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arxiv: 2604.14017 · v1 · submitted 2026-04-15 · 🧮 math.OC · cs.LG

Stochastic Trust-Region Methods for Over-parameterized Models

Aike Yang , Hao Wang This is my paper

Pith reviewed 2026-05-10 12:28 UTC · model grok-4.3

classification 🧮 math.OC cs.LG
keywords stochastic optimizationtrust-region methodsover-parameterized modelsstrong growth conditioninterpolation assumptionsconstrained optimizationKKT points
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The pith

A stochastic trust-region method reaches ε-stationary points with O(ε^{-2} log(1/ε)) complexity under interpolation assumptions without manual step-size tuning.

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

The paper introduces a unified stochastic trust-region framework for optimization in over-parameterized models. It aims to match the convergence rates of full-batch methods while removing the need for careful step-size selection that often destabilizes stochastic gradient descent. Under the strong growth condition, the first-order version finds an ε-stationary point with iteration and oracle complexity O(ε^{-2} log(1/ε)). A quadratic-penalty extension handles equality-constrained problems and reaches an O(ε)-approximate KKT point with O(ε^{-4} log(1/ε)) complexity. A sympathetic reader would care because the approach promises more reliable training of deep networks and stable handling of hard constraints.

Core claim

Under the strong growth condition, the first-order stochastic trust-region algorithm achieves an iteration and stochastic first-order oracle complexity of O(ε^{-2} log(1/ε)) for finding an ε-stationary point. For equality-constrained problems, the quadratic-penalty-based stochastic trust-region method with penalty parameter μ establishes an iteration and oracle complexity of O(ε^{-4} log(1/ε)) to reach an ε-stationary point of the penalized problem, corresponding to an O(ε)-approximate KKT point of the original constrained problem.

What carries the argument

The stochastic trust-region framework, which adaptively adjusts step sizes via trust-region radii instead of fixed learning rates, extended by a quadratic-penalty formulation for equality constraints.

Load-bearing premise

The strong growth condition or similar interpolation-type assumptions must hold for the objective functions.

What would settle it

A concrete numerical test in which the algorithm requires substantially more than O(ε^{-2} log(1/ε)) iterations to reach an ε-stationary point while the strong growth condition is satisfied would falsify the complexity claim.

Figures

Figures reproduced from arXiv: 2604.14017 by Aike Yang, Hao Wang.

Figure 1
Figure 1. Figure 1: Comparison of training loss and test accuracy for SGD, Adam, SLS, and STR on CIFAR-10. 5.1. Unconstrained Setting: Multi-class Classification We consider unconstrained multi-class image classification on the CIFAR-10 dataset, a standard benchmark for evaluating stochastic optimization methods in over￾parameterized and highly nonconvex settings. As the learning model, we use a ResNet￾20 architecture, follow… view at source ↗
Figure 2
Figure 2. Figure 2: Constrained orthogonal subspace fitting on synthetic spiked model data (d = 100, k = 5, n = 500). Comparison of STR-P with projected SGD, Riemannian gradient descent, and stochastic augmented Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Constrained orthogonal subspace fitting on synthetic spiked model data (d = 500, k = 10, n = 1000). Comparison of STR-P with projected SGD, Riemannian gradient descent, and stochastic augmented Lagrangian. 6. Conclusion In this work, we proposed a unified framework for stochastic trust-region meth￾ods in both unconstrained and equality-constrained over-parameterized optimization problems. By leveraging int… view at source ↗
read the original abstract

Under interpolation-type assumptions such as the strong growth condition, stochastic optimization methods can attain convergence rates comparable to full-batch methods, but their performance, particularly for SGD, remains highly sensitive to step-size selection. To address this issue, we propose a unified stochastic trust-region framework that eliminates manual step-size tuning and extends naturally to equality-constrained problems. For unconstrained optimization, we develop a first-order stochastic trust-region algorithm and show that, under the strong growth condition, it achieves an iteration and stochastic first-order oracle complexity of $O(\varepsilon^{-2} \log(1/\varepsilon))$ for finding an $\varepsilon$-stationary point. For equality-constrained problems, we introduce a quadratic-penalty-based stochastic trust-region method with penalty parameter $\mu$, and establish an iteration and oracle complexity of $O(\varepsilon^{-4} \log(1/\varepsilon))$ to reach an $\varepsilon$-stationary point of the penalized problem, corresponding to an $O(\varepsilon)$-approximate KKT point of the original constrained problem. Numerical experiments on deep neural network training and orthogonally constrained subspace fitting demonstrate that the proposed methods achieve performance comparable to well-tuned stochastic baselines, while exhibiting stable optimization behavior and effectively handling hard constraints without manual learning-rate scheduling.

