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REVIEW 3 major objections 5 minor 2 cited by

Better Training Data Attribution via Better Inverse Hessian-Vector Products

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper shows that more accurate inverse Hessian-vector products, computed by using the EKFAC curvature estimate as a preconditioner for Neumann series iterations, substantially improve training data attribution across architectures and…

desk verdict ASTRA is a sensible, well-tested improvement for TDA that deserves review, with a few fixable gaps. read the letter →

arxiv 2507.14740 v1 pith:DHPN2GYZ submitted 2025-07-19 cs.LG stat.ML

Andrew Wang , Elisa Nguyen , Runshi Yang , Juhan Bae , Sheila A. McIlraith , Roger Grosse This is my paper
classification cs.LGstat.ML
keywords trainingdataattributioninfluencefunctionsinverseHessian-vectorproductEKFACNeumannseriespreconditioninglineardatamodelingscoreunrolleddifferentiation
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

Training data attribution asks which training examples drive a model's behavior. For gradient-based methods such as influence functions and unrolled differentiation, the hard step is computing an inverse Hessian-vector product (iHVP), an expensive operation usually approximated with either the cheap but biased EKFAC curvature model or slow iterative Neumann series. This paper introduces ASTRA, which uses the EKFAC approximation as a preconditioner for stochastic Neumann series iterations, yielding a more accurate iHVP in a few hundred iterations. The authors show that this improved iHVP accuracy raises influence-function and SOURCE attribution scores, particularly for convolutional networks and when ensembled over training seeds: LDS reaches about 0.5 for single models and 0.6 with ensembling, and on CIFAR-10 ResNet-9 ensembled ASTRA-IF jumps from 0.25 to 0.6. The paper's central claim is that the quality of the iHVP approximation is a major, addressable bottleneck in training data attribution.

What carries the argument

EKFAC-preconditioned Stochastic Neumann Series (ASTRA) is the central algorithm, defined by the update $\theta_{k+1} \leftarrow \theta_k - \alpha(P+\tilde{\lambda}I)^{-1}(\tilde{G}_k+\lambda I)\theta_k + \alpha(P+\tilde{\lambda}I)^{-1}\nabla f_{z_q}$, with preconditioner $P := G_{\mathrm{EKFAC}}$. The method reframes the iHVP as minimizing a high-dimensional quadratic objective, uses the EKFAC block-diagonal eigendecomposition to improve conditioning, and relies on the identity that a Neumann series approximates the inverse of a scaled damped GGN. The paper additionally derives that truncating the series at $J$ steps adds an effective damping term $1/(\alpha J)$ that hits low-curvature directions hardest, and uses the EKFAC eigendecomposition to bin eigenvalues and show that low-curvature components are crucial for influence-function performance.

What would settle it

On a small model where the full GGN eigendecomposition is feasible, compute the exact $(G+\lambda I)^{-1}v$ for a query gradient and compare the error norms of EKFAC, ASTRA, and SNI. If ASTRA's output is not consistently closer to the exact iHVP than EKFAC's one-step output, or if the LDS gap between ASTRA-IF and EKFAC-IF disappears when both use an exact iHVP, the central claim is false.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that accurately solving the iHVP typically yields better training-data attribution than the EKFAC one-shot solution, and ASTRA is the algorithm that provides that accuracy at acceptable cost. ASTRA reuses the EKFAC eigendecomposition, already computed for EKFAC-IF and EKFAC-SOURCE, as a preconditioner for Stochastic Neumann Series iterations on the quadratic objective in parameter space. One ASTRA step with learning rate equal to one reproduces the EKFAC approximation, and further iterations continue toward the true damped Gauss-Newton inverse. In the reported experiments, ASTRA-IF and ASTRA-SOURCE beat their EKFAC counterparts in LDS in almost all settings, with the largest gains on ResNet-9 CIFAR-10, and the authors demonstrate that Neumann series truncation adds implicit damping that disproportionately degrades low-curvature directions, which are shown to be essential for good influence estimates.

