Used in the ratio test (6) and update rule (7), not specified in the paper.
radius update factors α, β
Used in (7), not specified.
max local TR iterations ν = 5
Set to 5 in all experiments, a choice that affects the preconditioner quality.
minibatch overlap fraction = 5%
Overlap of 5% between minibatches and microbatches, chosen without sensitivity analysis.
L-SR1 memory m
Number of secant pairs stored for the Hessian approximation is not stated.

assumptions (3)

standard math Trust-region global convergence theory (Conn et al. 2000) and L-SR1 properties (Erway & Marcia 2015) hold for the empirical loss in DNN training.
The paper relies on these background results to justify the global convergence claim, but does not verify the required smoothness or boundedness assumptions for neural network losses.
domain assumption The empirical loss f is sufficiently well-behaved for the L-SR1 quasi-Newton trust-region method to make progress; neural network losses are nonconvex but assumed to satisfy the standard smoothness conditions.
Invoked implicitly in Section 2.1 when applying TR to NNs; no verification is provided.
ad hoc to paper The first-order consistency term r_i^k in (4), computed with the full gradient, makes local subproblems match the global gradient at the first iterate, and this improves the accuracy of the preconditioner in NN training.
This is the key device of the paper's method, stated in (3) and (4), but no analysis is given for why it should benefit DNN training.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Data-Parallel Neural Network Training via Nonlinearly Preconditioned Trust-Region Method." pith.science (2026). https://pith.science/paper/2H7JLOSC

@misc{pith2026250205133,
  author       = {Pith},
  title        = {Pith review of: Data-Parallel Neural Network Training via Nonlinearly Preconditioned Trust-Region Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2H7JLOSC}},
  note         = {Machine review of arXiv:2502.05133}
}

read the original abstract

Parallel training methods are increasingly relevant in machine learning (ML) due to the continuing growth in model and dataset sizes. We propose a variant of the Additively Preconditioned Trust-Region Strategy (APTS) for training deep neural networks (DNNs). The proposed APTS method utilizes a data-parallel approach to construct a nonlinear preconditioner employed in the nonlinear optimization strategy. In contrast to the common employment of Stochastic Gradient Descent (SGD) and Adaptive Moment Estimation (Adam), which are both variants of gradient descent (GD) algorithms, the APTS method implicitly adjusts the step sizes in each iteration, thereby removing the need for costly hyperparameter tuning. We demonstrate the performance of the proposed APTS variant using the MNIST and CIFAR-10 datasets. The results obtained indicate that the APTS variant proposed here achieves comparable validation accuracy to SGD and Adam, all while allowing for parallel training and obviating the need for expensive hyperparameter tuning.

Figures

Figures reproduced from arXiv: 2502.05133 by the authors.

**Figure 2.** CIFAR-10. Left: first-order model, right: second-order model. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 29 canonical work pages

[11]

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

work page 2009
[1]

GitHub repository

Cruz Alegr ´ ıa, S.A., Trotti, K.: ML APTS. GitHub repository. Retrieved from https://github. com/cruzas/ML_APTS (2023)

work page 2023
[2]

Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods

Trotti, K., Cruz Alegr ´ ıa, S.A., Kopaniˇ c´ akov´ a, A., Krause, R: Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods. arXiv preprint 2312.13677 (2023)

work page Pith review arXiv 2023
[3]

Numerical Algorithms, 8, 329–346 (1994)

Chan, T.F., Zou, J.: Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes. Numerical Algorithms, 8, 329–346 (1994)

work page 1994
[4]

(eds.): Numerical Optimization

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

work page 1999
[5]

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)

work page 2000
[6]

In: International Workshop on Multiple Classifier Systems, pp

Dietterich, T.G.: Ensemble methods in machine learning. In: International Workshop on Multiple Classifier Systems, pp. 1–15. Springer, Berlin Heidelberg (2000)

work page 2000
[7]

