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REVIEW 3 major objections 4 minor 65 references

Lazy training reduces deep-network feature-importance testing to a z-test.

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2026-08-01 03:57 UTC pith:6LY73U2J

load-bearing objection A plausible and well-motivated extension of lazy-training variable importance to binary classification, but the central theorem is not fully established because the Donsker condition (R3) is verified only for fixed network width W, which grows with n. the 3 major comments →

arxiv 2607.22979 v1 pith:6LY73U2J submitted 2026-07-25 stat.ML cs.LG

Variable Importance Identification Through Lazy Training for Binary Classification

Anand Singh , Luke Pennella , Eshan Kabir , Xiaoxi Shen This is my paper
classification stat.ML cs.LG MSC 62G1062G2068T07
keywords variable importancelazy trainingneural tangent kernellocal Rademacher complexitybinary classificationhypothesis testingdeep ReLU networksasymptotic normality
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 tries to establish that, for binary classification with deep ReLU networks, you can test whether a feature subset matters without retraining the network for every candidate subset. It shows that under a spectral decay condition on the neural tangent kernel and a slow-growth condition on the number of network weights, the estimated variable-importance measure is asymptotically linear with a remainder that is smaller than the usual n^{-1/2} sampling noise. This makes the importance estimate divided by its standard error asymptotically standard normal under the null, so feature significance becomes a classical z-test. The algorithm trains one network, linearizes it around the fitted parameters using the neural tangent kernel, and solves a penalized logistic regression for each feature subset, avoiding expensive retraining. Simulations and an MNIST experiment support the claimed behavior.

Core claim

The central claim is Theorem 25: for any candidate feature set S, under the null hypothesis and with W = o(sqrt(n)/log n) network weights, the estimated variable-importance difference psi-hat minus psi-0 equals a sample average of influence-function contrasts plus op(n^{-1/2}). If this holds, the lazy variable-importance statistic is asymptotically normal with a variance that can be estimated from the same fitted network, so a valid z-test for feature importance can be run without refitting deep networks for each feature set. The paper's key claimed improvement is the op(n^{-1/2}) remainder rather than the Op(n^{-1/2}) rate achieved in prior work, attributed to the use of local Rademacher co

What carries the argument

The neural tangent kernel (NTK) matrix K_{-S}, formed from the gradients of the trained network evaluated at inputs with the candidate feature subset replaced by its mean. The lazy-regime linearization h_theta approximately equal to h_theta-f plus the gradient inner product with (theta - theta-f) turns each feature-removal fit into a ridge-regularized logistic regression in the RKHS of the NTK, solved by Newton-Raphson. The convergence argument rides on the local Rademacher complexity of the RKHS ball H_B, whose fixed point is controlled by Assumption (A1): the eigenvalues mu_j of K_{-S}/n decay as mu_j <= c j^{-alpha} with alpha > 1. That eigenvalue decay is what converts the estimation err

Load-bearing premise

The load-bearing premise, which the authors state they cannot currently prove for deep ReLU networks, is that the neural tangent kernel matrix at the fitted parameters has eigenvalues decaying as mu_j <= c j^{-alpha} with alpha > 1; without this spectral decay, the local-Rademacher rate and the op(n^{-1/2}) remainder collapse.

What would settle it

Compute the sorted eigenvalues of the NTK matrix K_{-S} for a deep ReLU network at converged parameters on a moderate-size nonlinear classification problem and fit the log-log slope of the tail. A slope with alpha <= 1, or a heavy tail that persists as network width grows, would invalidate the local-Rademacher rate and the claimed op(n^{-1/2}) result.

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

If this is right

  • Feature-importance inference for binary deep classifiers requires only one full-network training pass; every candidate feature subset is handled by a cheap linearized logistic regression.
  • Under the null hypothesis, the LazyVI test statistic is asymptotically standard normal, enabling p-values and confidence intervals without retraining or permutation tests.
  • The required regime is underparameterized: the number of network weights must grow slower than sqrt(n)/log n.
  • The likelihood-based setup extends to other exponential-family outcomes whenever the negative log-likelihood satisfies the convexity and Lipschitz conditions used in the proofs.

