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

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Stochastic Trust-Region Methods for Over-parameterized Models

Aike Yang , Hao Wang
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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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