Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$

Dawei Li; Mingyi Hong; Xiaotian Jiang

REVIEW 2 major objections 3 minor 1 cited by

Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read BB1 gradient method provably loses superlinear convergence on an open, positive-measure family of quadratics in every dimension n≥4, via a certified attracting seven-cycle.

desk verdict A real answer to a long-open BB question, but the entire proof leans on a single interval-arithmetic certificate that, as printed, is not independently runnable. read the letter →

arxiv 2607.21579 v1 pith:ESY6GGZ4 submitted 2026-07-23 math.OC cs.AIcs.LG

Dawei Li , Xiaotian Jiang , Mingyi Hong This is my paper · ORCID

classification math.OCcs.AIcs.LG MSC 65K0590C3037N4037C70

keywords Barzilai-BorweinBB1superlinearconvergencestrictlyconvexquadraticsperiodicorbitcomputer-assistedproofintervalarithmeticpositivemeasure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to settle a long-standing question: does the Barzilai–Borwein (BB1) gradient method converge superlinearly for almost every strictly convex quadratic problem and initialization? The authors answer no by constructing, for every dimension n≥4, a nonempty open (hence positive-Lebesgue-measure) set of quadratics and initial points on which BB1 converges but only at a guaranteed linear rate, with gradient components bounded between 10^{-6k} and 0.61^k times their initial values. The slow behavior is not a fragile exception: it persists under perturbations of both the Hessian spectrum and the starting point. The proof identifies a concrete dynamical mechanism—a nonresonant, attracting period-seven orbit of the projectivized BB dynamics in dimension four, validated by a computer-assisted interval-arithmetic certificate—and then uses stability, translation, and eigenvalue-clustering arguments to lift the example to every finite dimension.

What carries the argument

The engine of the proof is a certified nonresonant attracting seven-cycle of the projectivized BB1 dynamics in dimension four. The projective state (a_k, p_k), where a_k is the reciprocal step size and p_k the normalized squared spectral weights, evolves under a rational map F_λ; the authors find an exact root of eight rational equations—a seven-cycle F^7(z)=z together with a fifteen-step landing F^{15}(D(r))=z from admissible initial data—by interval Newton/Banach contraction inside a 10^{-55} box. A Lyapunov pair P≻0, P−A_7^T P A_7≻0 proves the cycle is Schur stable and hence attracting in an open basin; a translation by four makes the spectral width small enough to force the 0.61 upper ra

What would settle it

Re-run the supplied interval_cert.py script in arbitrary-precision interval arithmetic and check that the printed Krawczyk containment, contraction bound <7.200691e-27, positive pivots of P, positive diagonal-dominance margins of Q_7, and all separation and denominator certificates reproduce exactly; a single violated assertion would falsify the certified root and with it the theorem. Independently, a long high-precision BB1 simulation on the central translated spectrum with the described initial weights should show the four gradient components decaying in parallel between the two geometric en

Watch

Extended reading notes

Core claim

The central claim, Theorem 2.1, is that for every n≥4 and every b∈R^n there exist nonempty open sets A_n⊂S^n_{++} of simple-spectrum strictly convex quadratics and open sets X_n(A) of initial points such that every admissibly initialized BB1 trajectory satisfies the two-sided geometric bound 10^{-6k}∥Π_i(A)g_0∥ ≤ ∥Π_i(A)g_k∥ ≤ 0.61^k∥Π_i(A)g_0∥ for each spectral component i and all k. Consequently the gradient norm, energy norm of the error, and objective gap all decay at a rate bounded below by a positive geometric sequence, giving liminf ∥g_k∥^{1/k} ≥ 10^{-6} > 0. This rules out root-superlinear and Q-superlinear convergence and refutes any almost-everywhere superlinearity statement for BB

Load-bearing premise

The whole theorem rests on the computer-assisted certificate (Certificate 5.1): the interval-arithmetic script must genuinely prove that the exact root exists, that the cycle is attracting (P≻0 and P−A_7^T P A_7≻0), and that the separation bounds hold; any error in the hard-coded rational preconditioner, Lyapunov matrix, or arithmetic evaluation would collapse the open-family construction.

