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REVIEW 4 major objections 4 minor 103 references

Tracing sources of epistemic uncertainty in deep learning predictions: homo- and hetero-scedastic linearized estimators

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

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Pith's one-line read Using only a single trained network, the paper derives two linearized variance estimators—Ho for homoscedastic response noise and HeC3 for finite-sample variability—and shows empirically that their ratio tells whether a test point's…

desk verdict A solid, honest adaptation of classical HC variance estimators to deep learning; the source-attribution ratio is heuristic, but the experiments support it. read the letter →

arxiv 2608.07630 v1 pith:4LDTHVKZ submitted 2026-08-07 cs.LG stat.ML

Pierre Nodet , Thomas George This is my paper · ORCID
classification cs.LGstat.ML MSC 62J0762F4068T07
keywords epistemicuncertaintyaleatoriclinearizedneuralnetworkstangentfeaturesjackknifeestimatorheteroskedasticity-consistentEKFACFisherinformationmatrix
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's goal is to make a practical, per-test-point distinction between two sources of uncertainty in deep learning predictions: aleatoric uncertainty from response noise and epistemic uncertainty from finite training data. It derives two estimators from linear-regression theory, Ho and HeC3, after linearizing the network around its trained weights and treating tangent features as fixed. Ho estimates the variance a prediction would have if only response noise were resampled under homoscedastic noise; HeC3 estimates the variance from resampling both inputs and responses, using jackknife residuals. Experiments on toy problems, a regression dataset, and controlled MNIST and Fashion-MNIST subpopulations show the two estimators disagree where the sources differ, so their ratio indicates whether to collect more data or clean labels. EKFAC approximation makes the estimators scalable to modern architectures, giving uncertainty rankings competitive with ensembles at a fraction of the cost.

What carries the argument

The load-bearing object is the tangent-feature linearization: replace the network by its first-order Taylor expansion around the trained weights $\hat w$, with tangent features $\phi_{\hat w}(x) = \partial f_w(x)/\partial w|_{w=\hat w}$ treated as fixed. This makes the deep network formally identical to linear ridge regression, with design matrix $\Phi_{\hat w}$ and regularized Fisher information matrix $F_{\hat w\lambda} = \Phi_{\hat w}^\top \Phi_{\hat w} + \lambda I$, so textbook variance formulas apply. The two estimators are a homoscedastic ridge estimator, Ho, and a jackknife/sandwich estimator, HeC3, whose middle term uses leave-one-out residuals rather than raw residuals. To scale the inverses to deep architectures, the paper approximates $\Phi_{\hat w}^\top \Phi_{\hat w}$ with block-diagonal EKFAC, an eigenvalue-corrected Kronecker factorization that makes inverse Fisher-vector products efficient.

What would settle it

Take one architecture and one data distribution, draw many bootstrap training sets, retrain from scratch on each, and record the true per-test-point prediction variance; if the ranks produced by the Ho and HeC3 estimates diverge from this true variance as width shrinks or as test inputs move outside the training support, the fixed-tangent-features linearization is the assumption to reject.

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Extended reading notes

Core claim

The central claim is that the distinction between aleatoric and finite-sample epistemic uncertainty can be estimated per test point from a single trained model. For a network trained by ridge-regularized mean squared error, the paper linearizes the predictor as $f_w(x) \approx f_{\hat w}(x) + \phi_{\hat w}(x)^\top \Delta w$, where $\phi_{\hat w}(x)$ are the tangent features, the gradients of the output with respect to the weights at the trained parameters. This turns deep-learning prediction into linear ridge regression with design matrix $\Phi_{\hat w}$ and pseudo-responses $r = y - f_{\hat w}(X)$, so the classical estimators carry over directly. Equation 16, $\hat{\mathrm{var}}_{\mathrm{Ho}} = \hat\sigma^2 \phi_{\hat w}(x)^\top F_{\hat w\lambda}^{-1} F_{\hat w} F_{\hat w\lambda}^{-1} \phi_{\hat w}(x)$, estimates the variance that would remain if the training inputs were fixed and only response noise were resampled under a homoscedastic assumption. Equation 17, $\hat{\mathrm{var}}_{\mathrm{HeC3}} = \phi_{\hat w}(x)^\top F_{\hat w\lambda}^{-1} \Phi_{\hat w}^\top \mathrm{diag}(\hat u^2) \Phi_{\hat w} F_{\hat w\lambda}^{-1} \phi_{\hat w}(x)$, is a jackknife and sandwich estimator built from leave-one-out residuals $\hat u_i = \hat e_i / (1 - h_{ii})$, and estimates the total variance from resampling both inputs and responses. The paper argues that the two estimators respond differently to the two sources: HeC3 is relatively larger around overlapping classes or noisy labels, while both are large in data-sparse regions, making the ratio a practical source-attribution tool.

