Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimization

Fran\c{c}ois Glineur; Pierre Vernimmen

REVIEW 5 major objections 4 minor 1 cited by

Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimization

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Dividing every step size by 1/(1+δ) makes long-step and accelerated gradient methods robust to relative gradient error.

desk verdict Useful, honest paper on relative inexactness; the shortening factor is plausible and partly supported by PEP curves, but the headline claim that FGM is robust rests on one specific corruption model and needs stronger empirical support. read the letter →

arxiv 2506.09730 v3 pith:FAJLNDFK submitted 2025-06-11 math.OC cs.LG

Pierre Vernimmen , François Glineur This is my paper · ORCID

classification math.OCcs.LG MSC 90C2590C22

keywords performanceestimationrelativeinexactnesscompressedgradientdescentlong-stepmethodssilverstepsizeacceleratedmethodsmoothconvexoptimizationrobustness

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 asks whether first-order optimization methods keep their convergence guarantees when gradient computations are only relatively accurate, meaning the error in each gradient is bounded by a fraction δ of the gradient norm. Using worst-case performance estimation, it shows that long-step schedules (dynamic and Silver) and the accelerated fast gradient method lose robustness quickly as δ grows, sometimes producing a worst-case rate worse than taking no iteration at all. The paper then proposes a semi-heuristic repair: divide every step size by the shortening factor 1/(1+δ), a value motivated by a divergence threshold for constant-step gradient descent. In worst-case rates this recovers useful convergence, and in experiments on a logistic regression problem the shortened schedules, especially Silver steps, become competitive again, while FGM proves far more robust empirically than the worst-case theory predicts.

What carries the argument

The carrying mechanism is the performance estimation problem (PEP), a reformulation of worst-case analysis as a semidefinite program that computes tight convergence rates for a given algorithm and function class; the relative inexactness condition ‖d_k−∇f(x_k)‖≤δ‖∇f(x_k)‖ is added as an extra constraint. The second key object is the shortening factor 1/(1+δ), which rescales every step size of a schedule. It is motivated by the observation that constant-step gradient descent with step h>2/(1+δ) diverges on the one-dimensional quadratic f(x)=Lx²/2, while the standard proof of convergence on smooth convex functions survives as long as h≤2/(1+δ). Applying this uniform rescaling to the dynamic, Silver, and FGM schedules is the semi-heuristic repair whose worst-case and empirical effects the paper measures.

What would settle it

Run the shortened Silver and dynamic schedules on a different smooth convex problem with an adversarial relative error at δ=0.5 and compare accuracy against the unshortened versions; if shortening no longer improves accuracy, or if PEP worst-case rates for shortened FGM exceed those of the unshortened method, the uniform-repair claim loses support.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the factor 1/(1+δ) acts as a uniform repair for step-size schedules under relative gradient inexactness, and that accelerated methods are much more robust in practice than their worst-case guarantees suggest. The performance-estimation analysis shows that without shortening, Silver and dynamic schedules degrade sharply as δ increases, and even FGM's rate worsens; with every step size divided by 1+δ, worst-case rates improve substantially. In the compressed-gradient experiment, no method is significantly harmed by mantissa truncation, and the exact-case ordering of methods persists. In the adversarial-corruption experiment, FGM keeps high accuracy even at large δ, and shortened Silver steps rival FGM for δ≤0.3, while the shortening factor slightly hurts constant-step and FGM. The paper concludes with a robustness ranking that recommends FGM, and shortened Silver steps when a memoryless method is wanted.

Load-bearing premise

The shortening factor is derived from a divergence threshold for constant-step gradient descent, and the paper assumes that dividing every step size of the Silver, dynamic, and FGM schedules by 1+δ preserves or restores their convergence; this is validated experimentally only on one logistic-regression dataset with six starting points.

Editorial extensions

If this is right

Shortening by 1/(1+δ) turns the theoretically fragile long-step schedules into methods with usable worst-case rates, making them candidates for memory-limited inexact settings.
Compressed gradient descent with mantissas truncated to a few bits can reduce gradient storage by more than a factor of three without harming empirical convergence, since the induced relative error stays below the robustness threshold.
The practical ordering FGM beats Silver beats dynamic and constant-step (matching $O(1/N^2)$, $O(1/N^{1.27})$, and $O(1/N)$) persists under compression, so method choice in inexact settings can follow the exact-case ranking.
Under adversarial corruption the paper recommends FGM as the default, with shortened Silver steps as the memoryless alternative.
Worst-case analysis, although pessimistic for accelerated methods, retains value because it is what suggested the shortening factor that later proved useful empirically.

