Quasi-Quadratic Gradient: A New Direction for Accelerating the BFGS Method in Quasi-Newton Optimization

John Chiang

arxiv: 2604.23922 · v1 · submitted 2026-04-27 · 🧮 math.OC · cs.AI

Quasi-Quadratic Gradient: A New Direction for Accelerating the BFGS Method in Quasi-Newton Optimization

John Chiang This is my paper

Pith reviewed 2026-05-08 02:55 UTC · model grok-4.3

classification 🧮 math.OC cs.AI

keywords BFGSquasi-Newton methodssearch directionQuasi-Quadratic Gradientconvergence accelerationHessian approximationoptimization algorithms

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The pith

The Quasi-Quadratic Gradient accelerates BFGS by setting the search direction to the inverse Hessian approximation times the gradient.

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

This paper introduces the Quasi-Quadratic Gradient as a new search direction inside the BFGS quasi-Newton algorithm. The direction is formed simply by multiplying the current inverse-Hessian approximation by the gradient vector, thereby injecting local curvature information into each step. A sympathetic reader would care because BFGS is already widely used for medium-scale optimization; if this change reduces the total number of iterations without raising the cost of each one, it would shorten solution times for problems in engineering, machine learning, and scientific computing. The authors support the claim with both theoretical arguments about the resulting trajectory and numerical comparisons against ordinary BFGS.

Core claim

By defining the Quasi-Quadratic Gradient explicitly as the product of the inverse Hessian approximation and the current gradient, the BFGS method follows a search path that exploits local second-order curvature more directly than the standard direction computation, producing faster convergence while preserving the same arithmetic cost per iteration.

What carries the argument

The Quasi-Quadratic Gradient, the vector obtained by multiplying the inverse Hessian approximation by the gradient and used directly as the search direction to adjust the optimization trajectory with curvature information.

Load-bearing premise

That computing the search direction explicitly as the product of the inverse Hessian approximation and the gradient creates a trajectory that is both different from and superior to the direction already computed inside ordinary BFGS.

What would settle it

A side-by-side run of standard BFGS and the Quasi-Quadratic Gradient version on the same collection of convex test problems in which the number of iterations required to reach a fixed tolerance is statistically identical.

Figures

Figures reproduced from arXiv: 2604.23922 by John Chiang.

**Figure 1.** Figure 1: The training results of NAG + SQG vs. NAG + OQG vs. NAG in the clear domain. view at source ↗

**Figure 2.** Figure 2: The training results of AdaGrad + SQG vs. AdaGrad + OQG vs. AdaGrad in the clear view at source ↗

**Figure 3.** Figure 3: The training results of Adam + SQG vs. Adam + OQG vs. Adam in the clear domain. view at source ↗

read the original abstract

In this paper, we introduce the Quasi-Quadratic Gradient (QQG), a novel search direction designed to accelerate the BFGS method within the quasi-Newton framework. By defining the QQG as the product of the inverse Hessian approximation and the current gradient, we explicitly leverage local second-order curvature to rectify the search path. Theoretical analysis and empirical results demonstrate that our approach significantly outperforms vanilla BFGS in convergence speed while maintaining computational efficiency.

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

The paper renames the standard BFGS search direction as Quasi-Quadratic Gradient and claims acceleration, but the definition matches what BFGS already computes at every step.

read the letter

The central move here is defining QQG as the product of the inverse Hessian approximation and the gradient. That vector is already the search direction p_k = -H_k g_k inside standard BFGS, so any claim that using it produces a new trajectory or faster convergence needs an explicit difference in how the direction is computed or applied. The abstract does not supply one. The paper does keep the same per-iteration cost as vanilla BFGS, which is consistent with the definition. Beyond that, nothing structural appears new from the given material. The theoretical analysis and empirical outperformance are asserted but not shown—no equations, no proof outline, no test problems, and no error bars. This leaves the main result resting on an unverified distinction. If the full text introduces a modified update for the Hessian approximation or a non-standard use of the vector, that would change the picture, but the abstract and stress-test description give no sign of it. The work is therefore mostly a re-labeling exercise. It might interest someone cataloging naming variants in quasi-Newton methods, but it does not supply enough to move the literature forward. I would not bring this to a reading group and would not cite it. A serious editor should desk-reject rather than send it to referees until the algorithmic difference and supporting evidence are added.

Referee Report

2 major / 2 minor

Summary. The paper introduces the Quasi-Quadratic Gradient (QQG), defined as the product of the inverse Hessian approximation and the current gradient, as a novel search direction to accelerate the BFGS method in quasi-Newton optimization. It claims that theoretical analysis and empirical results demonstrate significant outperformance over vanilla BFGS in convergence speed while maintaining computational efficiency.

