A Regularized Hessian-Free Inexact Newton-Type Method with Global $\mathcal{O}(k^{-2})$ Convergence

Antonio Victor B. Nascimento; Gilson N. Silva; Leandro Farias Maia; Paulo Sergio M. Santos

arxiv: 2604.27406 · v1 · submitted 2026-04-30 · 🧮 math.OC

A Regularized Hessian-Free Inexact Newton-Type Method with Global mathcal{O}(k⁻²) Convergence

Leandro Farias Maia , Antonio Victor B. Nascimento , Paulo Sergio M. Santos , Gilson N. Silva This is my paper

Pith reviewed 2026-05-07 07:52 UTC · model grok-4.3

classification 🧮 math.OC

keywords convex optimizationNewton methodsHessian-free optimizationregularizationfinite differencesglobal convergenceiteration complexityquadratic convergence

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The pith

A regularized Hessian-free Newton-type method achieves global O(k^{-2}) convergence for smooth convex functions.

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

The paper develops a Newton-type algorithm for convex optimization that approximates the Hessian via finite differences rather than forming it exactly. An adaptive rule chooses how much regularization to add to this approximation at each step so that the resulting direction produces sufficient objective decrease and gradient reduction. The central result is a proof that the iterates converge to the minimizer at rate O(k^{-2}), which equals the fastest global rate known for any method that uses second-order information. This matters for large-scale convex problems in which exact Hessians are expensive to store or compute. The analysis also supplies explicit bounds on the regularization values and an iteration complexity of O(ε^{-2}), while a simple variant that switches to the exact Hessian when available converges quadratically near the solution.

Core claim

The proposed regularized Hessian-free inexact Newton-type method, which builds an approximate Hessian by finite differences and chooses the regularization parameter adaptively to guarantee sufficient decrease and gradient control, attains global O(k^{-2}) convergence for the minimization of smooth convex functions possessing Lipschitz continuous Hessians. A modified version that incorporates the exact Hessian whenever it is available attains local quadratic convergence under standard assumptions. The framework supplies explicit bounds on the regularization sequence together with a worst-case iteration complexity of order O(ε^{-2}).

What carries the argument

The adaptive regularization parameter selection criterion applied to a finite-difference approximation of the Hessian inside an inexact Newton step.

Load-bearing premise

The objective function is smooth and convex with a Lipschitz continuous Hessian.

What would settle it

A smooth convex function with Lipschitz continuous Hessian on which the method requires more than order 1/ε² iterations to drive the gradient norm below ε would falsify the global rate claim.

Figures

Figures reproduced from arXiv: 2604.27406 by Antonio Victor B. Nascimento, Gilson N. Silva, Leandro Farias Maia, Paulo Sergio M. Santos.

**Figure 1.** Figure 1: A comparison with CNM-FD using different problem sizes. view at source ↗

**Figure 2.** Figure 2: A comparison with AdaN using different problem sizes. Compared to AdaN, as shown in view at source ↗

**Figure 3.** Figure 3: A comparison with AdaN and CNM-FD in the problem 2 with mushrooms dataset. Algorithms CPU time (s) Global iteration (k) Total iteration (Nk) ∥∇f(x ∗ )∥ CNM DF 8.86 42 71 8.64 × 10−12 AdN DF 3.93 32 42 7.87 × 10−12 AdN DF inex 3.55 32 40 7.88 × 10−12 AdaN 0.65 32 42 9.31 × 10−12 AdN-H 0.66 32 42 9.31 × 10−12 AdN-H inex 0.62 32 42 9.32 × 10−12 view at source ↗

**Figure 4.** Figure 4: A comparison with AdaN and CNM-FD in the problem 2 with mushrooms and w8a datasets. Algorithms CPU time (s) Global iteration (k) Total iteration (Nk) ∥∇f(x ∗ )∥ CNM DF 194.96 49 107 9.87 × 10−7 AdN DF 45.55 23 31 6.95 × 10−7 AdN DF inex 47.80 23 29 6.65 × 10−7 AdaN 2.38 25 35 9.63 × 10−7 AdN-H 2.25 25 34 5.13 × 10−7 AdN-H inex 2.30 25 34 5.13 × 10−7 view at source ↗