Where Pith is reading between the lines

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

  • If Assumption (A1) transfers from initialization to fitted parameters, the same framework may extend to overparameterized networks; the paper identifies this as an open bottleneck rather than a proven result.
  • A natural stress test is to run LazyVI on nonlinear simulations with eigenvalue decay alpha close to 1, where the local-Rademacher rate becomes borderline and the z-test approximation should visibly degrade.
  • The n-by-n NTK matrix is the computational bottleneck; Nystrom or block approximations to K_{-S} could plausibly scale the method to biobank-sized data, a direction the paper mentions but does not implement.

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 / 4 minor

Summary. The paper proposes LazyVI, a variable-importance testing procedure for binary classification with deep ReLU networks under the lazy-training / neural tangent kernel regime. The predictiveness measure is the expected logistic likelihood, and the estimator uses one fitted network plus a ridge-penalized linearized (NTK) fit for each feature subset, avoiding retraining. The main theoretical claim (Theorem 25) is that, under an eigenvalue decay condition on the NTK at the fitted parameters (A1), a regularization-rate condition (A2), and W = o(sqrt(n)/log n), the variable-importance estimator is asymptotically linear with an op(n^{-1/2}) remainder, yielding a valid z-test. The paper also reports simulations, MNIST experiments, an ADNI gene-association application, and a software implementation. The appendices contain detailed proofs of local Rademacher bounds, NTK-RKHS estimates, and the verification of the variable-importance framework conditions.

Significance. If Theorem 25 were established, it would be a practically useful result: it would provide first-order inference for feature importance in binary ReLU classifiers without retraining for each feature subset, improving on the Op(n^{-1/2}) rate in prior lazy-training work. The paper is also transparently written: it ships code, gives explicit conditions, includes large-scale simulation support, and lays out the local-Rademacher machinery in detail. These are genuine strengths. However, the central theoretical claim is not supported as written. The verification of Williamson et al.'s condition (R3) is incomplete because the relevant function class grows with W_n, and the main rate improvement rests on an unproven spectral assumption on the NTK at the fitted parameters. These are load-bearing gaps, not presentation issues.