Load-bearing premise

The load-bearing premise is that the network's prediction depends linearly on its weights across the random training sets being averaged over, so tangent features computed once at the trained weights stay representative; if higher-order terms matter under retraining or distribution shift, both variance estimates become biased and the source attribution may fail.

Editorial extensions

If this is right

  • At a test point where HeC3 is large and Ho is small, the dominant source is local label noise; the paper's controlled experiments indicate that cleaning noisy labels improves those predictions.
  • At points where both HeC3 and Ho are large, the dominant source is scarce local data; adding training examples in that region is the indicated remedy.
  • The ratio of HeC3 to Ho nearly identifies a noisy subpopulation in controlled settings, with AUROC up to about 0.99, so a single number per test point can flag heteroscedastic noise.
  • Computed with EKFAC, the estimators need only inverse Fisher-vector products and are competitive with a ten-model ensemble for misclassification detection on CIFAR-10, at a fraction of the compute.

Reading between the lines

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

  • Because the HeC3/Ho ratio is a linear smoother of leave-one-out squared residuals with tangent-feature alignment weights, it naturally doubles as a local conditional-variance estimator; a testable extension is to calibrate a threshold on this ratio for selective prediction.
  • The fixed-tangent-features premise is most credible for wide networks after the initial representation-learning phase; comparing these estimates against full-retraining Monte Carlo variances on narrow networks or shifted test distributions would map where the linearization breaks.
  • For very large models, using only last-layer features is cheaper, and the paper finds HeC3 more stable in that regime; an implicit extension is to deploy HeC3 on last-layer features when full EKFAC is too costly.
  • A natural next step, flagged by the authors, is to derive analogous estimators through logistic-regression diagnostics, which would give source-aware uncertainty for softmax classifiers and generative transformers.
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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 / 4 minor

Summary. The paper adapts two classical linear-regression variance estimators to deep neural networks. The homoscedastic estimator Ho (Eq. 16) is derived from the fixed-design, homoscedastic-noise variance formula after replacing inputs by tangent features; the heteroscedastic-consistent estimator HeC3 (Eq. 17) is derived from the jackknife estimator for random designs. The authors scale both estimators with an EKFAC approximation of the Fisher information matrix, and evaluate them on 1D/2D toy tasks, MNIST/Fashion-MNIST meta-class tasks with controlled noisy and missing subpopulations, and CIFAR-10/IMDB benchmarks. The central claim is that Ho and HeC3 quantify distinct sources of predictive uncertainty: response noise versus finite sampling of (X, ε), and that their ratio identifies test points whose uncertainty stems mainly from local label noise or from locally scarce data.

Significance. If the source-attribution claim holds, the paper provides a practical, single-model alternative to ensembles for decomposing predictive uncertainty into aleatoric and finite-sample components, with an approximation that scales to modern architectures. The paper has clear strengths: the linear-regression derivations are clean and the toy experiments include Monte Carlo ground truth, which supports the behavior of both estimators for small models. The controlled meta-class experiments are a good design for probing subpopulation-level source attribution, and the comparison against Deep Ensemble on CIFAR-10 is informative. The main weakness is that the deep-learning version of the estimators relies on an unquantified linearization assumption: tangent features are anchored at a single fitted ŵ that is itself a random function of the training data, and the authors explicitly state in Section 2.3 that this requires higher-order terms to be negligible across different draws of X and ε. Since the source-attribution claim is exactly about variability across training samples, this assumption is load-bearing and is not yet validated by a retraining-based ground truth on the real-scale tasks.