Reading between the lines

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

If the shortening heuristic transfers beyond logistic regression, the same 1/(1+δ) rescaling could be applied to other fixed-schedule methods, such as restarted or adaptive variants, with similar robustness gains; this is a testable extension the paper does not perform.
The large empirical gap between FGM's observed robustness and its worst-case rate suggests the relative-inexactness model is pessimistic for realistic compression, pointing toward stochastic or average-case analyses of compressed gradients.
A natural next experiment is to measure the actual δ induced by mantissa truncation in neural-network training; the paper's bound δ=(1/2)^{nbit+1} likely overstates typical error, implying that even larger memory savings could be safe.
The stability threshold h≤2/(1+δ) offers a rule of thumb for any gradient method: keep effective step sizes below the inexactness-adjusted limit, which may also explain why some adaptive methods tolerate low-precision gradients.

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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

The paper is worth reading. It gives the first PEP-based worst-case curves I know of for long-step and accelerated methods under relative gradient inexactness, and it introduces a simple shortening factor, 1/(1+δ), that makes the worst-case rates substantially better for Silver, dynamic, and constant-step schedules. The motivation from the divergence threshold h > 2/(1+δ) on a quadratic is clean and independently checkable. The authors are honest that the factor is semi-heuristic, and the GitHub code makes the numerics reproducible in principle. The compression experiments are also useful: they show that mantissa truncation barely hurts any method, which is a practically relevant observation for large-scale GPU training.

The main soft spot is the empirical claim that FGM is "much more robust than expected." In the adversarial corruption experiment, the error is chosen to point away from x*, which is a valid perturbation but not chosen adversarially with respect to FGM. For an accelerated method that overshoots, that direction can be benign. The paper's own PEP worst-case curves in Figure 2a show FGM degrading substantially for δ ≥ 0.4, so the empirical robustness is a statement about a specific attack, not about worst-case behavior. The authors do phrase it as "this kind of adversarial error," which limits the overclaim, but the practical recommendation to prefer FGM leans on this experiment. The shortening-factor recommendation is on firmer ground because Figure 2b provides independent PEP evidence for long-step methods; the same cannot be said for FGM.

The empirical section is otherwise thin in the usual ways: one presented dataset, six initial points, no error bars, and a second dataset mentioned only in parentheses. That is acceptable for a first look but not enough to support strong practical preferences. The compression results even show compression sometimes helping, which the authors attribute to truncation effectively shrinking step sizes; that is plausible but worth a more careful test.

Overall, the central idea is sound, the writing is clear, and the weaknesses are in the empirical support rather than in the logic. This deserves a serious referee and probably a conditional accept after the authors either strengthen the FGM robustness evidence or soften the recommendation. I would cite this paper for the shortening factor and the PEP methodology application, and I would bring it to a reading group as a good example of computer-aided analysis done honestly.

Send it to peer review.

Referee Report

5 major / 4 minor

Summary. This paper studies the behavior of four first-order optimization methods—constant step-size gradient descent, dynamic step sizes, Silver step sizes, and Nesterov's fast gradient method (FGM)—when gradients are subject to relative inexactness of the form ||d_k - ∇f(x_k)|| ≤ δ||∇f(x_k)||. Using the Performance Estimation Problem (PEP) methodology, the authors numerically compute worst-case convergence rates in squared gradient norm for N = 50 iterations and find that long-step and accelerated methods degrade as δ grows. They then propose a 'shortening factor' 1/(1+δ), motivated by a divergence threshold for constant-step gradient descent on a quadratic, and show via PEP that shortened schedules have substantially better worst-case rates. The empirical part considers a logistic regression problem with two inexactness models: compressed mantissa gradients and an 'adversarial' corruption that adds the longest perturbation pointing away from the optimizer. The reported experiments indicate that compression is largely harmless, that FGM is surprisingly robust under the corruption model, and that shortening helps the long-step methods in that model. The paper recommends FGM, or shortened Silver steps when a memoryless method is needed.

