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REVIEW 4 major objections 5 minor 47 references

Where Did Your Model Learn That? Label-free Influence for Self-supervised Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper introduces Influence-SSL, a label-free influence score for self-supervised visual models that measures each training image's contribution to its own augmented-view consistency, and reports that removing the highest-scoring…

desk verdict Useful label-free influence score for SSL, but Theorem 1 is proven for a single-example Hessian, not the dataset Hessian the method actually uses, so the theory needs rework before the empirical claims are supported. read the letter →

arxiv 2412.17170 v1 pith:QWDOHCJH submitted 2024-12-22 cs.LG cs.CV

Nidhin Harilal , Amit Kiran Rege , Reza Akbarian Bafghi , Maziar Raissi , Claire Monteleoni This is my paper · ORCID
classification cs.LGcs.CV
keywords influencefunctionsself-supervisedlearningdataattributionrepresentationaugmentationinvarianceoutlierdetectionfairnessanalysisduplicate
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

Self-supervised models learn representations without labels, so classical influence functions—which trace a training point's contribution to a labeled prediction—do not directly apply. This paper defines a label-free influence score for SSL that asks how much a training image stabilizes its own representation when the image is augmented; the score is the negative quadratic form of the augmentation-consistency gradient with the inverse Hessian. The authors prove that in a two-layer linear network with small additive perturbations this influence is exactly a scaled version of the per-sample SSL loss, and they show empirically that the scores are stable across training runs, flag duplicates and uniform-background outliers, reveal an uneven racial profile in a face dataset, and—counterintuitively—that removing the highest-influence training images improves downstream classification accuracy in three SSL methods.

What carries the argument

The load-bearing object is the SSL self-influence score $I(f,i) = -\nabla_\theta L(f_\theta(x_i), f_\theta(\hat{x}_i))^\top H_\theta^{-1} \nabla_\theta L(f_\theta(x_i), f_\theta(\hat{x}_i))$, where $L$ is the cosine distance between a training image and an augmented view and $H_\theta$ is the Hessian of the model's loss over the dataset. The paper's theoretical engine is the reduction of this score in a two-layer linear network with $x_{\mathrm{aug}} = x + \varepsilon\delta(x)$, $|\delta|=1$, $\varepsilon\ll1$: there the score collapses to $-2\varepsilon^2|W\delta|^2$, making influence exactly proportional to the per-sample SSL loss. In practice the Hessian inverse is approximated with a low-rank inverse-Hessian-vector-product method, and the augmentation is implemented as Gaussian noise, which the appendix shows is the perturbation choice that best preserves score rankings.

What would settle it

Retrain a small SSL model after removing each of the top-1000 Influence-SSL points and compare the actual leave-one-out change in downstream accuracy with the score's predictions; if removing high-influence points does not consistently improve accuracy across seed runs, the central removal claim fails. A second check is to compute Equation 6 in the exact two-layer linear network of Theorem 1 with $\varepsilon\ll1$ and verify that the score equals $-2\varepsilon^2|W\delta|^2$ within numerical precision.

Watch

Extended reading notes

Core claim

The central claim is that a training example's influence in self-supervised learning can be measured without any labels by looking at how much the learned representation changes when that example is augmented, and that this score is informative about what the model learned. In the simplified two-layer linear setting the score has the closed form $I_{\mathrm{ssl}}(x) = -2\varepsilon^2 |W\delta(x)|^2$, so influence is proportional to the per-sample augmentation loss and measures how far the model's representation moves under that input's perturbation. Empirically, high-influence CIFAR images are dominated by examples with uniform backgrounds, low-influence images include near-duplicates, and deleting the highest-influence examples improves downstream accuracy for SimCLR, BYOL, and Barlow Twins on CIFAR-10 and CIFAR-100, the opposite of the supervised baseline. On FairFace, the top-100 influential faces over-represent Black and Middle Eastern groups, which the paper presents as evidence that the score can surface biases hidden from conventional metrics.

Load-bearing premise

The whole argument assumes that what is true for a two-layer linear network with tiny perturbations is still true for deep nonlinear SSL models; if the inverse-Hessian term stops behaving like the constant it becomes in the linear proof, the theoretical foundation no longer covers the experiments.

