Slowing Learning by Erasing Simple Features

Lucia Quirke; Nora Belrose

REVIEW 4 major objections 4 minor 15 references

Slowing Learning by Erasing Simple Features

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper derives QLEACE, a closed-form method that removes all class information available to quadratic classifiers, and shows it reliably slows feedforward learning while expressive vision models recover the labels from higher-order…

desk verdict Solid closed-form quadratic erasure with a real backfiring finding, but the empirical support is thinner than the causal claims. read the letter →

arxiv 2502.02820 v1 pith:BIARRDC3 submitted 2025-02-05 cs.LG

Lucia Quirke , Nora Belrose This is my paper · ORCID

classification cs.LG

keywords concepterasurequadraticmomentsLEACEQLEACEdistributionalsimplicitybiasoptimaltransportprequentialMDLlearningdynamics

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

The paper tries to establish that erasing simple, low-order statistical features makes learning harder, and that how much harder reveals how networks learn. It derives QLEACE, a closed-form map that equalizes each class's mean and covariance with the dataset-wide mean and covariance, thereby removing all information available to quadratic classifiers. Training on QLEACE'd data reliably slows feedforward networks, but expressive vision models eventually exploit higher-order statistics that QLEACE inadvertently injects, sometimes learning faster than on the original data. The paper also studies approximate label-free erasure, which avoids injection but acts like data augmentation on some datasets. These results matter because they test the distributional simplicity bias and warn practitioners that moment erasure is not the same as information erasure.

What carries the argument

The load-bearing machine is QLEACE's per-class affine transport map $T_i(x) = \Sigma_i^{-1/2}(\Sigma_i^{1/2}\bar{\Sigma}\Sigma_i^{1/2})^{1/2}\Sigma_i^{-1/2}(x-m_i)+\bar{m}$, with target mean $\bar{m}$ and covariance $\bar{\Sigma}$ defined as the Gaussian Wasserstein barycenter of the class means and covariances. This map is proven to minimize the total expected squared edit among all transports that equalize class means and covariances, and moment equalization is proven sufficient and, under conditions on the loss, necessary for a degree-$N$ polynomial classifier to be no better than the constant predictor. The other machinery includes LEACE linear erasure, ALF-QLEACE's rank-$d-r$ projection that removes the directions of largest class variance differences, L-BFGS gradient-based moment erasure, and prequential MDL as the measure of learning difficulty.

What would settle it

Run QLEACE on a dataset whose class-conditional covariance is rank-deficient, then fit a quadratic classifier to the erased data; if it beats chance, the full-rank assumption is violated and the closed-form guarantee is not met. Equivalently, on any real dataset, compare the empirical mean and covariance of each erased class to the barycenter target; any significant mismatch is direct evidence the method did not equalize second moments.

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Extended reading notes

Core claim

The paper's central discovery is that there is a closed-form, provably minimal way to erase every piece of class information that a quadratic, degree-2 polynomial classifier could use, and that this erasure behaves very differently from linear erasure. QLEACE applies, per class, the optimal transport map from the class-conditional distribution to a common target distribution whose mean and covariance are the Gaussian Wasserstein barycenter of the class means and covariances; Theorems 2.2 and 2.3 show matching first- and second-order moments makes the constant predictor optimal among polynomial predictors, and Theorem 2.6 proves this particular target minimizes the expected squared edit. On CIFAR-10, CIFARNet, and SVHN, QLEACE consistently slows feedforward MLPs and LeNets more than LEACE does, but in ConvNeXt and Swin models it backfires after about 16 epochs: the models exploit higher-order Shannon information QLEACE injected about class labels and can reach lower final loss than on unerased data. The approximate label-free variant ALF-QLEACE avoids injection because it applies the same projection to every point, yet it sometimes makes learning easier, acting as data augmentation. The paper concludes that quadratic concept erasure is unreliable in practice, while LEACE is a reliable way to make features less salient.

Load-bearing premise

The guarantee that QLEACE removes all quadratically available information assumes every class has a full-rank covariance matrix and an exact deterministic transport map exists; on real image data those conditions often fail, so the computed erasure may fall short of the guarantee.