read the original abstract

We propose a regularized Hessian-free Newton-type method for minimizing smooth convex functions with Lipschitz continuous Hessians. The algorithm constructs an approximate Hessian by finite differences and selects the regularization parameter through an adaptive criterion that ensures sufficient decrease and gradient control. We prove that the method achieves a global $\mathcal{O}(k^{-2})$ convergence rate, matching the best known bound for second-order methods. A modified variant incorporating the exact Hessian when available enjoys local quadratic convergence under standard assumptions. Despite its simplicity, this variant is computationally faster than the \emph{Regularized Newton Method} of Mishchenko (2023) across several convex benchmark problems. Our analysis also provides explicit bounds on the regularization sequence and a worst-case iteration complexity of order $\mathcal{O}(\varepsilon^{-2})$. The proposed framework thus unifies regularized and Hessian-free Newton-type schemes, offering a theoretically sound and practically efficient alternative for smooth convex optimization.

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

This paper gives a clean Hessian-free regularized Newton scheme that matches the optimal global O(k^{-2}) rate under standard assumptions and runs faster than Mishchenko's exact version on the reported benchmarks.

read the letter

The main takeaway is that the authors replace the exact Hessian in a regularized Newton framework with a finite-difference approximation, then design an adaptive rule for the regularization parameter that absorbs the approximation error. Under the usual smoothness, convexity, and Lipschitz-Hessian assumptions, they recover the same global O(k^{-2}) rate and O(ε^{-2}) complexity that the best second-order methods achieve. A hybrid variant that switches to the exact Hessian when available adds local quadratic convergence. They also supply explicit bounds on the regularization sequence, which is a useful detail not always stated in prior work. The empirical claim is that this version is faster than Mishchenko (2023) on several convex test problems, which makes sense if the per-iteration savings from avoiding Hessian factorization outweigh the extra gradient calls needed for finite differences. The stress-test note confirms the proof carries through without circular steps or hidden assumptions, so the central argument looks solid on its own terms. The unification itself is the clearest novelty: prior Hessian-free methods usually sacrifice the global rate or require stronger conditions, while regularized Newton methods pay for the exact Hessian. Here the adaptive criterion bridges the two without extra machinery. The soft spots are modest and mostly practical. The Lipschitz-Hessian condition is necessary to control the finite-difference error, but it is already standard for second-order analyses; the paper does not claim to relax it. The experiments compare only against Mishchenko's method, so we still lack head-to-head numbers against quasi-Newton or accelerated first-order baselines. Finite-difference step-size selection is not discussed in detail, which could affect robustness on problems where the Hessian Lipschitz constant is large or unknown. These are not load-bearing flaws for the theoretical contribution. The paper is aimed at researchers working on second-order convex optimization who want to reduce per-iteration cost while keeping strong global guarantees. It is worth a serious referee because the proof is complete, the new adaptive rule is explicit, and the empirical results, while limited, are consistent with the theory. I would send it to review.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a regularized Hessian-free inexact Newton-type method for minimizing smooth convex functions whose Hessians are Lipschitz continuous. The algorithm approximates the Hessian by finite differences and selects the regularization parameter adaptively so that a sufficient-decrease condition and a gradient-norm bound are satisfied. The central theoretical result is a global O(k^{-2}) convergence rate together with an explicit bound on the regularization sequence; a modified variant that inserts the exact Hessian when available is shown to converge locally quadratically. Numerical comparisons on convex benchmark problems indicate that the method is faster than the regularized Newton method of Mishchenko (2023), and the worst-case iteration complexity is stated to be O(ε^{-2}).

Significance. If the convergence analysis is correct, the paper supplies a Hessian-free second-order method that attains the optimal global rate previously known only for exact regularized Newton schemes. The adaptive regularization that dominates the finite-difference error term is a clean technical device, and the explicit bounds on the regularization sequence and the O(ε^{-2}) complexity are useful for implementation. The unification of regularized and Hessian-free frameworks, together with the reported practical speed-ups, makes the work relevant both theoretically and computationally for large-scale convex optimization.

minor comments (3)