Editorial extensions

If this is right

  • Removing the highest-influence training points improves downstream top-1 accuracy for SimCLR, BYOL, and Barlow Twins on CIFAR-10 and CIFAR-100, while the same removal degrades a supervised baseline.
  • Influence-SSL scores are reproducible: Pearson rank correlations exceed 0.96 across independent training runs for all three SSL frameworks.
  • Low-influence examples tend to be semantic duplicates and high-influence examples tend to have uniform backgrounds, so the score can serve as a label-free signal for deduplication and outlier detection.
  • On FairFace, the top-100 influential faces are disproportionately Black and Middle Eastern, indicating the score can reveal demographic skew in SSL representation learning.
  • In the linear model, influence decomposes into a perturbation scale factor and a representation-sensitivity term, so the score directly measures how strongly augmentations move a point in representation space.

Reading between the lines

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

  • A testable corollary of the linear theory is that in deep networks the score's ranking may be dominated by the per-sample augmentation loss; comparing Influence-SSL rankings against that loss alone would show whether the inverse-Hessian term adds signal.
  • The uniform-background result suggests SSL pretraining can encode background as a shortcut; influence scores could be used as a pre-deployment audit to find images whose backgrounds, not objects, drive their representations.
  • Because low-influence examples are near-duplicates, Influence-SSL could double as a label-free deduplication filter before pretraining; one extension would test whether dropping low-influence images reduces pretraining cost without hurting downstream accuracy.
  • The paper deliberately leaves cross-point influence (a training point's effect on a held-out query's representation) for future work; defining query-conditional influence would let practitioners ask which training images best explain a specific test prediction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces Influence-SSL, a label-free influence score for self-supervised learning defined in Eq. (6) as I(f,i) = -∇L(fθ(xi), fθ(x̂i))^T Hθ^{-1} ∇L(fθ(xi), fθ(x̂i)), where L is cosine distance and Hθ is described as the Hessian of the model's loss over the dataset. The authors provide a linear-network analysis (Theorem 1) claiming the score reduces to -2ε²|Wδ|², prove structural properties such as rotation invariance and scaling, and present experiments on CIFAR-10/100 and FairFace showing score stability across runs, detection of duplicates and uniform-background images, improved downstream accuracy after removing high-influence points, and uneven demographic representation among high-influence faces.

Significance. If the proposed score were a valid influence function, it would fill a real gap: supervised influence functions require labels and cannot be applied directly to SSL embeddings, and existing label-free attribution tools are mostly heuristic. The paper is commendable for releasing code, using three SSL frameworks (SimCLR, BYOL, Barlow Twins), reporting training configurations, and for the clean algebraic properties of the linearized score (Propositions 1-2, Appendix C). However, the theoretical bridge from Eq. (6) to the closed form in Theorem 1 is broken by the Hessian mismatch, and the simplified closed form is a scalar multiple of the per-sample SSL loss. The empirical removal results therefore currently read as augmentation-loss pruning rather than as downstream effects of causal example influence. These issues affect the central claim, so the paper in its present form is not ready for acceptance.