Editorial extensions

If this is right

Because QLEACE transforms each class with a different map, it can inject label-dependent geometry into higher-order moments; the observed backfiring means moment-matching is not information-erasure when the downstream network is expressive.
LEACE remains the reliable erasure: it raises prequential MDL and final loss across every architecture, dataset, width, and depth tested, with roughly constant effect as networks widen.
The backfiring transition, which appears around 16 epochs in ConvNeXt and Swin on CIFAR-10, is direct evidence that distributional simplicity bias breaks down once networks have had enough time to exploit higher-order statistics.
ALF-QLEACE and gradient-based erasure do not backfire because they do not depend on the class label, but their effects are smaller and sometimes negative, acting as data augmentation, so quadratic erasure is dataset- and architecture-dependent.
Any practitioner wanting to make a feature harder to learn should prefer LEACE; quadratic erasers should be used with caution and validated for backfiring on the target architecture.

Reading between the lines

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

A testable extension the authors leave implicit: if backfiring is caused by class-dependent geometry, then a deep network trained on QLEACE'd data should show the same late-epoch loss collapse on any dataset where the barycenter maps are non-isometric, while ALF-QLEACE should never show it.
We infer that moment erasure is insufficient as a fairness or safety intervention in deployment, since a trained adversarial network, not a polynomial classifier, is the relevant threat model; a label-independent or Shannon-information erasure would be safer.
The sharp transition around epoch 16 suggests that probing intermediate layers of ConvNeXt or Swin before and after that epoch could reveal exactly where higher-order statistics are decoded, making the mechanism observable rather than inferred from loss curves.
An open design question follows: can the data-augmentation effect of ALF-QLEACE be harnessed deliberately? Since it removes high-variance class-specific directions, it may act as a regularizer; comparing it to standard augmentation on the same architectures would separate the two effects.

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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 useful thing here is QLEACE—the closed-form method for erasing all class-conditional mean and covariance differences via a Gaussian Wasserstein barycenter—and the backfiring observation in convolutional architectures. The math is the strong part: Theorems 2.2, 2.3, and 2.6 are coherent, the proof in the appendix checks out, and the extension from LEACE's linear erasure to quadratic moments is a genuine contribution. The experimental finding that QLEACE consistently slows learning in MLPs but later triggers a sharp loss collapse in ConvNeXts and Swins is real and worth understanding.

That said, the paper overreaches in its causal story. The claim that sophisticated architectures 'learn to use injected higher order Shannon information' is not directly measured. The guarantee of exact quadratic guardedness rests on full-rank covariance assumptions and a deterministic Monge map; on CIFAR-10 pixel space, sample covariances are extremely ill-conditioned. The paper never says whether the barycenter solve is regularized, and it reports no diagnostic of residual class-conditional covariance mismatch. If residual second-order signal remains, the backfiring could be simple quadratic features, not higher-order Shannon information. That is a load-bearing gap, not a nit.

Other soft spots are smaller. No code or data is released, so the numbers are not checkable. The ALF-QLEACE rank is specified inconsistently—rank d−k in the text, rank d−1 a few paragraphs later, rank d−15 in the figure caption. The gradient-based erasure loss weights are tuned but not reported. All of these are fixable in revision.

I don't think the paper is fatally flawed. The mathematical contribution is solid, and the backfiring phenomenon deserves publication. But I'd want a serious referee to push on the empirical support: require code, a description of the covariance solver, a residual-moment diagnostic, and either measurement of the leaked Shannon information or softer causal language. A competent editor should send this to review, not desk-reject.

Referee Report

4 major / 4 minor

Summary. The paper derives QLEACE, a closed-form optimal-transport-based erasure method that equalizes class-conditional means and covariances, and compares it with LEACE, an approximate label-free variant (ALF-QLEACE), and a gradient-based erasure method on CIFAR-10, CIFARNet, and SVHN. The main empirical findings are that LEACE consistently slows learning across architectures, while QLEACE slows learning in feedforward networks but backfires in ConvNeXt and Swin models, and approximate variants can act as data augmentation. The authors interpret backfiring as evidence that expressive architectures exploit injected higher-order Shannon information about class labels.

Significance. If the QLEACE guarantee holds in practice, the paper provides a clean minimal-distortion method for removing all second-order class information and a novel intervention-based demonstration of the distributional simplicity bias. The theoretical core (Theorems 2.2, 2.3, and 2.6) is clearly stated with proofs in the appendix; the use of maximal update parametrization and prequential MDL is appropriate, and the width-scaling comparison is informative. The backfiring phenomenon is a striking and potentially influential result. However, the empirical verification of the second-moment guarantee is absent, which leaves the main interpretation underdetermined.

major comments (4)