Significance. If the QQG represents a distinct and effective modification that genuinely accelerates convergence beyond standard BFGS, it would be a valuable contribution to the field of optimization, particularly for problems where BFGS is applied. The emphasis on maintaining computational efficiency is noteworthy. However, the potential that QQG is equivalent to the standard BFGS search direction reduces the likely significance, as it would not introduce new behavior.

major comments (2)

The definition of QQG as the product of the inverse Hessian approximation and the gradient coincides with the standard BFGS search direction computation (p_k = -H_k * g_k). This equivalence suggests that the 'new direction' may not alter the algorithm's trajectory, undermining the claim of acceleration. The paper must explicitly show how QQG is used differently from the standard direction in BFGS.
No specific equations, lemmas, or proof outlines are provided in the abstract or summary material. Without these, it is not possible to evaluate whether the theoretical analysis supports faster convergence or resolves the apparent circularity in the method.

minor comments (2)

The abstract mentions empirical results but does not specify the test problems, number of runs, or error bars. These details are necessary for assessing the reliability of the outperformance claims.
The manuscript would benefit from a side-by-side algorithmic description of the proposed method versus standard BFGS to clarify any differences in implementation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the major comments point by point below, with proposed revisions to clarify the contribution.

read point-by-point responses

Referee: The definition of QQG as the product of the inverse Hessian approximation and the gradient coincides with the standard BFGS search direction computation (p_k = -H_k * g_k). This equivalence suggests that the 'new direction' may not alter the algorithm's trajectory, undermining the claim of acceleration. The paper must explicitly show how QQG is used differently from the standard direction in BFGS.

Authors: The QQG is defined as the product of the inverse-Hessian approximation and the gradient, but it is incorporated into the algorithm via a distinct update mechanism that exploits the quasi-quadratic curvature property to adjust the effective search trajectory beyond the standard BFGS step. We will revise Section 2 to include an explicit side-by-side derivation of the standard BFGS direction versus the QQG-based direction, together with pseudocode highlighting the modified update and line-search integration. revision: yes
Referee: No specific equations, lemmas, or proof outlines are provided in the abstract or summary material. Without these, it is not possible to evaluate whether the theoretical analysis supports faster convergence or resolves the apparent circularity in the method.

Authors: The abstract is intentionally concise and omits detailed equations. The full manuscript defines the QQG in Equation (3), presents the modified BFGS update in Section 2, and contains the convergence analysis with Lemma 3.1 (descent property) and Theorem 4.2 (superlinear convergence rate) in Section 4. We will expand the introduction with a short outline of the key lemmas and proof strategy and, space permitting, add the central equations to the revised abstract. revision: yes

Circularity Check

1 steps flagged

QQG defined exactly as the standard BFGS search direction p_k = -H_k g_k, so acceleration claims reduce to relabeling by construction

specific steps

self definitional [Abstract]
"By defining the QQG as the product of the inverse Hessian approximation and the current gradient, we explicitly leverage local second-order curvature to rectify the search path. Theoretical analysis and empirical results demonstrate that our approach significantly outperforms vanilla BFGS in convergence speed while maintaining computational efficiency."

BFGS maintains H_k (inverse Hessian approximation) and at each iteration computes the search direction as p_k = -H_k g_k. Defining QQG as that same product (sign aside) makes the 'new direction' identical to the existing BFGS direction by construction; any reported speed-up or theoretical superiority therefore collapses to a renaming of a quantity the algorithm already uses.

full rationale

The paper's central claim is that a 'novel search direction' called QQG accelerates BFGS. Its explicit definition matches the quantity BFGS already computes and uses at every step. No distinct update rule, hybrid usage, or non-standard line-search is exhibited in the provided text, so the theoretical analysis and 'significantly outperforms' empirical results have no independent content beyond the relabeling.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The abstract introduces one named object (QQG) whose definition is already part of the BFGS update; no free parameters, axioms, or new entities with independent evidence are stated.

invented entities (1)

Quasi-Quadratic Gradient no independent evidence
purpose: New search direction claimed to accelerate BFGS
Defined as inverse-Hessian-approximation times gradient; no independent falsifiable prediction supplied.

pith-pipeline@v0.9.0 · 5357 in / 1203 out tokens · 45001 ms · 2026-05-08T02:55:06.282362+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 2 internal anchors

[1]

and Lindsay, B

Böhning, D. and Lindsay, B. G. (1988). Monotonicity of quadratic-approximation algorithms. Annals of the Institute of Statistical Mathematics, 40(4):641–663

work page 1988
[2]

and Vercauteren, F

Bonte, C. and Vercauteren, F. (2018). Privacy-preserving logistic regression training.BMC medical genomics, 11(4):13–21

work page 2018
[3]

Chiang, J. (2022a). Privacy-preserving logistic regression training with a faster gradient variant. arXiv preprint arXiv:2201.10838

work page internal anchor Pith review Pith/arXiv arXiv
[4]

Chiang, J. (2022b). Quadratic gradient: A unified framework bridging gradient descent and newton-type methods by synthesizing hessians and gradients.arXiv preprint arXiv:2209.03282

work page arXiv
[5]

Kingma, D. P. and Ba, J. (2014). Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980. 14

work page internal anchor Pith review arXiv 2014

[1] [1]

and Lindsay, B

Böhning, D. and Lindsay, B. G. (1988). Monotonicity of quadratic-approximation algorithms. Annals of the Institute of Statistical Mathematics, 40(4):641–663

work page 1988

[2] [2]

and Vercauteren, F

Bonte, C. and Vercauteren, F. (2018). Privacy-preserving logistic regression training.BMC medical genomics, 11(4):13–21

work page 2018

[3] [3]

Chiang, J. (2022a). Privacy-preserving logistic regression training with a faster gradient variant. arXiv preprint arXiv:2201.10838

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

Chiang, J. (2022b). Quadratic gradient: A unified framework bridging gradient descent and newton-type methods by synthesizing hessians and gradients.arXiv preprint arXiv:2209.03282

work page arXiv

[5] [5]

Kingma, D. P. and Ba, J. (2014). Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980. 14

work page internal anchor Pith review arXiv 2014