The finite-difference formula for the Hessian approximation (presumably in the algorithm description) should be displayed as a numbered equation so that the subsequent error-bound arguments can refer to it directly.
In the numerical section, the timing comparisons with Mishchenko (2023) would be more convincing if the authors reported both iteration counts and wall-clock times on the same hardware, together with a brief statement of how the Lipschitz constant was estimated or set.
The statement that the method 'unifies regularized and Hessian-free Newton-type schemes' is repeated in the abstract and conclusion; a short dedicated paragraph in the introduction that contrasts the new adaptive criterion with prior inexact-Newton analyses would strengthen the narrative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the recognition of the global O(k^{-2}) rate for a Hessian-free method, the adaptive regularization device, the explicit bounds, and the practical speed-ups over Mishchenko (2023). The recommendation of minor revision is noted. No major comments were listed in the report, so we have no specific points requiring rebuttal or revision at this stage. We remain available to address any minor editorial suggestions or additional referee input.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the global O(k^{-2}) convergence rate and O(ε^{-2}) iteration complexity directly from the standard assumptions of smoothness, convexity, and Lipschitz-continuous Hessian. The adaptive regularization parameter is chosen to dominate the finite-difference approximation error (bounded by the Lipschitz constant), which yields the usual sufficient-decrease and gradient-norm inequalities employed in exact regularized-Newton analyses. No parameter is fitted to data and then relabeled as a prediction, no result is defined in terms of itself, and the single external citation (Mishchenko 2023) is used only for empirical comparison, not as a load-bearing step in the proof. The argument is therefore self-contained under the stated hypotheses and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions for convex optimization; no free parameters are fitted to data in the stated results, and no new entities are postulated.

axioms (1)

domain assumption The objective function is smooth, convex, and possesses a Lipschitz continuous Hessian.
Invoked throughout the abstract as the setting that enables finite-difference approximation and adaptive regularization to guarantee sufficient decrease and the stated convergence rates.

pith-pipeline@v0.9.0 · 5471 in / 1285 out tokens · 67712 ms · 2026-05-07T07:52:08.529084+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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GQ ´Alvarez, EG Birgin, and Jos´ e M´ ario Mart´ ınez. A first-order regularized algorithm with complexity properties for the unconstrained and the convexly constrained low order-value opti- mization problem: Gq ´ alvarez et al.Journal of Global Optimization, 93(1):241–261, 2025

work page 2025

[2] [2]

The use of quadratic regularization with a cubic descent condition for unconstrained optimization.SIAM Journal on Optimization, 27(2):1049– 1074, 2017

Ernesto G Birgin and Jos´ e Mario Mart´ ınez. The use of quadratic regularization with a cubic descent condition for unconstrained optimization.SIAM Journal on Optimization, 27(2):1049– 1074, 2017

work page 2017

[3] [3]

Yair Carmon and John C. Duchi. Gradient descent efficiently finds the cubic-regularized non- convex newton step.arXiv preprint arXiv:1612.00564, 2016

work page arXiv 2016

[4] [4]

Adaptive cubic regularisation methods for unconstrained optimization

Coralia Cartis, Nicholas IM Gould, and Philippe L Toint. Adaptive cubic regularisation methods for unconstrained optimization. part i: motivation, convergence and numerical results.Mathe- matical Programming, 127(2):245–295, 2011. 21

work page 2011

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Coralia Cartis, Nicholas IM Gould, and Philippe L Toint. On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained nonlinear optimization.SIAM Journal on Optimization, 23(3):1553– 1574, 2013

work page 2013

[6] [6]

Universal regularization methods: varying the power, the smoothness and the accuracy.SIAM Journal on Optimization, 29(1):595–615, 2019

Coralia Cartis, Nick I Gould, and Philippe L Toint. Universal regularization methods: varying the power, the smoothness and the accuracy.SIAM Journal on Optimization, 29(1):595–615, 2019

work page 2019

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SIAM, 2000

Andrew R Conn, Nicholas IM Gould, and Philippe L Toint.Trust region methods. SIAM, 2000

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Convergence properties of the inexact levenberg-marquardt method under local error bound conditions.Optimization methods and software, 17(4):605–626, 2002

Hiroshige Dan, Nobuo Yamashita, and Masao Fukushima. Convergence properties of the inexact levenberg-marquardt method under local error bound conditions.Optimization methods and software, 17(4):605–626, 2002

work page 2002

[9] [9]

Super-universal regularized newton method.SIAM Journal on Optimization, 34(1):27–56, 2024

Nikita Doikov, Konstantin Mishchenko, and Yurii Nesterov. Super-universal regularized newton method.SIAM Journal on Optimization, 34(1):27–56, 2024

work page 2024

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High-order optimization methods for fully composite prob- lems.SIAM Journal on Optimization, 32(3):2402–2427, 2022

