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REVIEW 3 major objections 5 minor 17 references

The paper claims that the top-r gradient subspace used by GaLore-type optimizers is not a trackable object: two same-step minibatch estimates disagree as much as estimates taken a refresh apart, so the apparent rotation is estimator noise.

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2026-08-02 08:21 UTC pith:GTD5WN2P

load-bearing objection Clever same-step probe and a useful k* diagnostic, but the headline 0.725 vs 0.742 comparison likely mismatches batch sizes; load-bearing gap. the 3 major comments →

arxiv 2607.05872 v2 pith:GTD5WN2P submitted 2026-07-07 cs.LG math.OCstat.ML

No Subspace to Track: Non-Identifiability and Optimizer State in Low-Rank Training

Noel Thomas This is my paper
classification cs.LG math.OCstat.ML
keywords gradient subspace non-identifiabilityGaLorelow-rank trainingAdam second momentchordal distancereproducible rankoptimizer memoryspectral tail scaling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tests the premise behind memory-saving optimizers like GaLore: that the top-r gradient subspace they recompute every T steps is a slowly drifting object worth tracking. It finds that premise false. At the same training step, two estimates of the top-128 subspace from disjoint minibatches disagree almost exactly as much as estimates taken a full refresh interval apart (0.725 versus 0.742 of the maximum chordal distance), so the refresh-to-refresh rotation is dominated by sampling noise rather than motion. Only about 39 of 128 directions survive resampling, and N-fold gradient averaging does not open the spectral gap that would make the subspace identifiable: the deep spectral tail shrinks as N^(-1/4), not the N^(-1/2) of pure noise. The paper turns this into a concrete prescription: do not average the basis, transport Adam's state through each refresh's rotation, and shorten the second-moment memory to beta2=0.99, with controlled perplexity gains at 1B parameters.

Core claim

Beyond a reproducible core of about 39 of 128 directions, the gradient's top-r subspace is statistically non-identifiable at the ranks GaLore-family optimizers use. The paper demonstrates this with a same-step control: two disjoint split-batch estimates disagree as much as two time-separated estimates. It explains the cause as a gapless gradient singular spectrum with a power-law tail that averaging shrinks only as N^(-1/4), so no finite averaging budget makes the top-r frame well defined. The practical consequence is that Adam's second moment, carried blindly across a refresh, is provably about (r-k*)/2 worse than the best rotation-blind estimator, while first-moment transport through the r

What carries the argument

The load-bearing tool is the sign-corrected chordal distance between orthonormal frames, sqrt(2r - 2*sum cos(theta_i)), which removes the per-column sign artifact and saturates toward the geometric maximum sqrt(2r) when frames are nearly orthogonal. Applied as a same-step split-batch probe, it separates estimator noise from true motion; combined with the gradient singular spectrum and N-fold averaging scaling, it yields the reproducible rank k*. The theoretical cost analysis uses a uniform random rotation model to bound the blind second-moment carry, plus an optimality argument showing that first-moment transport by the rotation matrix R = U_new^T U_old is the best linear map under isotropic

Load-bearing premise

The conclusion that no averaging budget can make the top-r subspace identifiable rests on extrapolating the measured N^(-1/4) spectral-tail decay from N up to 64 to all N and from finite samples to a permanently gapless population spectrum; if the tail is finite-sample noise that would decay as N^(-1/2) at larger N, a large enough budget could open a gap.

What would settle it

Take a pretrained Pythia-1B checkpoint, run the same split-batch probe with N=1024 or larger averaged minibatches instead of stopping at N=64, and measure the same-step chordal distance. If the disagreement starts decaying as N^(-1/2) beyond N=64 and the adjacent singular-value ratio at r=128 develops a clear gap, the no-averaging-budget claim fails. A second falsifier: if two same-step estimates from disjoint minibatches agree well below the across-time rotation (for example, below 0.5 sqrt(2r)) at some layer during training, the subspace is identifiable at that layer.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • At each refresh, the GaLore top-r frame is redrawn nearly orthogonal to the previous one; the apparent rotation is mostly estimator noise, not drift of a stable object.
  • No N-fold gradient averaging budget opens a spectral gap at rank r, because the deep tail is power-law signal shrinking as N^(-1/4), not random noise shrinking as N^(-1/2).
  • Averaging the basis over time does not help: at 1B parameters, even a 10,000-step covariance average leaves the per-refresh rotation at 84% of its geometric maximum and leaves final perplexity essentially flat.
  • Transporting Adam state through the rotation (first moment by R, second moment by the squared entries of R) and lowering beta2 toward 0.99 improves final perplexity: 18.73 versus 19.28 for the best untransported configuration at the default beta2.
  • Before trusting any low-rank assumption, measure the reproducible rank k* by splitting a batch and counting the directions that agree; if r exceeds k*, the excess directions are noise being refreshed.

