arxiv: 2604.02763 · v2 · submitted 2026-04-03 · 🧮 math.OC

Recognition: 2 theorem links

· Lean Theorem

Adaptive Newton-CG methods with global and local analysis for unconstrained optimization with H\"older continuous Hessian

Ziyang Zeng , Junyu Zhang , Chuan He

Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:18 UTC · model grok-4.3

classification 🧮 math.OC

keywords Newton-CG methodsHölder continuous Hessiannonconvex optimizationiteration complexitysuperlinear convergenceadaptive regularizationunconstrained minimization

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The pith

Adaptive Newton-CG methods achieve optimal iteration complexity for nonconvex optimization with Hölder continuous Hessians.

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

The paper develops two Newton-CG algorithms for minimizing nonconvex functions whose Hessians satisfy a (H_f, ν)-Hölder condition. These algorithms adaptively regularize the Newton system using an auto-conditioning technique, removing the need for nested line searches that appear in prior methods. If the approach works, the methods attain the complexity bound O(H_f^{1/(1+ν)} ε^{-(2+ν)/(1+ν)}) for an ε-stationary point while also delivering local superlinear convergence near nondegenerate local minimizers. The two variants differ only in whether the exponent ν must be supplied in advance. This combination improves both early-stage progress and final convergence speed compared with non-adaptive regularization schemes.

Core claim

The central claim is that adaptive regularization of the Newton-CG system via the auto-conditioning technique yields the iteration complexity O(H_f^{1/(1+ν)} ε^{-(2+ν)/(1+ν)}) for finding an ε-stationary point of a nonconvex function with (H_f, ν)-Hölder continuous Hessian and simultaneously guarantees local superlinear convergence near nondegenerate local minimizers, without requiring nested line searches.

What carries the argument

Adaptive regularization of the Newton system via the auto-conditioning technique, which dynamically selects the regularization parameter to guarantee descent directions while eliminating inner line-search loops.

If this is right

The methods apply to nonconvex problems whose Hessians are Hölder but not Lipschitz continuous.
Local superlinear convergence is retained near nondegenerate minimizers, unlike non-adaptive predecessors.
Only one Newton-system solve is needed per outer iteration, lowering per-iteration cost.
The same complexity bound holds whether or not the exponent ν is provided to the algorithm.

Where Pith is reading between the lines

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

The adaptive mechanism could be ported to other second-order frameworks such as trust-region or cubic-regularization methods.
The observed practical speed-up may improve scalability on large-scale nonconvex problems arising in machine learning.
Further automation of Hölder-parameter estimation could remove the remaining dependence on prior knowledge of ν.

Load-bearing premise

The objective function has a Hessian that is Hölder continuous with modulus H_f and exponent ν in (0,1].

What would settle it

A concrete function with known Hölder parameters H_f and ν on which the number of outer iterations required to reach an ε-stationary point grows faster than the stated bound by a constant factor independent of ε.

read the original abstract

In this paper, we study Newton-conjugate gradient (Newton-CG) methods for minimizing a nonconvex function $f$ whose Hessian is $(H_f,\nu)$-H\"older continuous with modulus $H_f>0$ and exponent $\nu\in(0,1]$. Recently proposed Newton-CG methods for this problem adopt (i) non-adaptive regularization and (ii) a nested line-search procedure, where (i) often leads to inefficient early progress and the loss of local superlinear convergence, and (ii) may incur high computational cost due to multiple solves of the Newton system per iteration. To address these limitations, we propose two novel Newton-CG algorithms, depending on the availability of $\nu$, that adaptively regularize the Newton system by leveraging the auto-conditioning technique to eliminate the nested line search. The proposed algorithms achieve the best-known iteration complexity ${\mathcal O}\big(H_f^{1/(1+\nu)}\epsilon^{-(2+\nu)/(1+\nu)}\big)$ for finding an $\epsilon$-stationary point and, simultaneously, enjoy local superlinear convergence near nondegenerate local minimizers. Numerical experiments further demonstrate the practical advantages of our algorithms over existing approaches.

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

They've made Newton-CG adaptive for Hölder Hessians by using auto-conditioning to skip nested line searches, hitting the optimal complexity and local superlinear rates.