Editorial extensions

If this is right

If correct, the longstanding conjecture of almost-everywhere superlinear convergence for BB1 on strictly convex quadratics is false in every dimension n≥4; the slow behavior is open and positive-measure, not a null-set exception.
Linear-rate behavior for BB1 is caused by a stable recurrent attractor (a period-seven orbit), not merely by invariance of coordinate subspaces; four spectral modes remain active, producing a seven-periodic normalized spectral measure rather than collapse to two endpoint modes.
The construction answers the specific open question raised by prior R-linear upper bound work: generic initialization does not automatically accelerate BB1, even on an open set of spectra.
The rate constants 10^{-6} and 0.61 are uniform across all finite dimensions n≥4, even though the size of the open neighborhood is dimension-dependent, so the obstruction to superlinearity is authentically n-dimensional.
The method of proof—computer-assisted certification of a periodic orbit plus analytic continuation—offers a template for proving robust lower bounds on convergence rates in other history-dependent gradient methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the construction is open in joint problem-data space (A,x_0), a natural testable extension would be to check whether small random perturbations of the certified spectrum and initial point, in simulations of BB1 on n=4 quadratics, actually display the predicted geometric decay with the certified seven-cycle as an attractor.
The same projective-attractor mechanism may explain observed slow phases of BB-type steps in nonquadratic problems and in spectral/safeguarded variants: wherever the delayed Rayleigh quotient locks onto a resonant cycle of step sizes, superlinear escape is precluded.
The gap between the certified lower rate 10^{-6} and the observed asymptotic factor β^{1/7}≈0.144 along the cycle suggests the true obstruction may be stronger than the conservative theorem states; sharper constants might be obtained by optimizing the Lyapunov certificate.
The translation trick reveals how the lower rate emerges from separation (nonresonance) while the upper rate emerges from spectral width; this split may transfer to other spectral gradient rules whose projective dynamics is translation-invariant.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper answers a question that has been open for a while: for every n≥4, there is a nonempty open set of strictly convex quadratics and initializations where BB1 converges only linearly, with each spectral component between 10^-6 and 0.61 per step, so superlinear convergence fails on a positive-measure family. That is genuinely new — Li and Sun's lower-rate example was Lebesgue-null, and Dai and Fletcher's dimension-four transition was asymptotic and numerical. If correct, it settles the almost-everywhere superlinearity question negatively on an open set.

The analytic machinery is well built. The reduction of BB1 to a projective rational map, the certified seven-cycle via interval Newton, attraction via a Lyapunov inequality, continuation by the implicit-function theorem, embedding into higher dimensions with transverse Floquet bounds, and the lift to original problem data are all standard moves, and I could not find a gap in the pen-and-paper parts. The paper is also appropriately scoped and transparent about its constants and limitations.

The soft spot is exactly the certificate. Theorem 2.1's existence, stability, and openness all rest on Certificate 5.1's strict inequalities, which come from interval_cert.py in Appendix A. As printed, that script is not runnable: the long string constants RSTR and PSTR are broken across lines with inserted arrow and guillemet artifacts, so the exact rational entries of the preconditioner and Lyapunov matrix are not unambiguously recoverable. No independent run is reported, and there is no formal verification. A single mistyped digit in R or P would destroy the contraction or the positive-definiteness and collapse the proof. That is a mechanical red flag, not a detected mathematical error, but it is load-bearing. The paper's disclosure that a language model produced the initial sketch is honest and beside the point; it just makes a clean, versioned certificate more necessary.

Minor notes: the rate constants are conservative and the result is an existence theorem, both stated clearly. The reference list is appropriate.

Who should read it: optimization researchers interested in BB methods or in computer-assisted proofs of dynamical systems. It deserves a serious referee, but only if the certificate is supplied as a clean, runnable, versioned artifact and ideally independently checked. I wouldn't cite it in my own work until that happens, and I'd suggest the authors provide the script and outputs as supplementary material. Engage with the work — this is the strongest lower bound on BB1's generic behavior I've seen — but engage with the certificate first.

Referee Report

2 major / 3 minor

Summary. The paper proves that for every dimension n≥4 and every linear term b, there exists a nonempty open set of strictly convex quadratic problems and initial points on which the Barzilai–Borwein method BB1 converges, but every spectral component of the gradient decays at a two-sided geometric rate with explicit constants 10^{-6} and 0.61. Consequently liminf ||g_k||^{1/k} ≥ 10^{-6}, ruling out root-superlinear and Q-superlinear convergence on a positive-measure family. The proof reduces BB1 on quadratics to a projective rational map, certifies by interval arithmetic an attracting seven-cycle with an exact fifteen-step admissible connection in dimension four, proves Schur stability via a Lyapunov inequality, translates the spectrum to obtain the rate constants, continues the construction by the implicit-function theorem, embeds it into every n>4 via transverse Floquet bounds, and finally lifts the projective family to problem data by changes of variables. Theorem 2.1 states the result; Sections 4–9 supply the proof.