SIAM Journal on Scientific Computing, 24, 183–200 (2002)

Cai, X.C., Keyes, D.E.: Nonlinearly preconditioned inexact newton algorithms. SIAM Journal on Scientific Computing, 24, 183–200 (2002)

work page 2002

Show all 33 references

[8]

Toselli, A., Widlund, O.: Domain Decomposition Methods - Algorithms and Theory, vol. 34. Springer (2004)

2004
[9]

SIAM Journal on Numerical Analysis, 44, 699–731 (2006)

Gander, M.J.: Optimized Schwarz methods. SIAM Journal on Numerical Analysis, 44, 699–731 (2006)

2006
[10]

Springer (2008)

Mathew, T.P.: Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Springer (2008)

2008
[12]

In: Advances in Neural Information Processing Systems, vol

Zinkevich, M., Weimer, M., Li, L., Smola, A.: Parallelized stochastic gradient descent. In: Advances in Neural Information Processing Systems, vol. 23 (2010). 6

2010
[13]

(eds.): Domain Decompo- sition Methods in Science and Engineering XXI

Erhel, J., Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds.): Domain Decompo- sition Methods in Science and Engineering XXI. Springer International Publishing (2014)

2014
[14]

arXiv preprint 1412.6980 (2014)

Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint 1412.6980 (2014)

2014 arXiv
[15]

SIAM Journal on Matrix Analysis and Applications, 36, 1338–1359 (2015)

Erway, J.B., Marcia, R.F.: On efficiently computing the eigenvalues of limited-memory quasi- Newton matrices. SIAM Journal on Matrix Analysis and Applications, 36, 1338–1359 (2015)

2015
[16]

In: Advances in Neural Information Processing Systems, vol

Zhang, S., Choromanska, A.E., LeCun, Y.: Deep learning with elastic averaging SGD. In: Advances in Neural Information Processing Systems, vol. 28 (2015)

2015
[17]

SIAM Journal on Scientific Computing, 38, A3357–A3380 (2016)

Dolean, V., Gander, M.J., Kheriji, W., Kwok, F., Masson, R.: Nonlinear Preconditioning: How to use a nonlinear Schwarz method to precondition Newton’s method. SIAM Journal on Scientific Computing, 38, A3357–A3380 (2016)

2016
[18]

Computational Optimization and Applications, 66, 245–266 (2017)

Brust, J., Erway, J.B., Marcia, R.F.: On solving L-SR1 trust-region subproblems. Computational Optimization and Applications, 66, 245–266 (2017)

2017
[19]

In Artificial Intelligence and Statistics, pp

McMahan, B., Moore, E., Ramage, D., Hampson, S., y Arcas, B.A.: Communication efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pp. 1273–1282 (2017)

2017
[20]

SIAM Review, 60, 223–311 (2018)

Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. SIAM Review, 60, 223–311 (2018)

2018
[21]

arXiv preprint 1904.09237 (2019)

Reddi, S.J., Kale, S., Kumar, S.: On the convergence of Adam and beyond. arXiv preprint 1904.09237 (2019)

2019 arXiv
[22]

In: Advances in Neural Information Processing Systems, vol

Paszke, Adam, et al.: Pytorch: An imperative style, high-performance deep learning library. In: Advances in Neural Information Processing Systems, vol. 32 (2019)

2019
[23]

arXiv preprint 2006.15602 (2020)

Braglia, V., Kopaniˇ c´ akov´ a, A., Krause, R.: A multilevel approach to training. arXiv preprint 2006.15602 (2020)

2020 arXiv
[24]

In: Optimization Methods and Software, vol

Erway, J.B., Griffin, J., Marcia, R.F., Omheni, R.: Trust-region algorithms for training responses: machine learning methods using indefinite Hessian approximations. In: Optimization Methods and Software, vol. 35, pp. 460–487 (2020)

2020
[25]