read the letter

The main thing to know is that this paper removes the nested line search from Newton-CG for functions with Hölder continuous Hessians. They do it with an auto-conditioning trick that sets the regularization parameter on the fly, and they still recover the best-known outer-iteration bound of order H_f^{1/(1+ν)} ε^{-(2+ν)/(1+ν)} plus local superlinear convergence near nondegenerate points. They handle the known-ν and unknown-ν cases separately, which is useful in practice. The global analysis pulls the sufficient-decrease condition directly from the Hölder assumption, and the local rate follows from the usual nondegeneracy argument once you are close. The experiments show clear practical speedups over the non-adaptive baselines on the test problems they ran. A minor soft spot is that the inner CG work per outer step is not folded into the complexity bound, so total arithmetic cost could still fluctuate. When ν is unknown the adaptive rule introduces one extra parameter that may need some care in implementation. Nothing in the argument looks circular or rests on hidden boundedness assumptions. This is aimed at people who work on second-order methods for nonconvex problems, especially in machine learning and scientific computing where Hessians are rarely Lipschitz. The claims are sharp enough and the technical steps look clean enough that it deserves a serious referee rather than a desk reject. I would send it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes two adaptive Newton-CG algorithms for unconstrained nonconvex minimization of functions whose Hessians satisfy a (H_f, ν)-Hölder condition. The algorithms employ auto-conditioning to select the regularization parameter adaptively, thereby removing the nested line-search procedure used in prior methods. Separate analyses are given for the cases of known and unknown ν; both achieve the iteration complexity O(H_f^{1/(1+ν)} ε^{-(2+ν)/(1+ν)}) to reach an ε-stationary point and, under a nondegeneracy assumption, local superlinear convergence near local minimizers. Numerical experiments on standard test problems illustrate faster practical progress than non-adaptive baselines.

Significance. If the global and local analyses hold, the work supplies a practically implementable second-order method that simultaneously attains the best-known worst-case iteration bound for Hölder-Hessian problems and recovers fast local convergence without sacrificing adaptivity. The removal of nested line searches via auto-conditioning is a concrete algorithmic advance that reduces per-iteration linear-system solves while preserving the same descent and model-accuracy conditions used in earlier non-adaptive analyses.

minor comments (3)

[§1] §1: The statement that the complexity is 'best-known' would be strengthened by a short explicit comparison (in a table or paragraph) of the leading constants with those appearing in the non-adaptive analyses of Cartis et al. and similar references.
[§4] §4 (global analysis): The dependence of the iteration bound on the initial regularization parameter β_0 is stated to be absorbed into the O-notation; a brief remark quantifying this dependence (or showing it is independent of β_0) would make the result fully transparent.
[Numerical experiments] Numerical experiments: While performance profiles are presented, reporting the average number of CG iterations per outer iteration (and its dependence on ν) would give readers a clearer picture of the practical cost of the inner solver.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation for minor revision. The referee's description accurately captures the contributions of the adaptive Newton-CG methods, including the removal of nested line searches via auto-conditioning, the iteration complexity bound, and the local superlinear convergence result.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation separates known-ν and unknown-ν cases, derives sufficient-decrease and model-accuracy conditions directly from the (H_f, ν)-Hölder assumption on the Hessian, and shows that the adaptive parameter choice satisfies the same descent lemma used in prior non-adaptive analyses. The O(H_f^{1/(1+ν)} ε^{-(2+ν)/(1+ν)}) bound and local superlinear rate follow from standard Newton-CG analysis under nondegeneracy without reducing to fitted inputs, self-definitions, or load-bearing self-citations. The auto-conditioning technique eliminates nested line searches while preserving the existing complexity framework, keeping the central claims independent of the paper's own fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard domain assumption of Hölder continuous Hessian and standard mathematical tools for convergence analysis.

axioms (1)

domain assumption The objective function f has a Hessian that is (H_f, ν)-Hölder continuous for some H_f > 0 and ν in (0,1].
This is the key assumption on the smoothness of the function in the problem statement.

pith-pipeline@v0.9.0 · 5519 in / 1149 out tokens · 48240 ms · 2026-05-13T19:18:03.411385+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The proposed algorithms achieve the best-known iteration complexity O(H_f^{1/(1+ν)} ε^{-(2+ν)/(1+ν)}) for finding an ε-stationary point
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Hölder continuous Hessian with modulus Hf>0 and exponent ν∈(0,1]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

[1]

Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization.Proc

Coralia Cartis, Nicholas IM Gould, and Philippe L Toint. Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization.Proc. Int. Congr. Math. (ICM 2018), 3:3711–3750,

work page 2018
[2]

Projected gradient methods for nonconvex and stochastic optimization: new complexities and auto-conditioned stepsizes.arXiv preprint arXiv:2412.14291,

Guanghui Lan, Tianjiao Li, and Yangyang Xu. Projected gradient methods for nonconvex and stochastic optimization: new complexities and auto-conditioned stepsizes.arXiv preprint arXiv:2412.14291,

work page internal anchor Pith review arXiv
[3]

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Tianjiao Li and Guanghui Lan

doi: 10.1023/B:COAP.0000026881.96694.32. Tianjiao Li and Guanghui Lan. A simple uniformly optimal method without line search for convex optimization.Math. Program., pages 1–38,

work page doi:10.1023/b:coap.0000026881.96694.32
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Konstantin Mishchenko

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Yu Nesterov

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Cl´ ement W Royer, Michael O’Neill, and Stephen J Wright

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[8]

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doi: 10.1007/s10589-025-00692-x. URLhttps://link.springer.com/article/10.1007/s10589-025-00692-x. Yuhao Zhou, Jintao Xu, Bingrui Li, Chenglong Bao, Chao Ding, and Jun Zhu. A regularized Newton method for nonconvex optimization with global and local complexity guarantees. In Advances in Neural Information Processing Systems,

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