major comments (4)
  1. [§2.3, Eq. (12) and Eqs. (16)–(17)] The central claim that Eq. 16 and Eq. 17 estimate variance from response noise ε and from finite sampling of (X, ε) relies on the validity of the first-order expansion around a single fitted ŵ across different training draws. The paper states in Section 2.3 that this is 'equivalent to making the assumption that orders greater than 1 are negligible when considering different samples X and ε,' but no evidence is provided that the tangent features φ_{ŵ} remain representative when ŵ changes under retraining. The missing-subpopulation experiments in Section 5 are precisely the regime where test points lie far from the training manifold, so quadratic terms can be large. I recommend adding a bootstrap-retraining comparison on the meta-class tasks: retrain the model on multiple subsamples (or with different label-noise draws), compute the empirical variance of predictions, and compare it with the Ho/HeC3 estimates. This directly tests whether the fixed-feature approximation preserves the source attribution.
  2. [§2.3 and Appendix A.2] The deep-learning HeC3 estimator uses the Sherman–Morrison leave-one-out formula with tangent features anchored at the full-data optimum ŵ, but the leave-one-out prediction of the actual nonlinear network under retraining is not equal to this linearized quantity. In linear regression the jackknife formula is exact; in the nonlinear setting it is an additional approximation whose error is not measured. The 1D toy experiment provides Monte Carlo ground truth for one small MLP, but the meta-class and CIFAR experiments do not compare HeC3 against retrained ensembles or jackknife-by-retraining. A concrete experiment with a small number of retrained models on the meta-class tasks (or on a subset of MNIST) would clarify whether the estimator tracks the true leave-one-out variability.
  3. [§5, Table 1, and Appendix A.5] The 'source disentanglement' is presented as a heuristic ratio analysis rather than a formal decomposition. Appendix A.5 shows that HeC3/Ho is a linear smoother of squared jackknife residuals normalized by the global variance; it is therefore a conditional variance estimator, not a direct estimator of the aleatoric component of the total predictive variance. The empirical support for source attribution is strong only for the noisy subpopulation (HeC3/Ho AUROC 0.97–0.99), while the missing subpopulation AUROCs are moderate (0.58–0.80 for Ho and HeC3 individually). The text says 'both variances are equally high' for the missing subpopulation, but the reported AUROC values do not clearly demonstrate that both estimators reliably identify the missing subpopulation. The claims in this section should be softened or supplemented with additional statistics, such as the actual separation of variance distributions for the two subpopulations.
  4. [§2.2 and §2.3] The jackknife estimator is described as 'free of any assumptions,' but this is only true in the linear-regression setting from which it is borrowed. After linearization, the deep-learning HeC3 estimator inherits the pseudo-response substitution r = y − f_{ŵ}(X), the fixed-feature approximation, and the EKFAC approximation with leverage clipping described in Appendix B.6. The pseudo-responses are not centered and include model misspecification, so interpreting Eq. 17 as the variance due to finite sampling of (X, ε) requires an additional assumption that the misspecification term is absorbed into the noise. This should be stated explicitly and, if possible, tested by comparing the deep-learning estimators with true retraining-based variances on a small-scale task.
minor comments (4)
  1. [§2.3, Eq. (15)–(16)] The symbol F_{ŵ} is used in Eqs. (15) and (16) but is not defined; it should be defined as F_{ŵ} = Φ_{ŵ}^T Φ_{ŵ}, to distinguish it from F_{ŵλ} = Φ_{ŵ}^T Φ_{ŵ} + λI.
  2. [Abstract and throughout] There are several formatting and spelling errors, including 'uncertaintytomoderndeeplearning' in the abstract and 'each test points' instead of 'each test point'. The text also has missing spaces in a few places (e.g., 'experimentsaremeant' and 'GalandGhahramani2016'). A careful copyedit is needed.
  3. [§4 and Appendix A.4] The EKFAC approximation of the meat matrix Φ^T diag(û^2) Φ is not jointly diagonalizable with Φ^T Φ, and the paper considers two approximations: one using the same basis U and one using a different basis V. The accuracy of these approximations is not evaluated. A small experiment comparing exact (or Cholesky-based) and EKFAC-based HeC3 variances would help assess how much of the source-attribution signal is lost in the approximation.
  4. [Appendix B.6] The clipping of approximate leverages to [0, 1−ε] with ε = 1×10⁻⁴ is an ad hoc correction that may affect the jackknife residuals for high-leverage points. The sensitivity of the HeC3/Ho ratio to this clipping choice is not reported.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: estimators are adapted from classical linear-regression statistics, validated against independent Monte Carlo ground truth, and the residual-smoother identity is explicitly derived rather than hidden.

full rationale

The paper's derivation chain is self-contained and non-circular. Equations (8)-(11) reproduce textbook linear-regression variance estimators with proofs in Appendix A, and Equations (12)-(17) are obtained by an explicitly stated linearization ansatz (Section 2.3: 'This is equivalent to making the assumption that orders greater than 1 are negligible when considering different samples X and ε'), not by fiat or by fitting the target result. The toy experiments of Section 3 validate Ho and HeC3 against Monte Carlo estimates computed by independent resampling procedures, so the agreement is not imposed by construction. The Appendix A.5 identity (HeC3/Ho as a linear smoother of jackknife residuals) is an explicit algebraic reduction proved by the authors; it transparently explains the mechanism behind the noisy-subpopulation result instead of covertly assuming it, and the cited econometrics literature is acknowledged. The EKFAC approximation is a self-citation to George et al. (2018), but it is a published, independently evaluated algorithm used as an implementation tool, and the paper additionally reports last-layer (Dense) results that bypass EKFAC, so no load-bearing claim reduces to the self-citation. The main limitation—the quality of fixing tangent features at the trained weights under different training samples—is flagged by the authors in Section 6 as needing further quantification; it is an accuracy/robustness assumption, not a circular dependence. No circular step meeting the quoted-evidence bar was found.