Load-bearing premise

The 200 to 300 ASTRA iterations used in the experiments land closer to the true damped Gauss-Newton inverse than the one-step EKFAC solution, and that improvement, rather than ensembling or another factor, is what raises the attribution scores.

Editorial extensions

If this is right

  • Influence functions computed with ASTRA's more accurate iHVP replace EKFAC-based influence functions in settings where LDS performance matters, at an incremental cost of a few hundred iterations per query.
  • Ensembling multiple training seeds becomes substantially more effective with accurate iHVPs, raising LDS from around 0.5 to 0.6 in many settings.
  • For convolution architectures, where EKFAC makes extra structural assumptions, ASTRA closes a large performance gap, lifting ensembled CIFAR-10 ResNet-9 LDS from 0.25 to 0.6.
  • ASTRA converges in fewer iterations and with a single practical learning-rate heuristic across settings, easing the notorious tuning burden of vanilla stochastic Neumann series iterations.

Reading between the lines

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

  • A direct head-to-head of ASTRA against a high-accuracy reference iHVP (for example, conjugate gradient or a full eigendecomposition on a small model) would settle whether the claimed 'more accurate' property is strictly true; the paper only measures downstream LDS, not iHVP error itself.
  • The same preconditioning recipe, a cheap biased curvature estimate plus a few correction iterations, could transfer to any iHVP-heavy task outside TDA, such as second-order optimization or Laplace approximations, where neither the biased one-step solution nor slow unpreconditioned iterations is satisfactory alone.
  • The low-curvature analysis implies that projection-based attribution methods that discard low-curvature directions may be throwing away exactly the signal ASTRA recovers; quantifying ASTRA's benefit under random or PCA gradient projections would test that implication.
  • For ASTRA-SOURCE, the paper leaves open how to compute the average GGN per segment; better averaging schemes could further improve unrolled-differentiation attribution, since only the $r_\ell$ term is refined while the $S_{\ell'}$ term remains EKFAC-based.
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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

3 major / 5 minor

Summary. The paper proposes ASTRA, an algorithm for computing inverse Hessian-vector products (iHVPs) in training data attribution (TDA). ASTRA combines the EKFAC-decomposed GGN as a preconditioner with stochastic Neumann series iterations (SNI), aiming to produce a more accurate iHVP approximation than the one-step EKFAC solution at modest additional cost. The authors apply ASTRA to influence functions (ASTRA-IF) and to the approximate unrolled differentiation method SOURCE (ASTRA-SOURCE), and evaluate TDA performance with Linear Datamodeling Score (LDS) against ground-truth retraining in seven settings, including MLPs, ResNet-9, GPT-2, and a non-converged benchmark. They report that ASTRA outperforms EKFAC-based counterparts in most settings, especially for convolution architectures, improves the benefit of ensembling, and converges in fewer iterations than vanilla SNI. The paper also analyzes the role of low-curvature directions and argues that truncating Neumann series introduces implicit damping, which motivates the preconditioned approach.

Significance. If the central claim is validated, ASTRA is a useful and practical contribution to TDA: it shows that iHVP solver accuracy is a key bottleneck, provides an algorithm with modest incremental cost over EKFAC, and demonstrates substantial LDS gains on convolution architectures and with ensembling. The evaluation is anchored to external ground truth through LDS with counterfactual retraining, and the fixed point of the preconditioned iteration is exactly the target iHVP, which gives a sound theoretical starting point. The paper also ships a clean ablation of low-curvature directions. However, the absence of a direct accuracy comparison to a high-accuracy iHVP reference and the incorrect 'unbiased' claim leave the interpretation of the gains partially open, so the significance of the accuracy improvement is not yet fully established.