Significance. If the claims are validated, the paper offers practically relevant guidance for using low-precision or compressed gradients on GPUs, and it demonstrates how PEP can be used to diagnose robustness issues beyond standard convergence-rate analysis. The authors release their code, which is a genuine strength, and the shortening-factor idea is natural and potentially useful. However, the central empirical claims are currently supported by a narrow experimental base—one dataset shown, six initial points, no error bars—and by a corruption model that is adversarial only in a limited sense. The PEP results are numerical SDP outcomes rather than certified proofs. The paper's contribution is therefore promising but not yet fully established; the gap between the strength of the recommendations and the evidence is the main issue.

major comments (5)

[Section 3.2, Figures 2a and 4] The headline empirical finding that FGM is 'much more robust than expected' and the recommendation to prefer FGM rest entirely on the corruption model that adds the longest feasible perturbation pointing away from x*. That perturbation is one specific point on the boundary of the relative-inexactness ball and is not chosen adversarially with respect to FGM; for a method whose extrapolated point y_k can overshoot x*, this direction may be comparatively benign. The PEP worst-case rates in Figure 2a, which are adversarial over both the function and the error sequence, show FGM degrading substantially for δ ≥ 0.4. To make the robustness claim load-bearing, the authors should add corruption directions that are adversarial for each method (for example, aligned with the momentum or extrapolation direction) or report the worst performance over randomized boundary perturbations.
[Section 2, shortening factor paragraph] The shortening factor 1/(1+δ) is introduced from a divergence threshold h > 2/(1+δ) for constant-step gradient descent on the quadratic f(x)=Lx^2/2, citing references [15,16]. Its extension to the Silver, dynamic, and FGM schedules is an assumption rather than a derived result. The PEP rates in Figure 2b provide numerical support for worst-case rates, but the practical recommendation that shortened Silver is a robust memoryless method is validated only on one logistic regression dataset, with only a mention of a phishing dataset and no shown results. The authors should either provide additional datasets, give a theoretical justification for why the factor transfers to these schedules, or explicitly restrict the claim to the tested setting.
[Abstract and Sections 3.1–3.2] The abstract states that 'the shortening factor significantly helps the long-step methods,' but this is observed only in the adversarial corruption experiment of Section 3.2. In the compressed-gradient experiment of Section 3.1, the text explicitly says that shortened schedules 'do not seem to make much difference' (page 9). The conclusions should distinguish between the two inexactness models, because the current wording overgeneralizes the evidence.
[Section 2, Figures 2a–2b] The paper refers to the PEP results as 'exact convergence rates' and states that long-step and accelerated methods are 'shown to be theoretically not robust.' These rates are outputs of a semidefinite program solved numerically with MOSEK, and no solver tolerance or certificate verification is reported. The authors should state the numerical nature of these results and either verify them with a rigorous post-processing step or soften the wording from 'shown' to something like 'numerically observed.'
[Section 3, Figures 3 and 4] All empirical curves are means over six initial points on a single dataset, with no error bars, no per-seed results, and no statistical tests. The claims that compression 'does not seem to cause any significant harm' and that shortening 'greatly improves' performance are quantitative; without variability information, the reader cannot assess how robust these conclusions are to initialization or to the choice of dataset. The authors should report error bars or additional dataset results before drawing practical recommendations.

minor comments (4)

[Figures 3 and 4] The axis labels in the submitted figures are visually corrupted (for example, 'Mean of min k ∈[N] --∇(f(x_k)--' and 'Mean of be() )rainin accuracie('). The figures should be regenerated with proper mathematical notation.
[Section 3] The smoothness constant L used for normalized step sizes is estimated numerically, whereas the theoretical PEP analysis assumes L is known exactly. The paper should state clearly that the experiments use an estimate, and discuss how errors in this estimate interact with the relative inexactness level δ.
[Table 1] The qualitative ratings (✗, ∼, ✓) in Table 1 are presented without explicit thresholds or a precise definition of when a method is considered 'bad,' 'intermediate,' or 'good.' A short criterion would make the table more informative.
[Section 2] The derivation of the divergence threshold h > 2/(1+δ) referenced to [15,16] is not reproduced. Since this threshold is the motivation for the central shortening factor, a brief proof sketch in an appendix would improve the self-containedness of the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PEP rates are exact reformulations, the shortening factor is an explicitly heuristic prior step, and experiments do not feed back into the analysis.