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

0 major / 3 minor

Summary. The paper claims to develop a unified stochastic trust-region framework for over-parameterized models. Under the strong growth condition, the first-order stochastic trust-region algorithm achieves an iteration and stochastic first-order oracle complexity of O(ε^{-2} log(1/ε)) for ε-stationary points in the unconstrained case. For equality-constrained problems, the quadratic-penalty stochastic trust-region method attains O(ε^{-4} log(1/ε)) complexity for an ε-approximate KKT point. The methods are demonstrated on deep neural network training and orthogonally constrained subspace fitting tasks, showing performance comparable to tuned stochastic baselines with stable behavior.

Significance. If the results hold, this work is significant for providing automatic step-size adaptation via trust regions in stochastic settings, achieving rates that match or improve upon standard methods under interpolation assumptions common in over-parameterized models. The extension to constrained problems is particularly valuable, and the empirical results suggest practical applicability without extensive hyperparameter tuning. Strengths include the explicit complexity bounds and the handling of hard constraints.

minor comments (3)
  1. Clarify in the introduction or algorithm description how the trust-region parameters (e.g., initial radius and acceptance thresholds) are set in practice, as this could still involve some user choice despite the claim of eliminating manual step-size tuning.
  2. The description of the numerical experiments would benefit from specifying the number of independent runs or random seeds used and any statistical measures of variability, to better support the claims of stable optimization behavior and comparability to baselines.
  3. Ensure the strong growth condition is formally defined with its equation number in the main text before its use in complexity statements, for improved readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and recommendation for minor revision. The referee's summary accurately captures the key contributions of our work on the stochastic trust-region framework for over-parameterized models, including the complexity bounds under the strong growth condition and the empirical demonstrations on DNN training and constrained subspace fitting. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity in derivation of complexity bounds under explicit assumptions.

full rationale

The paper presents a theoretical analysis deriving iteration and stochastic first-order oracle complexity bounds for a stochastic trust-region algorithm and its quadratic-penalty extension. These bounds (O(ε^{-2} log(1/ε)) and O(ε^{-4} log(1/ε))) are obtained by applying standard trust-region analysis steps to the strong growth condition (SGC) as an input assumption, without any reduction of the claimed results to fitted parameters, self-definitions, or self-citation chains. No load-bearing steps equate outputs to inputs by construction, and the derivation remains self-contained once SGC and other stated assumptions are granted. This is the typical non-circular outcome for complexity proofs in optimization theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the strong growth condition as the key domain assumption for achieving the stated rates; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption strong growth condition (interpolation-type assumption)
    Invoked to obtain the O(ε^{-2} log(1/ε)) and O(ε^{-4} log(1/ε)) complexity bounds for the proposed algorithms.

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Works this paper leans on

43 extracted references · 43 canonical work pages · 1 internal anchor

  1. [1]

    Belkin, D

    M. Belkin, D. Hsu, S. Ma, and S. Mandal,Reconciling modern machine-learning practice and the classical bias–variance trade-off, Proceedings of the National Academy of Sciences 116 (2019), pp. 15849–15854

    work page 2019

  2. [2]

    Bellavia, B

    S. Bellavia, B. Morini, and S. Rebegoldi,On the convergence properties of a stochastic trust-region method with inexact restoration, Axioms 12 (2023), p. 38

    work page 2023

  3. [3]

    Bertsekas,Nonlinear programming, Journal of the Operational Research Society 48 (1997), pp

    D.P. Bertsekas,Nonlinear programming, Journal of the Operational Research Society 48 (1997), pp. 334–334

    work page 1997

  4. [4]

    Bertsekas,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 2014

    D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 2014

    work page 2014

  5. [5]

    Birgin and J.M

    E.G. Birgin and J.M. Mart´ ınez,Practical Augmented Lagrangian Methods for Constrained Optimization, SIAM, 2014

    work page 2014

  6. [6]

    D. Boob, Q. Deng, and G. Lan,Stochastic first-order methods for convex and nonconvex functional constrained optimization, Mathematical Programming 197 (2023), pp. 215–279

    work page 2023

  7. [7]