Significance. If the computer-assisted certificate is correct, this is a substantial negative answer to the almost-everywhere superlinearity question for BB1 on strictly convex quadratics. The paper identifies a robust dynamical mechanism — an attracting nonresonant seven-cycle — rather than a null-measure exceptional example, and it gives explicit, if conservative, rate constants. The analytic parts of the proof are careful and mostly standard: the reduction to the projective map, the translation conjugacy, the IFT continuation, and the Floquet embedding into higher dimensions are coherent and appear sound. A particular strength is the honest statement of limitations and the explicit acknowledgement that the construction is an existence result, not a universal rate formula. The main weakness is entirely at the computer-assisted certificate: the supplied script is the sole evidence for the strict inequalities that feed every later implication, and as printed it is not reproducible. The paper also discloses AI-assisted proof development, which is not a scientific defect but increases the need for machine-checkable artifacts.

major comments (2)

[Appendix A, interval_cert.py; Certificate 5.1(C1)–(C7)] The central claim of Theorem 2.1 is conditional on the interval certificate in §5.2. As printed, the script is not executable: the triple-quoted strings RSTR and PSTR contain line-break artifacts ('↪→' and '⌋') and split numeric literals across lines, so the intended exact entries of the preconditioner R and Lyapunov matrix P cannot be recovered unambiguously. No output transcript is included. A single digit error in R could destroy the contraction bound (38), and an error in P could destroy P≻0 or P−A7ᵀPA7≻0. The certificate therefore currently functions as an unverifiable black box. Please supply a clean ancillary source file and the full output of the script (Krawczyk ratios, contraction bound, LDLT pivots, Gershgorin margins, separation values, and the transverse τ interval).
[§5.2, (C5)–(C7); §6, Lemma 6.2] The rigorous existence and stability of the seven-cycle depend on strict inequalities that are asserted from the script but not independently reproduced. In particular, P≻0 and P−A7ᵀPA7≻0 are certified by interval LDLT pivots and Gershgorin margins, and the transverse Floquet bound (45) is certified by one interval computation, but the paper does not report the script's actual printed values for these quantities. Because every downstream conclusion — the open basin, the rate bounds, and the n>4 embedding — consumes these inequalities as black boxes, the authors should either include a full verification transcript or an independent cross-check (e.g., interval arithmetic in another language or a formal proof assistant) before the theorem can be considered established.

minor comments (3)

[Abstract; Eq. (6)] The two rate constants are both rendered as ρ in several places; in the main text they are denoted with under/overbars. Please ensure the final PDF clearly distinguishes \underline{\rho}=10^{-6} from \overline{\rho}=0.61.
[Figs. 1 and 3] The caption of Fig. 1 states that the transient minimum of the one-step multiplier is approximately 1.7×10^{-5}, while Certificate 5.1(C4) reports a separation lower bound of 1.388×10^{-4}. This is not contradictory if Fig. 1 refers to the shifted spectrum and to the ratio |a_k−λ_i|/a_k, but the captions should say so explicitly to avoid apparent inconsistency.
[§1; Appendix B] The statement that Li and Sun's 'generic initialization' question is 'answered in the negative on an open set of matrices' could be misread as disproving almost-everywhere fast convergence for a fixed matrix. The theorem actually gives a positive-measure set of joint (A,x0) pairs. Please phrase this as 'on a positive-measure set of joint problem-data pairs' to match the theorem. The LLM prompt in Appendix B is long and not part of the mathematical proof; consider moving it to supplementary material or an acknowledgments note.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rate bounds are derived from a certified periodic orbit and separation inequalities, not assumed or fitted.

full rationale

The derivation chain is self-contained. §4 reduces BB1 on a quadratic to the exact projective multiplier identity d^i_{k+1}=d^i_k(a_k−λ_i)/a_k, so the desired two-sided geometric bound (6) follows as soon as one proves 10^{-6} ≤ |a_k−λ_i|/a_k ≤ 0.61 for all i,k. The paper proves this by (a) solving, via interval Newton, eight rational equations G(u*)=0 that express exact seven-cycle periodicity F^7(z*)=z* and an exact fifteen-step admissible landing F^15(D(r*))=z* (Lemma 5.2, Certificate 5.1); (b) proving Schur stability of A_7 from the certified Lyapunov inequality P−A_7^T P A_7≻0 (Lemmas 6.1–6.3); and (c) deriving the numerical constants from translation by 4 and certified separation: λ4−λ1<3.02, λ1>4.99, and |a_k−λ_i|>10^{-5}, giving upper 3.02/4.99<0.61 and lower 10^{-5}/8.01>10^{-6}. The constants 10^{-6} and 0.61 are not inputs to the root-finding or to the construction; they are consequences of the certified strict inequalities and of free, explicitly stated spectral choices. The open sets A_n and X_n(A) are obtained by continuity and the implicit-function theorem from the certified cycle, not defined as the set where (6) is assumed. References to Li–Sun and other prior work are contextual and are not used to establish the cycle, stability, or rates. The one significant risk — that interval_cert.py as printed contains line-break artifacts and is not independently run — is a verification/reproducibility concern about Certificate 5.1, not a circularity: even if the certificate failed, the failure would be a missing or wrong proof component, not an equation equivalent to its own conclusion. No step in the paper reduces a claimed prediction to a fitted parameter, a renamed empirical pattern, or a load-bearing self-citation. Under the proportionality rule, the correct circularity score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central proof is an existence construction: its constants are hand-chosen or search-found to realize a seven-cycle and a fifteen-step connection, not empirical fits or physical inputs. The only nonstandard assumptions are correctness of the interval certificate and of the hard-coded rational matrices; these are verifiable in principle but not independently verified here. There are no invented physical entities, and no parameter was fitted to the target rate inequalities.