[Section 2.2 (Theorem 2.6) and Section 4] The exact second-moment guarantee is not verified empirically. The theorem assumes full-rank class-conditional covariance matrices and an exact solution to the barycenter equation (4), but on raw-pixel CIFAR-10/SVHN (d=3072) the empirical covariance is severely ill-conditioned; the paper does not state whether QLEACE uses shrinkage, a pseudoinverse, or truncated SVD, and it reports no post-erasure diagnostic of residual class-conditional covariance differences. Since the central backfiring interpretation attributes the late ConvNeXt/Swin performance to injected higher-order Shannon information, the missing quadratic-residual check leaves the plausible alternative that residual second-order signal drives the effect. Please report a diagnostic such as max_k ||Σ_k − Σ̄||_F (and the mean/covariance deviations for all erasers) and specify the numerical solver, regularization, and convergence tolerance.
[Section 2.3, Eq (11), and Figure 1] The rank of the ALF-QLEACE projection is specified inconsistently. Eq (11) says 'rank d−k', the next paragraph says 'rank d − 1', Figure 1 says 'rank d − 15', and the iterative-erasure paragraph says 'up to d rank 1 erasures' can be added. In addition, the claimed min-max optimality of truncating the singular-value decomposition of the worst-class difference matrix is asserted but not proved; the objective (11) is over a projection matrix P and is not obviously solved by that heuristic. Please give the exact algorithm (rank parameter, iteration count, class selection rule) and a proof or citation for the projection optimality.
[Section 2.4] The gradient-based erasure method is not reproducible. The three loss-term weights are said to be 'tuned' to ensure near-zero quadratic information while staying close to the original images, but their final values, the L-BFGS hyperparameters, and the optimization budget are not reported, and no code or data are released. Because the comparison across erasers is the empirical backbone of the paper, please provide code/data or a complete hyperparameter table (including per-dataset values) for all erasure methods.
[Section 4, QLEACE paragraph] The causal claim that backfiring reflects learned use of 'injected higher order Shannon information' is not directly supported by any quantitative measurement of higher-order statistics. The paper shows sample images and learning curves, but does not report third- or fourth-order conditional moment differences, mutual-information estimates, or the accuracy of a nonlinear probe on QLEACE'd data. Without such a probe, the alternative explanation that the backfiring is due to residual quadratic information (or some other artifact) is not excluded. Please add a higher-order statistic probe or soften the causal interpretation.

minor comments (4)

[Section 4, CIFARNet] 'Backfiring occurs in all two- and three-layer MLPs on the CIFARNet dataset' conflicts with the Figure 8 caption, which says 'for some model dimensions'; please reconcile the text with the figure.
[Section 2.3, Eq (11)] The notation 'k∼K' is nonstandard and should be 'k ∈ K'; also, the sentence invoking the min-max theorem should state explicitly which matrix norm is being used.
[Section 4 and Table 16] The z-score normalization analysis is described as 'ambiguous' but no sample sizes, standard errors, or test details are given; please clarify how the ambiguity was assessed.
[Section 2.2, footnote 2] The footnote acknowledges that the Monge problem may not have a solution; since real image covariances may be low-rank, a brief discussion of a Kantorovich relaxation or a regularized solver would help readers judge when QLEACE's guarantee applies.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QLEACE is derived from external optimal-transport lemmas and its erasure guarantee is proved, while the backfiring observations are empirical rather than fitted.

full rationale

The derivation chain in Section 2 is self-contained. Theorem 2.2 and Theorem 2.3 extend Belrose et al. (2023) by a direct Jensen/first-order-optimality argument, and the extension does not assume the target conclusion. Theorem 2.6 solves the moment-equalization and edit-distance tradeoff using external optimal-transport results (Rüschendorf & Uckelmann 2002; Cuesta-Albertos et al. 1996), so the QLEACE map's second-moment-erasure property is a proved consequence of its construction rather than a fitted prediction. ALF-QLEACE likewise follows from a stated min-max and SVD argument. The experiments report measured MDL and loss differences, and the backfiring phenomenon is an observed outcome; the higher-order-information interpretation is not used to fit the erasers. The paper does cite the authors' own LEACE and DSB work for method and motivation, but neither citation is load-bearing: the QLEACE and ALF-QLEACE guarantees and the learning curves stand on the paper's own equations and data. One caveat belongs under correctness rather than circularity: Theorem 2.6 assumes full-rank class-conditional covariances and a feasible Monge map (stated in Section 2.2), and the paper reports no diagnostic confirming that post-erasure class-conditional second moments are equal on CIFAR-10/SVHN, so residual quadratic signal is a possible alternative explanation for backfiring.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No invented physical entities are proposed; the new objects are algorithms. The free parameters are the tuned gradient-erasure loss weights and the ALF-QLEACE rank. The central derivation relies on standard optimal transport lemmas and on domain assumptions about full-rank covariance, Monge feasibility, and the distributional simplicity bias.