    Bottou,Large-scale machine learning with stochastic gradient descent, Proceedings of COMPSTAT (2010), pp

    L. Bottou,Large-scale machine learning with stochastic gradient descent, Proceedings of COMPSTAT (2010), pp. 177–186

    work page 2010

  8. [8]

    Bottou, F.E

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

    work page 2018

  9. [9]

    Cartis and K

    C. Cartis and K. Scheinberg,Global convergence rate analysis of unconstrained optimiza- tion methods based on probabilistic models, Mathematical Programming 169 (2018), pp. 337–375

    work page 2018

  10. [10]

    R. Chen, M. Menickelly, and K. Scheinberg,Stochastic optimization using a trust-region method and random models, Mathematical Programming 169 (2018), pp. 447–487

    work page 2018

  11. [11]

    Conn, N.I

    A.R. Conn, N.I. Gould, and P.L. Toint,Trust region methods, SIAM, 2000

    work page 2000

  12. [12]

    Curtis, D.P

    F.E. Curtis, D.P. Robinson, and M. Samadi,A stochastic sequential quadratic optimization algorithm for nonlinear equality constrained optimization, Mathematical Programming 188 (2021), pp. 235–292

    work page 2021

  13. [13]

    Curtis and R

    F.E. Curtis and R. Shi,A fully stochastic second-order trust region method, Optimization Methods and Software 37 (2022), pp. 844–877

    work page 2022

  14. [14]

    Y. Dar, P. Mayer, L. Luzi, and R. Baraniuk,Subspace fitting meets regression: The effects of supervision and orthonormality constraints on double descent of generalization errors, inInternational Conference on Machine Learning. PMLR, 2020, pp. 2366–2375

    work page 2020

  15. [15]

    Dzahini and S.M

    K.J. Dzahini and S.M. Wild,Stochastic trust-region algorithm in random subspaces with convergence and expected complexity analyses(2022). arXiv preprint

    work page 2022

  16. [16]

    Ghadimi and G

    S. Ghadimi and G. Lan,Stochastic first- and zeroth-order methods for nonconvex stochas- tic programming, SIAM Journal on Optimization 23 (2013), pp. 2341–2368

    work page 2013

  17. [17]

    Gratton, C.W

    S. Gratton, C.W. Royer, L.N. Vicente, and Z. Zhang,Complexity and global rates of trust-region methods based on probabilistic models, IMA Journal of Numerical Analysis 36 (2015), pp. 1662–1695

    work page 2015

  18. [18]

    Y. Ha, S. Shashaani, and R. Pasupathy,Complexity of zeroth- and first-order stochastic trust-region algorithms(2024). arXiv preprint

    work page 2024

  19. [19]

    K. He, X. Zhang, S. Ren, and J. Sun,Deep Residual Learning for Image Recognition, 24 inProceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 2016, pp. 770–778

    work page 2016

  20. [20]

    T. Lin, C. Jin, and M.I. Jordan,Near-optimal algorithms for minimax optimization, in Conference on Learning Theory (COLT). 2020, pp. 2738–2779

    work page 2020

  21. [21]

    Nemirovski, A

    A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro,Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization 19 (2009), pp. 1574– 1609

    work page 2009

  22. [22]

    Nocedal and S.J

    J. Nocedal and S.J. Wright,Numerical Optimization, 2nd ed., Springer, 2006

    work page 2006

  23. [23]

    Nouiehed, M

    M. Nouiehed, M. Sanjabi, T. Huang, J.D. Lee, and M. Razaviyayn,Solving a class of nonconvex min–max games using iterative first-order methods, SIAM Journal on Opti- mization 29 (2019), pp. 3011–3040

    work page 2019

  24. [24]

    Paquette and K

    C. Paquette and K. Scheinberg,Stochastic cubic regularization with variance reduction, Mathematical Programming 179 (2020), pp. 67–99

    work page 2020

  25. [25]

    Rafique, M

    H. Rafique, M. Liu, Q. Lin, and T. Yang,Weakly-convex concave minimax optimization: provable algorithms and applications in machine learning, Journal of Machine Learning Research 22 (2021), pp. 1–45

    work page 2021

  26. [26]

    Recht and C

    B. Recht and C. R´ e,Parallel stochastic gradient algorithms for large-scale matrix com- pletion, Mathematical Programming Computation 5 (2013), pp. 201–226

    work page 2013

  27. [27]