Where Pith is reading between the lines

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

  • If the N^(-1/4) tail scaling extends beyond N=64, then any tracking algorithm relying on subspace persistence has an identifiability floor set by k*, suggesting incremental subspace trackers cannot recover more than k* directions from fresh SVDs.
  • A cheap pre-flight check for other optimizers and modalities would be the same split-batch probe: comparing r to k* predicts when state transport or a lower beta2 matters and when a random projection would suffice.
  • The paper's energy-capture account suggests a testable extension: a random projection that still includes the dominant spike should match exact-SVD GaLore whenever k* is much smaller than r, and should diverge as k* approaches r.
  • Because the full-rank control reverses the beta2 preference, the sign of the beta2 effect could serve as a diagnostic for whether an optimizer is implicitly refreshing a subspace, even when the refresh is hidden.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper measures the geometric premise of GaLore-family optimizers: that the top-r gradient subspace is a slowly drifting, trackable object. Using a same-step control at Pythia-160M, it reports that two top-r subspace estimates from disjoint minibatches at the same step disagree by 0.725·sqrt(2r), nearly equal to the 0.742·sqrt(2r) rotation observed across a full refresh interval, and concludes that the apparent rotation is dominated by estimator noise. It further reports that the gradient spectrum is gapless at r, that only k*≈39 of r=128 directions are reproducible, that N-fold averaging shrinks the deep spectral tail only as N^{-1/4}, and that this non-identifiability holds across scales and architectures. The paper then derives optimality statements for transporting Adam state across the refresh (Theorem 1, Proposition 3) and presents a 2×2 experiment at Pythia-1B comparing carry vs. transport and β2=0.999 vs. 0.99, plus a β2 sweep across optimizers and a full-rank control.

Significance. If the central same-step claim holds, this is an important and largely novel diagnosis of why GaLore-family optimizers work: not by tracking a persistent subspace but by capturing gradient energy in a re-drawn frame. The paper's strengths are genuine: the same-step split-batch control is a clean, time-removed measurement that is independent of the paper's own theory; Theorem 1 and Proposition 3 are parameter-free derivations with stated assumptions; the cross-architecture scaling (70M to 6.9B, plus ViT) is a valuable generality check; and the paper repeatedly and explicitly labels its own limitations, including the inability to separate transport from error feedback and the recipe-dependence of the β2 effect. The main empirical claims are falsifiable and the proposed k* probe is actionable. However, the headline conclusion depends on a matched-batch comparison that is not fully documented, and the empirical transport result is confounded by error feedback; these issues are load-bearing and require revision.