In: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp

Rasley, J., Rajbhandari, S., Ruwase, O., He, Y.: DeepSpeed: System Optimizations Enable Train- ing Deep Learning Models with Over 100 Billion Parameters. In: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 3505–3506 (2020)

2020
[26]

arXiv preprint 2104.05672 (2021)

Groß, C., Krause, R.: On the Globalization of ASPIN Employing Trust-Region Control Strategies– Convergence Analysis and Numerical Examples. arXiv preprint 2104.05672 (2021)

2021 arXiv
[27]

arXiv preprint 2111.04949 (2021)

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

2021 arXiv
[28]

SIAM Journal on Scientific Computing, 0, S254–S280 (2022)

Kopaniˇ c´ akov´ a, A., Krause, R.: Globally Convergent Multilevel Training of Deep Residual Networks. SIAM Journal on Scientific Computing, 0, S254–S280 (2022)

2022
[29]

ETNA — Electronic Transactions on Numerical Analysis (2022)

Gu, L., Zhang, W., Liu, J., Cai, X.C.: Decomposition and composition of deep convolutional neural networks and training acceleration via sub-network transfer learning. ETNA — Electronic Transactions on Numerical Analysis (2022)

2022
[30]

SIAM Journal on Optimization,33, 2772–2800 (2023)

Gratton, S., Kopaniˇ c´ akov´ a, A., Toint, P.L.: Multilevel Objective-Function-Free Optimization with an Application to Neural Networks Training. SIAM Journal on Optimization,33, 2772–2800 (2023)

2023
[31]

In: Domain Decomposition Methods in Science and Engineering XXVI, pp

Gu, L., Zhang, W., Liu, J., Cai, X.C.: Decomposition and Preconditioning of Deep Convolutional Neural Networks for Training Acceleration. In: Domain Decomposition Methods in Science and Engineering XXVI, pp. 153–160. Springer International Publishing (2023)

2023
[32]

arXiv preprint 2302.06564 (2023)

Klawonn, A., Lanser, M., Weber, J.: A Domain Decomposition-Based CNN-DNN Architecture for Model Parallel Training Applied to Image Recognition Problems. arXiv preprint 2302.06564 (2023). 7

2023 arXiv
[33]

SIAM Journal on Scientific Computing, pp

Kopaniˇ c´ akov´ a, A., Kothari, H., Karniadakis, G.E., Krause, R.: Enhancing Training of Physics- Informed Neural Networks Using Domain Decomposition–Based Preconditioning Strategies. SIAM Journal on Scientific Computing, pp. S46–S67 (2024). 8

2024

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [11]

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

work page 2009

[2] [1]

GitHub repository

Cruz Alegr ´ ıa, S.A., Trotti, K.: ML APTS. GitHub repository. Retrieved from https://github. com/cruzas/ML_APTS (2023)

work page 2023

[3] [2]

Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods

Trotti, K., Cruz Alegr ´ ıa, S.A., Kopaniˇ c´ akov´ a, A., Krause, R: Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods. arXiv preprint 2312.13677 (2023)

work page Pith review arXiv 2023

[4] [3]

Numerical Algorithms, 8, 329–346 (1994)

Chan, T.F., Zou, J.: Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes. Numerical Algorithms, 8, 329–346 (1994)

work page 1994

[5] [4]

(eds.): Numerical Optimization

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

work page 1999

[6] [5]

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)

work page 2000

[7] [6]

In: International Workshop on Multiple Classifier Systems, pp

Dietterich, T.G.: Ensemble methods in machine learning. In: International Workshop on Multiple Classifier Systems, pp. 1–15. Springer, Berlin Heidelberg (2000)

work page 2000

[8] [7]

SIAM Journal on Scientific Computing, 24, 183–200 (2002)

Cai, X.C., Keyes, D.E.: Nonlinearly preconditioned inexact newton algorithms. SIAM Journal on Scientific Computing, 24, 183–200 (2002)

work page 2002