free parameters (5)

central spectrum λ = (1, λ2, λ3*, 4) = λ2=1.8786699041860466; λ3*≈2.7127814910653372
Hand-chosen, with search assistance, so that the projectivized BB map has a nonresonant seven-cycle; the theorem's existence proof uses this spectrum as the center of A_n.
admissible initial weights r* = c6≈1.34e-5, c7≈0.98917, c8≈3.49e-6, r4≈0.01081
Chosen to land exactly on the seven-cycle after fifteen steps; this is the center of the open initial-weight set.
interval-Newton preconditioner R = hard-coded 8×8 rational matrix in Appendix A
Constructed so H(u)=u−RG(u) is a contraction on the 1e-55 box; a search/design choice, not derived from general theory.
Lyapunov matrix P = hard-coded 4×4 rational matrix in Appendix A
Chosen to witness Schur stability of the seven-step derivative; not unique and not derived from the theorem.
extra-eigenvalue interval (7.49,7.51) for n>4 = central νℓ equally spaced in (7.49,7.51)
Chosen to lie in the transverse stability band certified by τ(ν)<1; this is part of the dimension-lifting construction.

assumptions (5)

standard math Interval arithmetic enclosure principle: directed rounding with 120-digit decimal precision produces valid real inclusions for every arithmetic expression over interval inputs.
Used to turn Appendix A script outputs into exact inequalities in Certificate 5.1 (Krawczyk 1969; Rump 2010).
standard math Banach fixed-point theorem in (R^8, ‖·‖∞).
Gives the exact root u* from H(X)⊂int X and the contraction bound ‖DH‖∞<1 in Lemma 5.2.
standard math Implicit function theorem for continuation of the seven-cycle under eigenvalue perturbations.
Used in Proposition 7.2 and Proposition 8.1 to turn one certified cycle into open parameter neighborhoods.
standard math Weyl eigenvalue continuity and the spectral projector formula Π_i(A)=∏_{j≠i}(A−λ_j I)/(λ_i−λ_j) on simple-spectrum matrices.
Used in §9 to lift projective weight neighborhoods to open matrix neighborhoods and then to initial points.
domain assumption The supplied Python script's hard-coded rational constants R and P are exactly as intended, and the printed transcription is faithful.
The certificate is load-bearing; a typo or transcription error in these constants would break the proof. No independent checksum or repository is provided.

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Cite this review

Pith. "Pith review of Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$." pith.science (2026). https://pith.science/paper/ESY6GGZ4

@misc{pith2026260721579,
  author       = {Pith},
  title        = {Pith review of: Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ESY6GGZ4}},
  note         = {Machine review of arXiv:2607.21579}
}

abstract

Barzilai--Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai--Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $\rho_{\min}=10^{-6},\rho_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.

Figures

Figures reproduced from arXiv: 2607.21579 by the authors.

**Figure 2.** Map of the proof of Theorem 2.1. The shaded box is the computer-assisted step (Certificate 5.1); every other stage is a pen-and-paper argument that consumes its certified strict inequalities. Arrow labels name the tool carrying each implication. Fix an orthogonal diagonalization A = QΛQ T , Λ = diag(λ1, . . . , λn), 0 < λ1 < · · · < λn, (16) and define the spectral gradient coordinates dk := Q T gk, di k = (Q T gk)i… view at source ↗

**Figure 3.** The certified projective orbit at the central parameter. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Sharp Worst-Case Asymptotic Rate of the Barzilai--Borwein Method in $\mathbb R^d$ and Hilbert Spaces
math.NA 2026-08 accept novelty 8.0 of 10

For either Barzilai-Borwein rule, the worst asymptotic gradient root factor on strongly convex quadratics is exactly (κ(H)-1)/(κ(H)+1), and the same constant governs local nonlinear convergence under strict differentiability.

Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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