    Robbins and S

    H. Robbins and S. Monro,A stochastic approximation method, Annals of Mathematical Statistics 22 (1951), pp. 400–407

    work page 1951

  28. [28]

    Fast Convergence of Stochastic Gradient Descent under a Strong Growth Condition

    M. Schmidt and N.L. Roux,Fast convergence of stochastic gradient descent under a strong growth condition(2013). Available at https://arxiv.org/abs/1308.6370

    work page Pith review arXiv 2013

  29. [29]

    Schmidt, N.L

    M. Schmidt, N.L. Roux, and F. Bach,Minimizing finite sums with the stochastic average gradient, Mathematical Programming 162 (2017), pp. 83–112

    work page 2017

  30. [30]

    Shalev-Shwartz and S

    S. Shalev-Shwartz and S. Ben-David,Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014

    work page 2014

  31. [31]

    Z. Shen, P. Zhou, C. Fang, J. Xie, and A. Ribeiro,A stochastic trust region method for non-convex minimization(2020). ICLR 2020, revised 2025

    work page 2020

  32. [32]

    Vapnik,An overview of statistical learning theory, IEEE transactions on neural networks 10 (1999), pp

    V.N. Vapnik,An overview of statistical learning theory, IEEE transactions on neural networks 10 (1999), pp. 988–999

    work page 1999

  33. [33]

    Vaswani, F

    S. Vaswani, F. Bach, and M. Schmidt,Fast and Faster Convergence of SGD for Over- Parameterized Models and an Accelerated Perceptron, inProceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, K. Chaudhuri and M. Sugiyama, eds., Proceedings of Machine Learning Research Vol. 89, 16–18 Apr. PMLR, 2019, pp. 1195–1204

    work page 2019

  34. [34]

    Vaswani, K

    S. Vaswani, K. Mishchenko, I. Laradji, M. Schmidt, and L. Bottou,Fast and faster conver- gence of SGD for over-parameterized models and an accelerated perceptron, inAdvances in Neural Information Processing Systems. 2019

    work page 2019

  35. [35]

    Vaswani, A

    S. Vaswani, A. Mishkin, I. Laradji, M. Schmidt, G. Gidel, and S. Lacoste-Julien,Painless stochastic gradient: Interpolation, line-search, and convergence rates, Advances in neural information processing systems 32 (2019)

    work page 2019

  36. [36]

    Wang and Y.x

    X. Wang and Y.x. Yuan,Stochastic trust region methods with trust region radius depending on probabilistic models, arXiv preprint arXiv:1904.03342 (2019)

    work page arXiv 1904

  37. [37]

    Z. Wang, K. Balasubramanian, S. Ma, and M. Razaviyayn,Zeroth-order algorithms for nonconvex–strongly-concave minimax problems with improved complexities, Journal of Global Optimization 87 (2023), pp. 709–740

    work page 2023

  38. [38]

    Wen and W

    Z. Wen and W. Yin,A feasible method for optimization with orthogonality constraints, Mathematical Programming 142 (2013), pp. 397–434

    work page 2013

  39. [39]

    C. Xie, C. Li, C. Zhang, Q. Deng, D. Ge, and Y. Ye,Trust Region Methods For Nonconvex Stochastic Optimization Beyond Lipschitz Smoothness, inAAAI. 2024. arXiv preprint

    work page 2024

  40. [40]

    Xu,First-order methods for constrained convex programming based on linearized aug- mented lagrangian function, INFORMS Journal on Optimization 3 (2021), pp

    Y. Xu,First-order methods for constrained convex programming based on linearized aug- mented lagrangian function, INFORMS Journal on Optimization 3 (2021), pp. 89–117

    work page 2021

  41. [41]

    Understanding deep learning requires rethinking generalization

    C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals,Understanding deep learning requires rethinking generalization, arXiv preprint arXiv:1611.03530 (2016). 25

    work page internal anchor Pith review arXiv 2016

  42. [42]

    Zheng,Trust-region stochastic optimization with variance reduction technique(2024)

    X. Zheng,Trust-region stochastic optimization with variance reduction technique(2024). arXiv preprint

    work page 2024

  43. [43]

    Y. Zhou, J. Yang, H. Zhang, Y. Liang, and V. Tarokh,Sgd converges to global minimum in deep learning via star-convex path, arXiv preprint arXiv:1901.00451 (2019). 26

    work page Pith review arXiv 1901