full rationale

The paper's theoretical contribution is the computation of exact worst-case convergence rates via Performance Estimation, by solving the stated PEP with the relative inexactness constraint (2) and fixed step-size schedules. These rates are outputs of semidefinite programs, not fitted to the empirical results. The shortening factor 1/(1+δ) is derived from a parameter-free divergence threshold for constant-step gradient descent on f(x)=L x^2/2 and from a proof-structure preservation observation, both cited to the authors' prior work [15,16]; this is a self-citation, but it is not load-bearing in a circular sense because the cited facts are independently checkable mathematical statements whose assumptions do not include the target result, and the factor is explicitly labelled 'semi-heuristic' before being evaluated on new schedules by PEP and on a separate logistic-regression experiment. The empirical claims about compression and adversarial corruption are observations under specified perturbation models, not predictions generated by fitting parameters to those same observations; no fitted input is renamed as a prediction. The concern that the adversarial corruption model is only one non-algorithm-specific direction is an external-validity caveat, not a circularity. Accordingly, the derivation chain is self-contained and no step reduces by construction to its own input.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the relative inexactness model (Assumption 2), the PEP interpolation framework, the choice of the shortening factor 1/(1+δ), and the numerically estimated Lipschitz constant used to normalize steps in the experiments. No invented entities are introduced; no free parameters are fitted to the empirical results, though the shortening factor is a hand-chosen heuristic.

free parameters (1)

shortening factor 1/(1+δ) = 1/(1+δ) for δ∈[0,1)
Hand-chosen heuristic derived from the divergence threshold h≤2/(1+δ) for constant-step gradient descent; applied uniformly to all methods in Section 2 (Figure 2b) and experiments.

assumptions (4)

domain assumption Relative inexactness model: ‖d_k−∇f(x_k)‖≤δ‖∇f(x_k)‖ with δ∈[0,1)
Defines the class of inexact oracles studied; appears as condition (2) in Section 1.1.
standard math Smooth convex interpolation conditions from [13] characterize the function class for the PEP
Used in Section 1.4 to reformulate (PEP) as an SDP; these are known results, not proved here.
domain assumption Estimated Lipschitz constant L_N from the preliminary run is a valid upper bound for the iterates of the inexact methods
Section 3 states 'the value of L_N is likely a good approximation'; the step-size normalizations and robustness conclusions depend on it.
ad hoc to paper The shortening factor preserves convergence behavior of silver/dynamic/FGM
Section 2 states the schedules are 'shortened schedules (divided by 1+δ)' and calls the factor semi-heuristic; no proof is given that it preserves the original convergence rates.

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Cite this review

Pith. "Pith review of Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimization." pith.science (2026). https://pith.science/paper/FAJLNDFK

@misc{pith2026250609730,
  author       = {Pith},
  title        = {Pith review of: Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FAJLNDFK}},
  note         = {Machine review of arXiv:2506.09730}
}

read the original abstract

This work assesses both empirically and theoretically, using the performance estimation methodology, how robust different first-order optimization methods are when subject to relative inexactness in their gradient computations. Relative inexactness occurs, for example, when compressing the gradient using fewer bits of information, which happens when dealing with large-scale problems on GPUs. Three major families of methods are analyzed: constant step gradient descent, long-step methods, and accelerated methods. The latter two are first shown to be theoretically not robust to inexactness. Then, a semi-heuristic shortening factor is introduced to improve their theoretical guarantees. All methods are subsequently tested on a concrete inexact problem, with two different types of relative inexactness, and it is observed that both accelerated methods are much more robust than expected, and that the shortening factor significantly helps the long-step methods. In the end, all shortened methods appear to be promising, even in this inexact setting.

Figures

Figures reproduced from arXiv: 2506.09730 by the authors.

**Figure 2.** Exact convergence rate versus inexactness level [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Empirical results of the compressed gradient descen [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Empirical results of gradient descent with corrupti [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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

Worst-case convergence analysis of relatively inexact gradient descent on smooth convex functions
math.OC 2025-06 conditional novelty 6.0 of 10

For smooth convex functions, relatively inexact gradient descent has a three-regime worst-case one-step rate, and the largest provably safe stepsize is 2/(1+delta).

Reference graph

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