major comments (3)
  1. [§3.2, Appendix F] The central same-step versus across-time comparison is not fully controlled, because the batch size of the across-time rotation measurement is not reported. Appendix F specifies the same-step probe as B=8 sequences of length 512, N=1, but the 'across-time reference' 0.742·sqrt(2r) is only described as 'the sign-corrected rotation across one refresh interval of 160M training' with no batch size, sequence length, or averaging fold. If the refresh-subspace gradient during training is computed on a larger effective batch — for example the equivalent of N=64 averaging — then Table 4 implies the matched same-step floor at 160M is 0.725×0.76 ≈ 0.55·sqrt(2r), not 0.725. In that case the across-time rotation of 0.742 would exceed the estimator-noise floor by a large margin, indicating substantial genuine rotation. Because this comparison is the primary evidence for 'no subspace to track,' the man
  2. [§5.2, Table 1, §7 Limitations] The empirical claim that 'transporting the state works' is not isolated from error feedback. The transport row of Table 1 is the full LDAdam update, which combines change-of-basis transport of both moments with error feedback. The paper explicitly acknowledges this in §5.2 and in the Limitations, but the contribution list in §1 and the conclusion still present 'transport beats the best untransported configuration' as a supported empirical result. The 18.73 versus 19.28 gap could be driven entirely by LDAdam's error-feedback mechanism. Since the paper's own theory (Proposition 3, Theorem 1) makes a sharp prediction about transport alone, the experiment should include an ablation that transports the moments without error feedback, or the empirical claims in the abstract and conclusion should be explicitly scoped to 'the full LDAdam update (transport plus error feedback).'
  3. [§3.5, Abstract] The abstract's claim that 'no averaging budget makes the subspace well defined' is an extrapolation beyond the data. The N^{-1/4} scaling is measured for N ≤ 64 at 160M and confirmed only qualitatively at larger scales (Appendix F); the paper explicitly disclaims a closed-form law and reports that the spiked-covariance prediction fails out of sample. It is possible that the apparent power-law tail is finite-sample noise that would cross over to N^{-1/2} at larger N, which the reader's own 'weakest assumption' note identifies. The measured scaling supports the weaker claim that averaging does not rescue identifiability at any tested budget; the stronger 'no finite averaging budget' claim requires either a derived scaling argument or an explicit statement that it is an extrapolation. I recommend softening the abstract and §3.5 conclusion to match the evidence.
minor comments (5)
  1. [Appendix F] Please add the batch size, sequence length, and averaging fold (if any) for the across-time rotation measurement. This is needed not only for the scientific control but also for reproducibility.
  2. [Figure 3b] The axis label 'deep-tail s500 N' is unclear; it should define the rank index (e.g., singular value at index 500) and the normalization explicitly in the caption.
  3. [Table 4] The entries are ratios of N=64 to N=1 disagreement. This is fine, but the caption should also give the absolute N=1 baseline (or point to Table 5) so a reader can reconstruct the absolute floor at each scale.
  4. [§3.4] The sentence 'Both readings derive from the same measurement, so this is a consistency check between two estimators rather than an independent prediction' is honest and useful; consider moving it into the main text of §3.4 rather than the appendix-like discussion, as it preempts a natural circularity objection.
  5. [§5.3] The full-rank AdamW control is one seed. This is stated, but the figure caption calls it 'directional'; consider adding a second seed or explicitly marking the margin as untested for seed variation, since it is already reversed by learning rate.

Circularity Check

0 steps flagged

No significant circularity: the central measurements are independent controls, the k* inversion is explicitly labeled a consistency check rather than a prediction, and the theoretical bounds are standalone derivations under stated assumptions.

full rationale

The paper's central claim, that per-refresh subspace rotation is dominated by estimator noise, rests on a same-step split-batch measurement (Appendix F) that is independent of any theory in the paper, compared with a direct across-time rotation measurement. The trajectory control (same-step disagreement 0.64-0.71√2r at five training states versus across-time 0.742√2r) addresses the checkpoint-state mismatch. The k* inversion is explicitly described in §3.4 as 'a consistency check between two estimators rather than an independent prediction,' so it is not a fitted input relabeled as a prediction. Theorem 1 (Appendix B), Proposition 3 (Appendix C), and Proposition 4 (Appendix D) are standalone derivations under stated assumptions (Haar rotation, isotropic gradients, spectral-mass identity); none assumes the conclusion it is used to support. The N^{-1/4} averaging scaling is reported as a measured scaling, with the failure of the out-of-sample fit stated plainly, making it an extrapolation rather than a circular derivation. The reader-raised batch-size mismatch concern is a validity/control issue about matching effective sample sizes, not a reduction of any step to its inputs by construction. There is no load-bearing self-citation chain or imported uniqueness theorem; the LDAdam transport rule is credited as an external rule and is then given an independent optimality proof. Overall, no significant circularity is present.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

Free parameters are limited to fitted scaling exponents and calibration noise levels. The central claims additionally rest on the Haar idealization for refresh frames and the assumption that the gradient tail is signal rather than sampling noise. No new physical entities are introduced; k* is a diagnostic quantity rather than a mechanistic object.