major comments (3)
  1. [Appendix E.3, Condition (R3)] The proof defines D = {Z ↦ ℓ(f(X),Y) − ℓ(f0(X),Y) − E_P0[ℓ(f(X),Y) − ℓ(f0(X),Y)] : f ∈ F} and bounds log N(ε,F,||·||_sup) ≤ W log(1 + C/ε) via Theorem 36. For fixed W this would give a finite entropy integral and a Donsker class. But Theorem 25 assumes W = W_n = o(√n/log n), so W_n grows with n. Condition (R3) of Williamson et al. requires a single fixed P0-Donsker class D0 such that P0(g_n ∈ D0) → 1. The proof never supplies such a fixed class: D_n changes with n, and its entropy integral is O(√W_n), which diverges as n → ∞. Thus the empirical process indexed by D_n need not converge to a tight Gaussian limit, and the op(n^{-1/2}) expansion in Theorem 25 does not follow. This gap is independent of Assumption (A1); even granting A1, the Donsker verification is incomplete.
  2. [§3.2, Assumption (A1)] The faster rate and Theorem 25 depend critically on Assumption (A1): the eigenvalues of the NTK matrix K_{-S} at the fitted parameters decay as μ_j ≤ c j^{-α} with α > 1. The paper states explicitly in §3.2: 'we are currently unable to provide a theoretical proof that Assumption (A1) holds for the class of deep ReLU neural networks' and extrapolates from NTK results at initialization. Since Lemma 23, Theorem 24, and the resulting o(n^{-1/4}) rates used for condition (R1) all rely on this spectral assumption, the paper's central claim is conditional on an unproved structural conjecture. The empirical check in §4.2 is a single simulation setup and does not establish the assumption for general ReLU networks. The abstract's statement that the method relies on 'only a minimal set of assumptions' overstates the situation when a key assumption is conjectural.
  3. [§E.3, Condition (R1)] The proof verifies condition (R1) using L2(P) convergence rates from Theorem 15 and Theorem 24. However, Theorem 7's condition (R1) is stated in terms of the function-class norm ||·||_F of F, and the paper never specifies which norm ||·||_F represents. If the authors intend ||·||_F to be the L2(P) norm, this should be stated explicitly and used consistently in conditions (D1) and (D2). If a different norm is intended, the L2(P) bounds do not establish the required function-class-norm convergence. This ambiguity is load-bearing because (R1) is one of the three random conditions needed for the asymptotic linear expansion.
minor comments (4)
  1. [§4.1.3] The runtime comparison sentence contains a typo: 'LazyVInonlinear required approximately 21,534 seconds in the linear setting and 33,766 seconds in the linear setting.' The second 'linear setting' should presumably read 'nonlinear setting.'
  2. [Lemma 34] The class of functions defined in equation (55) is denoted M, but the lemma statement uses M before its definition; the notation is confusing. Also, the covering-number bound should make explicit the dependence on the Lipschitz constant of the activation, which is stated but not reflected in the constants.
  3. [§4.3.3] Two consecutive paragraphs give conflicting summaries: the first says 'both halves were significant with p<0.001' and identifies region 10 as most important, while the second says the top half had p≈0.388 and the Bottom-Left quadrant was not significant. The text should be reconciled.
  4. [Algorithm 1] The line 'Ensure::' contains a double colon typo.

Circularity Check

0 steps flagged

No circular reduction found: Theorem 25 is a conditional application of Williamson et al.'s external framework, and the paper's own caveats identify unproved assumptions, not circularity.

full rationale

The derivation chain is not circular. Theorem 25 is obtained by verifying conditions (D1), (D2), (R1), (R2), (R3) from Williamson et al. (2023); the central conclusion is an asymptotic linear expansion of the same form as that framework's Theorem 1, but it is derived from stated assumptions rather than being assumed. Assumption (A1) on NTK eigenvalue decay is an explicit input, and the paper candidly states it cannot prove it for deep ReLU networks at fitted parameters: 'Although we are currently unable to provide a theoretical proof that Assumption (A1) holds...' Assumption (A2) fixes the regularization order; these are assumptions, not fitted constants renamed as predictions. The local Rademacher rates in Lemmas 23-24 conditionally use (A1) to produce op(n^{-1/2}); that is a derivation from assumptions, not an equivalence by construction. The self-citations (Shen et al.; Shen and Wang; Shen and Espinoza) appear in related-work or analogous-condition contexts and are not load-bearing. The manuscript itself flags the real technical bottleneck in Section 5: the entropy bound 'yields a growth rate of order O(log 1/epsilon), but at the cost of dependence on the total number of network parameters.' This corresponds to the incomplete Donsker verification under W_n -> infinity in condition (R3), but an incomplete proof or unproved assumption is a correctness risk, not circularity. No step in the paper reduces its claimed prediction to its own inputs by definition or by self-citation.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The central claim rests on two load-bearing assumptions that are not independently established: the NTK spectral decay at fitted parameters (A1) and the choice of lambda order (A2). No new physical or mathematical entities are introduced. The proof of the main theorem also relies on Williamson et al.'s conditions, and the verification of those conditions is incomplete for growing network width.