free parameters (2)

Gradient-based erasure loss weights = not reported
Section 2.4: weights for mean-matching, covariance-matching, and distance penalties are tuned to make the dataset have almost no quadratic information; the values are not given, so the gradient-erased dataset is a fitted artifact.
ALF-QLEACE projection rank = d-15 in Figure 1; text also states d-k and d-1
Section 2.3 and Figure 1 caption: the number of removed directions is a hand-chosen hyperparameter, and the paper does not justify the choice or report sensitivity to it.

assumptions (5)

domain assumption Class-conditional distributions have full-rank covariance matrices and finite second moment.
Assumed in Theorem 2.6 so the optimal transport map in Eq. 8 is valid; real image data often has near-singular empirical covariance, and no shrinkage is described.
domain assumption The Monge formulation of optimal transport is feasible on real-world datasets.
Footnote 2 acknowledges the Monge problem may not always have a solution; the method assumes deterministic maps exist and are computable.
domain assumption Neural networks exhibit a distributional simplicity bias, learning low-order moments before higher-order statistics.
Motivates the experimental inversion; taken from the same authors' prior DSB work (Belrose et al., 2024), and the MDL results are interpreted through this lens.
domain assumption Prequential MDL area under the learning curve is a valid proxy for learning difficulty.
Voita and Titov framework adopted without discussion of its limitations for comparing erasure methods.
standard math Gaussian barycenter equation (Lemma 2.4) and 2-Wasserstein lower bound (Lemma 2.5) hold as stated.
Imported from Ruschendorf and Uckelmann (2002) and Cuesta-Albertos et al. (1996); the QLEACE closed form depends on these results.

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Cite this review

Pith. "Pith review of Slowing Learning by Erasing Simple Features." pith.science (2026). https://pith.science/paper/BIARRDC3

@misc{pith2026250202820,
  author       = {Pith},
  title        = {Pith review of: Slowing Learning by Erasing Simple Features},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BIARRDC3}},
  note         = {Machine review of arXiv:2502.02820}
}

read the original abstract

Prior work suggests that neural networks tend to learn low-order moments of the data distribution first, before moving on to higher-order correlations. In this work, we derive a novel closed-form concept erasure method, QLEACE, which surgically removes all quadratically available information about a concept from a representation. Through comparisons with linear erasure (LEACE) and two approximate forms of quadratic erasure, we explore whether networks can still learn when low-order statistics are removed from image classification datasets. We find that while LEACE consistently slows learning, quadratic erasure can exhibit both positive and negative effects on learning speed depending on the choice of dataset, model architecture, and erasure method. Use of QLEACE consistently slows learning in feedforward architectures, but more sophisticated architectures learn to use injected higher order Shannon information about class labels. Its approximate variants avoid injecting information, but surprisingly act as data augmentation techniques on some datasets, enhancing learning speed compared to LEACE.

Figures

Figures reproduced from arXiv: 2502.02820 by the authors.

**Figure 1.** Ship from the CIFARNet training set edited with each eraser. LEACE and gradient-based quadratic erasure minimally affect intelligibility of the image, while QLEACE and ALF-QLEACE reduce intelligibility. The ALF-QLEACE intervention is rank d − 15. A random projection of equal rank is included for comparison. We are therefore free to select the target mean and covariance matrix Σ∗ so as to minimize the expected edit … view at source ↗

**Figure 2.** Images can be easily identified after gradient-based erasure. Non-cherrypicked examples from CIFAR-10, CIFARNet, and SVHN. no difference to the solution. Therefore the barycenter of N (m1, Σ1), . . . , N (mk, Σk) will have mean and covariance (m¯ , Σ¯ ) that minimize Eq. 9, thus completing the proof. To solve for the unconditional covariance matrix, we use the fixed-point algorithm derived in Álvarez-Esteban et al.… view at source ↗