free parameters (2)
  • power-law tail exponent alpha = 1.03 - 1.35
    Fitted to the de-censored spectral tail and the overlap-versus-N curve; used to support the N^{-1/4} no-gap conclusion in Section 3.5.
  • per-layer noise scale nu_1 (spectral inversion) = not reported numerically
    One calibrated noise scale per layer used to invert the spectrum into a k* estimate; authors describe it as a consistency check between two estimators rather than an independent prediction.
axioms (5)
  • standard math Davis-Kahan / Wedin sin(theta) perturbation bounds apply to the computed gradient subspaces
    Used in Section 3.3 to argue that near-degenerate singular directions are maximally sensitive to perturbation and resample freely.
  • domain assumption Refresh frames follow a Haar(O(r)) distribution in the degenerate tail
    Used in Theorem 1 and Appendix B to derive the (r-k*)/2 blind-carry suboptimality factor. This is an idealization; the actual refresh rotations are not proven Haar-distributed.
  • domain assumption The gradient spectral tail is genuine signal with a power-law spectrum, not finite-sample sampling noise
    Load-bearing for the claim that no averaging budget opens a gap. The paper measures N^{-1/4} decay up to N=64 but does not derive the law; the assumption is that the tail persists as signal.
  • domain assumption Isotropy of gradients in the tail for first-moment transport
    Used in Proposition 3 (Appendix C); the authors note isotropy holds exactly only in the near-degenerate tail that redraws.
  • standard math Weingarten calculus moment formulas for Haar orthogonal matrices
    Used in Appendix B to compute the second-moment lower bound.
invented entities (1)
  • reproducible rank k* independent evidence
    purpose: Diagnostic probe of how many top-r gradient directions are reproducible across minibatches; used to decide whether a chosen rank exceeds the identifiable core.
    k* is operationally measurable by splitting a batch and counting overlapping directions, so it has a falsifiable handle, but it is a new metric proposed by this paper with no external validation yet.

pith-pipeline@v1.3.0-alltime-deepseek · 23423 in / 16462 out tokens · 168060 ms · 2026-08-02T08:21:57.659850+00:00 · methodology

0 comments
read the original abstract

Memory-efficient optimizers such as GaLore train large language models by projecting gradients onto a rank-r subspace recomputed every T steps, assuming this subspace is a slowly drifting object that can be tracked. We show that beyond a small reproducible core, there is no such object. Two estimates of the top-r subspace computed at the same step from disjoint minibatches disagree as much as estimates computed T steps apart (0.73 vs 0.74 of the maximal chordal distance sqrt(2r), at Pythia-160M with r=128): the apparent rotation at each refresh is dominated by estimator noise. This holds across four model families in three architecture classes from 70M to 6.9B parameters, strengthening with scale, and more weakly in a vision transformer. Only ~39 of 128 directions are reproducible across minibatches, and averaging cannot recover the rest: under N-fold averaging the gradient's spectral tail shrinks as N^(-1/4) rather than the N^(-1/2) of pure noise, so no averaging budget makes the subspace well defined. What helps instead follows from treating each refresh as a change of coordinates for Adam's state. Carrying the second moment blindly is provably about (r-k*)/2 worse than the best rotation-blind estimator, while the first moment transports exactly through the rotation, the optimal linear map under isotropic gradients and the rule LDAdam uses. At 1B over 40k steps (3 seeds), full LDAdam reaches 18.7 perplexity at beta2=0.999, beating untransported GaLore after its best beta2 fix (19.3); shortening the second-moment memory to beta2=0.99 helps the refreshing optimizers, though for canonical GaLore the effect is small and a full-rank control reverses it. One measurable fact, subspace non-identifiability, clarifies why GaLore works, which patches work, and what to check before trusting a low-rank assumption: the reproducible rank k*.

Figures

Figures reproduced from arXiv: 2607.05872 by Noel Thomas.

Figure 1
Figure 1. Figure 1: Measured subspace rotation per refresh against rank (Pythia-1B, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Per-matrix gradient singular spectrum at Pythia-160M (left) and 1B (right): the reason [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Averaging cannot make the top-r subspace identifiable (Pythia-160M, r=128, four layers). (a) Same-step split-batch subspace disagreement ∥∆U∥F / √ 2r against averaging fold N. At N=1 it is 0.725, indistinguishable from the across-time rotation 0.742 (dashed), and it falls only slowly with N. (b) The deep tail singular value scaled by √ N. Pure noise would be flat; the measured tail grows 2.0–3.2× from N=1 … view at source ↗
Figure 4
Figure 4. Figure 4: Same-step subspace disagreement ∥∆U∥F / √ 2r at N=1 (split-batch, pretrained check￾points, r=128, mean over four layer types) across architectures, scales, and domains. It rises with scale in Pythia (70M–6.9B), sits in the same band for GPT-2-large, Qwen2.5-3B, and Llama-3-8B, and holds more weakly for a vision transformer on CIFAR. Every model is full-rank-Adam-pretrained, so the non-identifiability is a … view at source ↗
Figure 5
Figure 5. Figure 5: Lowering β2 from the 0.999 default to 0.99 at Pythia-1B (final validation perplexity, log scale). The axis is logarithmic and each optimizer sits at its own training recipe (steps, learning rate, and length differ across the three, and the full-rank control); the comparison is within-optimizer, where each β2 pair shares its recipe, not across optimizers. For each refreshing optimizer shown, the lower value… view at source ↗

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