major comments (4)
  1. [§3.1 / Appendix C.1, Theorem 1] The proof applies Sherman-Morrison to H_ssl^λ = 2ε²δδᵀ + λI, the regularized Hessian of a single training example, but Eq. (6) and the classical influence definition require the Hessian of the full training objective over the dataset. For a dataset with Hessian H = 2Σ_i ε_i² δ_i δ_iᵀ + λI, the self-influence of point j is I_j = -4ε_j⁴ |Wδ_j|² δ_jᵀ H^{-1} δ_j, not -2ε²|Wδ_j|². The factor δ_jᵀH^{-1}δ_j is a leverage score and is not generally 1/(2ε_j²); even in the scalar case d=1, n=2, with ε_1=ε_2=ε and δ_1=δ_2=1, Eq. (6) gives -ε² while Theorem 1 predicts -2ε². Since the experiments use the full training-set Hessian via LoGra/LogIX, the theorem does not characterize the quantity actually computed.
  2. [§3.1, Theorem 1 and Lemma 3] In the simplified setting, the derived influence I_ssl(x) = -2ε²|Wδ|² is exactly -2 times the per-sample SSL loss L_ssl(x)=ε²|Wδ|². Thus the proposed 'influence' reduces, by the paper's own equations, to a deterministic monotone transform of the loss of one augmented pair. The score therefore measures augmentation sensitivity of the current representation, not the effect of the example on the learned parameters; a point can be 'high-influence' without causing any change in θ. This is a conceptual issue, not just a presentational one: it means the theoretical argument does not establish that Eq. (6) is an influence function in the Koh-Liang sense.
  3. [§4.3, Figure 6] The main empirical claim is that removing high-Influence-SSL points improves downstream accuracy, but the experiments do not compare against random removal or against removal by the per-sample SSL/consistency loss. Given Major Comment 2, the observed improvement may be exactly the effect of removing the largest per-sample augmentation losses; such hard-example pruning is a known phenomenon and would not validate the proposed attribution method. Please add baselines (random, highest raw SSL loss, highest gradient norm) and report the rank correlation between Influence-SSL and raw SSL loss on the actual ResNet18 models.
  4. [§4 and Theorem 1] The theory is stated for a two-layer linear network with small perturbations δ satisfying δᵀx = 0 and Euclidean loss, yet the experiments use ResNet18, Gaussian noise with σ=0.2, cosine distance, and nonlinear SSL objectives. No argument or diagnostic is given that the Hessian inverse in Eq. (6) is approximately constant across examples or that the linear closed form is predictive of deep-network scores. The paper's own caveat that influence functions can deviate from leave-one-out effects in neural networks (Section 3.1, citing [5,2]) makes this gap load-bearing for the claim that the experiments validate the theory. Please provide, at minimum, a rank-correlation check between Eq. (6) with full Hessian and the linearized score, or explicitly reframe the theory as intuition only.
minor comments (5)
  1. [Eq. (6) and experimental ranking] Eq. (6) defines I(f,i) as a negative quadratic form, so scores are non-positive, but the paper reports positive 'log influence scores' and ranks 'top influential' without saying whether it uses |I| or -I; please clarify this for all figures and tables.
  2. [Appendix B.4, Figure 12] Duplicate identification is based on visual inspection; provide a quantitative duplicate-detection measure (e.g., fraction of nearest-neighbor pairs among lowest-influence images) and a threshold for declaring duplicates.
  3. [Figure 6] Appendix A lists seeds and says error bars are used, but the figure appears without visible error bars; add them or state explicitly that they are omitted.
  4. [§4.5, Figure 8] The racial imbalance claim needs a statistical test against the FairFace base rates and a control using the lowest-influence examples; currently it is a histogram without error bars or a significance statement.
  5. [Appendix A and C.2] There are several typos, including 'validation ses' in Appendix A, 'eac' in Appendix C.2, and 'Inluence' in the Figure 9 caption; also, reference [21] is the authors' own workshop paper and its relation to this submission should be clarified.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 1's closed-form influence is -2 times the per-sample SSL loss, obtained by swapping in the single-example Hessian; the theoretical derivation restates its own input.

  1. self definitional [Theorem 1 and Appendix C.1 (Lemma 3, Lemma 4, final influence formula)]
    "Lssl(W ; x) = |W x− W xaug|2 ... = ε2|W δ|2 ... Hssl = 2ε2δδ T ... I λ ssl(x) = −4ε4∥W δ∥2 1 λ + 2ε2"

    Eq. (6) defines I = −∇L^T H^{-1}∇L. In the proof H is taken to be Hssl = 2ε²δδᵀ, the Hessian of the same single-example loss whose value Lemma 3 gives as Lssl = ε²|Wδ|². For this homogeneous quadratic, −∇L^T(∇²L)^{-1}∇L = −2Lssl, so Theorem 1's Issl = −2ε²|Wδ|² is −2 times the loss already fed into the formula. This is an algebraic identity of the proposed quadratic form, not an independent characterization. Moreover, Eq. (6) defines Hθ as the dataset Hessian; with H = 2Σᵢ εᵢ²δᵢδᵢᵀ the self-influence of point j is −4εⱼ⁴|Wδⱼ|²δⱼᵀH^{-1}δⱼ, which equals the theorem only for n=1. The theorem therefore holds by replacing the dataset Hessian with the per-example Hessian, making the result equivalent to its input.

full rationale

The central theoretical step is circular: Theorem 1's closed form is just −2×Lssl(x) under a single-example Hessian, so the claimed 'characterization' restates the definition. The empirical studies (Figures 2, 4, 6; FairFace) are benchmark-based and would stand or fall independently; no load-bearing self-citation is used, and [21] is a prior workshop paper cited only for the unsurprising point that label-based influence is not directly applicable. I score 6 rather than 8-10 because the implemented score uses a LoGra/LogIX full-data Hessian, so the practical ranking is not literally the n=1 identity; however, the theoretical bridge claimed in Theorem 1 reduces by construction, and the paper offers no argument that the deep-network Hessian makes the score differ from a monotone transform of the per-sample augmentation loss.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the choice of cosine self-influence as the measure (a domain assumption), on the Hessian being well-defined and representative of the training loss (not fully specified), and on the transfer of the linear theoretical result to deep networks. The Gaussian perturbation and its parameters are ad hoc choices. No new physical or mathematical entities are introduced.