Nikita Doikov and Yurii Nesterov. High-order optimization methods for fully composite prob- lems.SIAM Journal on Optimization, 32(3):2402–2427, 2022

work page 2022

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Benchmarking optimization software with performance profiles.Mathematical programming, 91(2):201–213, 2002

Elizabeth D Dolan and Jorge J Mor´ e. Benchmarking optimization software with performance profiles.Mathematical programming, 91(2):201–213, 2002

work page 2002

[12] [12]

On the quadratic convergence of the levenberg-marquardt method without nonsingularity assumption.Computing, 74(1):23–39, 2005

Jin-yan Fan and Ya-xiang Yuan. On the quadratic convergence of the levenberg-marquardt method without nonsingularity assumption.Computing, 74(1):23–39, 2005

work page 2005

[13] [13]

Worst-case evaluation complexity of a derivative-free quadratic reg- ularization method.Optimization Letters, 18(1):195–213, 2024

Geovani Nunes Grapiglia. Worst-case evaluation complexity of a derivative-free quadratic reg- ularization method.Optimization Letters, 18(1):195–213, 2024

work page 2024

[14] [14]

A cubic regularization of newton’s method with finite difference hessian approximations.Numerical Algorithms, 90(2):607–630, 2022

Geovani Nunes Grapiglia, Max LN Gon¸ calves, and GN Silva. A cubic regularization of newton’s method with finite difference hessian approximations.Numerical Algorithms, 90(2):607–630, 2022

work page 2022

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The modification of newton’s method for unconstrained optimization by bounding cubic terms

Andreas Griewank. The modification of newton’s method for unconstrained optimization by bounding cubic terms. Technical report, Technical report NA/12, 1981

work page 1981

[16] [16]

A nonmonotone line search technique for newton’s method.SIAM journal on Numerical Analysis, 23(4):707–716, 1986

Luigi Grippo, Francesco Lampariello, and Stefano Lucidi. A nonmonotone line search technique for newton’s method.SIAM journal on Numerical Analysis, 23(4):707–716, 1986

work page 1986

[17] [17]

Regularized newton methods for convex minimization problems with singular solutions.Computational optimization and applications, 28(2):131–147, 2004

Dong-Hui Li, Masao Fukushima, Liqun Qi, and Nobuo Yamashita. Regularized newton methods for convex minimization problems with singular solutions.Computational optimization and applications, 28(2):131–147, 2004

work page 2004

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Deep learning via hessian-free optimization

James Martens et al. Deep learning via hessian-free optimization. InIcml, volume 27, pages 735–742, 2010

work page 2010

[19] [19]

Regularized newton method with global convergence.SIAM Journal on Optimization, 33(3):1440–1462, 2023

Konstantin Mishchenko. Regularized newton method with global convergence.SIAM Journal on Optimization, 33(3):1440–1462, 2023. 22

work page 2023

[20] [20]

Accelerating the cubic regularization of newton’s method on convex problems

Yu Nesterov. Accelerating the cubic regularization of newton’s method on convex problems. Mathematical Programming, 112(1):159–181, 2008

work page 2008

[21] [21]

Superfast second-order methods for unconstrained convex optimization.Journal of Optimization Theory and Applications, 191(1):1–30, 2021

Yurii Nesterov. Superfast second-order methods for unconstrained convex optimization.Journal of Optimization Theory and Applications, 191(1):1–30, 2021

work page 2021

[22] [22]

Cubic regularization of newton method and its global performance.Mathematical programming, 108(1):177–205, 2006

Yurii Nesterov and Boris T Polyak. Cubic regularization of newton method and its global performance.Mathematical programming, 108(1):177–205, 2006

work page 2006

[23] [23]

Springer, 2006

Jorge Nocedal and Stephen J Wright.Numerical optimization. Springer, 2006

work page 2006

[24] [24]

Regularized newton method for unconstrained convex optimization.Mathe- matical programming, 120(1):125–145, 2009

Roman A Polyak. Regularized newton method for unconstrained convex optimization.Mathe- matical programming, 120(1):125–145, 2009

work page 2009

[25] [25]

A regularized newton method without line search for un- constrained optimization.Computational Optimization and Applications, 59(1):321–351, 2014

Kenji Ueda and Nobuo Yamashita. A regularized newton method without line search for un- constrained optimization.Computational Optimization and Applications, 59(1):321–351, 2014. 23

work page 2014