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

The paper introduces no new physical or model entities. Its burden lies in the linearization and feature-fixity assumptions, the one-hot regression reformulation for classification, and the hand-chosen regularization and clipping constants.

free parameters (2)
  • ridge regularization lambda = weight decay value used during optimization, e.g. 1e-2 for MNIST/CIFAR/IMDB
    Lambda enters every estimator through F_{ŵλ} and must be chosen by hand. Appendix D studies its influence but does not provide a principled selection rule, so the reported results depend on this choice.
  • EKFAC leverage clipping epsilon = 1e-4
    In Appendix B.6, leverages computed with EKFAC are clipped to the range [0, 1-epsilon] with epsilon = 1e-4, to avoid negative denominators in 1/(1-h_ii). This is a hand-chosen constant.
assumptions (5)
  • domain assumption Training converges to a minimizer of the ridge-regularized MSE, and the loss landscape near ŵ is such that the linearization in Eq. 12 has negligible higher-order terms.
    Section 2.3 invokes the tangent-features linearization and explicitly assumes orders greater than 1 are negligible when considering different samples X and ε.
  • domain assumption The tangent features φ_{ŵ} are fixed and independent of the particular noise realization or training sample, so variance over X and ε can be computed as if the design matrix were fixed.
    The transfer from linear regression to deep learning in Eqs. 14-17 replaces X by Φ_{ŵ} and treats Φ_{ŵ} as a constant, which requires that feature learning does not change the relevant linearized dynamics.
  • standard math IID samples from p(x,y) and existence of a ridge penalty λ ≥ 0; the Bayes predictor y*(x) = E[y|x] exists.
    Stated in Section 2 as the data-generating process and the regression setup.
  • domain assumption For classification, training the network on one-hot encoded targets with MSE loss approximates the classification task well enough for uncertainty estimation.
    Appendix B.1 reformulates classification as least-squares regression with targets +1/-1; this is a nontrivial modeling choice that affects the meaning of residuals and hence both estimators.
  • domain assumption The EKFAC block-diagonal plus Kronecker approximation of the Fisher information matrix and of the meat matrix preserves the ranking properties of the exact estimators.
    Section 4 assumes EKFAC is accurate enough for uncertainty estimates; Appendix B.6 acknowledges that approximate leverages can exceed 1 and require clipping.

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Cite this review

Pith. "Pith review of Tracing sources of epistemic uncertainty in deep learning predictions: homo- and hetero-scedastic linearized estimators." pith.science (2026). https://pith.science/paper/4LDTHVKZ

@misc{pith2026260807630,
  author       = {Pith},
  title        = {Pith review of: Tracing sources of epistemic uncertainty in deep learning predictions: homo- and hetero-scedastic linearized estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4LDTHVKZ}},
  note         = {Machine review of arXiv:2608.07630}
}
read the original abstract

We adapt two classical statistical estimators for quantifying uncertainty to modern deep learning, in order to provide clearer insights into uncertainty attributable to two sources : aleatoric uncertainty, or locally scarce data. Our approach leverages recent advances in approximate Fisher Information Matrices, to enable scaling to actual architectures. Experimental results demonstrate how each test points is differentially impacted by both sources, highlighting the practical utility of our estimators in improving the robustness of real-world applications.

Figures

Figures reproduced from arXiv: 2608.07630 by the authors.

Figure 1
Figure 1. Regression on the California Housing dataset. The [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Estimated confidence interval of neural networks with different estimators. Ground truth is computed using Monte [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. (left) Log-variance estimated using Ho and HeC3. Ground truth variances are estimated using Monte Carlo (100 trials), by resampling a full new dataset or by adding homoscedastic label noise on a fixed dataset. (right) Disagreements between the log-variance computed using Ho and HeC3. Regions of the feature space are colored in blue if the model is uncertain because of the sampling of (X, ε) or in red because of ε. w… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: We rank test examples on MNIST using the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Nonlinear regression task with (from top to bottom) homoscedasticity, heteroscedasticity near [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: We compare the regularized Hoλ and the unregularized Ho0 of the Ho uncertainty estimator on test examples of MNIST (top) and Fashion-MNIST (bottom) and measure whether examples with high uncertainty are examples from the missing subpopulation, for a number of λ values …
Figure 7
Figure 7. Figure 7: We compare different uncertainty estimators on test examples of MNIST (top) and Fashion-MNIST (bottom) and [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: umap (McInnes, Healy, and Melville 2018) projections of last layer embeddings of the different neural network [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Test examples of MNIST, Fashion-MNIST and CIFAR-10 sorted by their variances. The 100 examples with the highest [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Test examples of MNIST, Fashion-MNIST and CIFAR-10 sorted by the agreements between their [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

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Pith tools

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