free parameters (2)
  • Ridge penalty lambda = selected by 3-fold CV from 10 values in [1e-4, 1e-2.25] in simulations, MNIST, and ADNI
    In theory Assumption (A2) fixes lambda's asymptotic order; in practice it is chosen by cross-validation with no guarantee that the selected value satisfies A2. The theorem's rates depend on lambda's order.
  • Network architecture and training hyperparameters = two hidden layers of 50 ReLU neurons, L-BFGS learning rate 5e-4, early stopping patience 10
    Used in all experiments. The theory only requires W = o(sqrt(n)/log n) and fixed kappa_i, b_i, so these exact values are implementation choices rather than fitted scientific parameters.
axioms (6)
  • domain assumption (X,Y) are iid from P0, X is bounded by kappa, Y is Bernoulli, and logit(P(Y=1|X)) belongs to the network class F.
    Section 3.1. Needed for Proposition 8 and for f0 to be the population risk minimizer.
  • domain assumption The deep ReLU network class has fixed weight bounds kappa_i, b_i with b_i >= kappa_i.
    Section 3.1 and Lemma 13. Needed for uniform boundedness and covering-number bounds.
  • ad hoc to paper Assumption (A1): eigenvalues of (1/n)K_{-S} decay as mu_j <= c j^{-alpha} with alpha > 1, where K_{-S} is the neural tangent kernel at the fitted parameters.
    Section 3.2. The authors state they cannot prove this for deep ReLU networks at fitted parameters; the faster local-Rademacher rate depends on it.
  • ad hoc to paper Assumption (A2): lambda = (1 or ||K_{-S}||_op/4) * xi_n with xi_n = omega(n^{-(alpha-1)/(2(alpha+1))} or n^{-1/4}).
    Section 3.2. The regularization order is chosen precisely so that the linearization error and estimation error reach the claimed rates.
  • domain assumption W = o(sqrt(n)/log n) and kappa_i, b_i are fixed constants.
    Equation (16) and Section 3.3. Needed for the o(n^{-1/4}) L2 rate of the fitted network.
  • standard math Background empirical-process tools: Bartlett et al. local Rademacher bounds, Ossiander's entropy-integral Donsker criterion, Hsu et al. concentration for quadratic forms, Ledoux-Talagrand contraction.
    Used throughout the proofs in Appendices B-F; these are standard results the paper does not re-derive.

pith-pipeline@v1.3.0-alltime-deepseek · 65072 in / 13954 out tokens · 133043 ms · 2026-08-01T03:57:20.882448+00:00 · methodology

0 comments
read the original abstract

Deep neural networks have been widely used in many applications (e.g., computer vision and natural language processing); however, understanding their explainability remains a challenging task. Recently, substantial research has been devoted to improving the explainability of deep neural networks, with most of this work focusing on the regression framework. In this paper, we instead focus on the binary classification framework and adopt a variable-importance framework combined with the idea of lazy training to propose an efficient algorithm for identifying important features. From a theoretical perspective, our method relies on only a minimal set of assumptions and achieves well-controlled error rates. The validity of the proposed method and algorithm is examined through extensive simulation studies and real-data applications.

Figures

Figures reproduced from arXiv: 2607.22979 by Anand Singh, Eshan Kabir, Luke Pennella, Xiaoxi Shen.

Figure 1
Figure 1. Figure 1: Graphical illustration of the three essential quantities in the main result. (I) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Log-log plot of the sorted NTK eigenvalues. The spectrum decays faster than the red [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representative samples of digits ‘8’ and ‘9’ from the test set. The overlay illustrates the [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heatmaps of Variable Importance (VI) across three granularity levels. Columns compare [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Heatmaps of Lazy Training p-values across three levels of granularity. The p-values are multiplied by 2, 4, or 16 for the Top vs Bottom, Quadrant, and 16-Region respectively for the Bonferroni correction with values capped at 1, then are transformed using − log10(max{0.0001, p}) to enhance visual contrast and highlight significance. A score at approximately 1.3 and above corresponds to p ≤ .05. 24 [PITH_F… view at source ↗

discussion (0)

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Reference graph

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