**Figure 3.** Increase in MDL from the erasure of the CIFAR-10 dataset over 5 random seeds for ReLU MLPs and ConvNeXt V2s of various widths. While LEACE slows learning similarly in both architectures, ConvNeXts are less affected by quadratic erasure and exhibit a backfiring effect on data modified with QLEACE, resulting in improved performance relative to unerased data. covariance matrix and the unconditional covariance matrix, a… view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: MDL over 5 random seeds for feedforward networks of various lengths and widths on linearly erased CIFAR-10. All models have a constant depth of 2 when width is varying and a constant width of 128 when depth is varying. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: MDL over 5 random seeds for Swin Transformers and ConvNeXts of various lengths and widths on the linearly erased CIFAR-10 dataset. 40 60 80 100 120 −2 0 2 4 6 2 3 4 5 6 7 8 −2 0 2 4 6 40 45 50 55 60 65 −2 0 2 4 6 2 2.5 3 3.5 4 −2 0 2 4 6 Quadratic Erasers QLEACE ALF-QL…

**Figure 7.** Figure 7: MDL over 5 random seeds for Swin Transformers and ConvNeXts of various lengths and widths on the quadratically erased CIFAR-10 dataset. The ConvNeXts exhibit backfiring, achieving lower mean MDLs on surgically quadratically erased data than on unerased data. 12 [PITH_…

**Figure 8.** Figure 8: MDL over 5 random seeds for MLPs of various lengths and widths on the linearly and quadratically erased CIFARNet and SVHN datasets. The MLPs exhibit backfiring on the CIFARNet dataset for some model dimensions. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: Cross-entropy loss over 5 random seeds for ReLU MLPs of various lengths and widths on the CIFAR-10 dataset. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: Cross-entropy loss over 5 random seeds for GELU MLPs of various lengths and widths on the CIFAR-10 dataset. MLPs with a varying depth have a constant width of 128 and MLPs with a varying width have a constant depth of 2. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: Cross-entropy loss over 5 random seeds for SwiGLUs of various lengths and widths on the CIFAR-10 dataset. GLUs with a varying depth have a constant width of 128 and GLUs with a varying width have a constant depth of 2. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: Cross-entropy loss over 5 random seeds for ConvNeXtV2s of various lengths and widths on the CIFAR-10 dataset. 1 2 3 1 2 3 1 2 3 1 2 3 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11 1 2 3 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11 1 2 3 Eraser type Control LE…

**Figure 13.** Figure 13: Cross-entropy loss over five random seeds for Swin transformers of various lengths and widths on the CIFAR-10 dataset. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]

**Figure 14.** Figure 14: Cross-entropy loss over 5 random seeds for ReLU MLPs of various lengths and widths on the SVHN dataset. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 15.** Figure 15: Cross-entropy loss over 5 random seeds for ReLU MLPs of various lengths and widths on the CIFARNet dataset. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

**Figure 16.** Figure 16: (Top) ReLU MLP MDL results on the CIFAR-10 dataset after applying z-score normalization. Normalization appreciably reduces MDL on linearly erased data, resulting in a lower mean MDL than for approximately quadratically erased data. (Bottom) Increases in MDL across arc…

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15 Slowing Learning by Erasing Simple Features 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 2.5 3 3.5 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 2.5 3 3.5 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11 2 2.5 3 3.5 Eraser type Control LEACE QLEACE ALF-QLEACE Gradient-Based Loss over 5 seeds...

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(15) 3We are using NumPy-style indexing here, so that A(n) ...j refers to the slice of A(n) where the final axis index is set to j. This yields a “square” order-n tensor whose axes are all of size d. 8 Slowing Learning by Erasing Simple Features If P(Z ̸= j) = 0, then all class-conditional moments are trivially equal to the unconditional moments. Otherwis...

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Cross-entropy loss over 5 random seeds for ReLU MLPs of various lengths and widths on the CIFAR-10 dataset. 14 Slowing Learning by Erasing Simple Features 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 2.5 3 3.5 4 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 2.5 3 3.5 4 2 0 2 1 ...

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The ConvNeXts exhibit backfiring, achieving lower mean MDLs on surgically quadratically erased data than on unerased data

MDL over 5 random seeds for Swin Transformers and ConvNeXts of various lengths and widths on the quadratically erased CIFAR-10 dataset. The ConvNeXts exhibit backfiring, achieving lower mean MDLs on surgically quadratically erased data than on unerased data. 12 Slowing Learning by Erasing Simple Features 0 500 1000 1500 2000 1 1.5 2 2 4 6 8 1 1.5 2 Width ...

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The MLPs exhibit backfiring on the CIFARNet dataset for some model dimensions

MDL over 5 random seeds for MLPs of various lengths and widths on the linearly and quadratically erased CIFARNet and SVHN datasets. The MLPs exhibit backfiring on the CIFARNet dataset for some model dimensions. 13 Slowing Learning by Erasing Simple Features 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 2.5 3 2 0 2 1 2 2 2 3 2 4...

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