free parameters (3)
  • Gaussian perturbation parameters (mu, sigma) = mu=0.05, sigma=0.2
    The perturbation used to define augmented views in Equation 6 is set to Gaussian noise with these values, chosen by hand and justified via ablation (Appendix B.2, B.3).
  • Perturbation magnitude epsilon = implicit small; set by sigma=0.2
    The theory assumes epsilon << 1, but the empirical sigma is 0.2; epsilon is not explicitly set in the experiments.
  • Hessian low-rank approximation rank = not specified
    The inverse Hessian is computed via LoGra/LogIX, but the rank of the low-rank approximation is not reported, affecting the influence scores and reproducibility.
assumptions (5)
  • domain assumption The influence of a training point on its own representation can be measured by the cosine distance between the point and an augmented view, with the Hessian of the dataset loss.
    This is the core definition of Influence-SSL (Equation 6); it sets g = L(z_m, theta*) in the classical influence formula, which the paper argues is natural for SSL.
  • domain assumption The Hessian in Equation 6 is invertible or regularized.
    Classical influence functions require an invertible Hessian; the paper uses regularization in the linear proof and a low-rank approximation in experiments.
  • domain assumption Small perturbation assumption: x_aug = x + epsilon delta with epsilon << 1.
    Used in Theorem 1 and the theoretical properties; it does not hold for standard SSL augmentations like random cropping, so the theory does not directly apply to the experimental setup.
  • ad hoc to paper Gaussian noise is representative of SSL augmentations.
    The method uses Gaussian noise to compute influence scores, but SSL models are trained with augmentations like random crop and color jitter; the paper shows correlations with other augmentations (Table 4) but does not justify why Gaussian noise is the right choice for defining influence.
  • ad hoc to paper The linear network analysis transfers to deep nonlinear SSL models.
    Section 3.1 states the simplified setting offers geometric insights, but no argument is given that the closed-form result I_ssl = -2epsilon^2|W delta|^2 holds approximately for ResNet18; this is the main gap between theory and practice.

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Pith. "Pith review of Where Did Your Model Learn That? Label-free Influence for Self-supervised Learning." pith.science (2026). https://pith.science/paper/QWDOHCJH

@misc{pith2026241217170,
  author       = {Pith},
  title        = {Pith review of: Where Did Your Model Learn That? Label-free Influence for Self-supervised Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QWDOHCJH}},
  note         = {Machine review of arXiv:2412.17170}
}
read the original abstract

Self-supervised learning (SSL) has revolutionized learning from large-scale unlabeled datasets, yet the intrinsic relationship between pretraining data and the learned representations remains poorly understood. Traditional supervised learning benefits from gradient-based data attribution tools like influence functions that measure the contribution of an individual data point to model predictions. However, existing definitions of influence rely on labels, making them unsuitable for SSL settings. We address this gap by introducing Influence-SSL, a novel and label-free approach for defining influence functions tailored to SSL. Our method harnesses the stability of learned representations against data augmentations to identify training examples that help explain model predictions. We provide both theoretical foundations and empirical evidence to show the utility of Influence-SSL in analyzing pre-trained SSL models. Our analysis reveals notable differences in how SSL models respond to influential data compared to supervised models. Finally, we validate the effectiveness of Influence-SSL through applications in duplicate detection, outlier identification and fairness analysis. Code is available at: \url{https://github.com/cryptonymous9/Influence-SSL}.

Figures

Figures reproduced from arXiv: 2412.17170 by the authors.

Figure 1
Figure 1. Illustration of scenarios for varying influence levels for [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Correlation of Influence-SSL scores for two indepen [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. t-SNE projection of CIFAR-10 training images shows all examples (left), the 2,000 lowest (middle), and highest (right) influence [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Five lowest influential images of the CIFAR-10 ‘auto [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Visualization of the 12 highest and 12 lowest influential [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Accuracy (y-axis) vs. number of removed samples (x-axis) for CIFAR-10 (top row) and CIFAR-100 (bottom row). Samples were [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: The top 10 and lowest 10 influential examples in the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: Distribution comparison of log Inluence-SSL scores be￾tween SSL methods like SimCLR, BYOL and Barlow Twins. Note on FairFace experiment: Several commercial com￾puter vision systems (Microsoft, IBM, Face++) have been criticized due to their asymmetric accuracy across su…
Figure 10
Figure 10. Figure 10: Distribution of influence scores with varying perturba [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Pearson ranked correlation between influence scores [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: More examples of lowest-18 influential images in ‘au [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Top 400 influential images for SimCLR on CIFAR100. [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: Lowest 400 influential images for SimCLR on CIFAR100. [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: Top 400 influential images for supervised ResNet-18 on CIFAR100. [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Lowest 400 influential images for supervised ResNet-18 on CIFAR100. [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.