major comments (3)
  1. [Section 5 (paragraph after Figure 2)] The statement that ASTRA 'computes an unbiased estimator of the iHVP' is not correct for the finite number of iterations used in the experiments. For any fixed J, the estimator contains the deterministic truncation bias (I - α(P+λ̃I)^{-1}(G+λI))^J θ_0, analogous to the unpreconditioned expression in Eq. (33). Consequently, the explanation that ASTRA enjoys a larger ensembling benefit because it is an unbiased estimator is unsupported; at most, the estimator is consistent as J grows. This claim should be corrected and the ensembling discussion adjusted accordingly.
  2. [Section 5, Figure 3 and Section 2.1] The central claim that ASTRA provides a more accurate iHVP approximation than EKFAC is not directly benchmarked against a reference solution. Although the quadratic-loss curves in Figure 3 show that ASTRA decreases the objective relative to its EKFAC initialization, the paper does not state whether the loss is computed with the full-batch G or a stochastic mini-batch estimate, and it never reports the distance of ASTRA's output to a high-accuracy reference (e.g., converged LiSSA with many iterations, CG on the full-batch G, or an exact solve on a small problem). Because truncation alters the effective damping (Appendix G, Eq. (35)), the LDS gains in Figure 2 could in principle be caused by a shift in implicit regularization rather than by solver accuracy. A direct accuracy comparison is needed to support the paper's title claim.
  3. [Appendix G, Eqs. (33)-(35); Section 6] The implicit-damping result is derived only for the unpreconditioned SNI update, yet the paper uses it to motivate why preconditioning accelerates convergence in low-curvature directions. No analogous effective-damping or truncation-bias characterization is provided for the preconditioned ASTRA update. As a result, the comparison between ASTRA-IF and EKFAC-IF at the same nominal λ (Section 5, Table 3) does not establish that both methods solve the same damped problem, and the role of α and J in ASTRA's effective regularization is left unquantified. This gap should be addressed by deriving the effective filter function for the preconditioned iteration or by empirically bounding the regularization shift.
minor comments (5)
  1. [Section 5, Figure 3 vs. Table 3] The learning rate for ASTRA in Figure 3 is reported as 10^{-2} for all settings, whereas Table 3 lists α = 0.1λ (or 0.01λ for CIFAR-10), which are generally different values. Please clarify whether the training-curve experiment uses different hyperparameters than the main LDS experiments, and state this explicitly if so.
  2. [Section 3, Eq. (5)] The text says Eq. (5) is a preconditioned gradient descent on the quadratic in Eq. (4). It would be helpful to state explicitly that the fixed point of Eq. (5) is the same as the target iHVP because the preconditioner is positive definite, since this is the theoretical basis for the algorithm.
  3. [Section 5, 'ASTRA speeds up iHVP approximation'] The speed comparison is reported in iteration counts (Figure 3). A wall-clock comparison, or at least an explicit statement of the per-iteration cost of applying the EKFAC preconditioner, would make the practical speed claim more precise.
  4. [Section 6, Figure 4] The subspace analysis uses the EKFAC eigendecomposition as a proxy for the true GGN eigenbasis. This is acknowledged, but it would be useful to note explicitly that the conclusions about low-curvature directions are only as strong as the EKFAC approximation.
  5. [Table 3 and Section F.2] The batch size for ASTRA is specified in Table 3, but it is not stated whether this same batch size is used for the mini-batch GGN estimator in the quadratic-loss curves of Figure 3. Please clarify the evaluation protocol for the loss curves.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: ASTRA's gains over EKFAC are benchmarked against external counterfactual retraining (LDS); a non-circular internal inconsistency about truncation bias is the main caveat.

full rationale

ASTRA's central claim is checked against an external ground truth: LDS (Eq. 6) correlates TDA scores with expected counterfactual retraining outcomes, where the ground truth comes from retraining models on random subsets with multiple seeds (Section 5 and Table 2, up to 10,000 retrains). Neither the ASTRA estimator nor the EKFAC approximation enters that ground-truth computation, so the comparison is not self-referential. Hyperparameters in Figure 3 are tuned to the quadratic objective h_fzq (Eq. 4), the same objective ASTRA is designed to minimize, but the Figure 2 LDS results use fixed heuristics from Table 3, and LDS (not merely the quadratic loss) is reported as the performance metric. SOURCE and EKFAC are prior works with overlapping authors (Bae, Grosse), but they serve as baselines and building blocks rather than as the validation target; no uniqueness theorem or load-bearing self-citation forces the ASTRA choice. The most notable internal tension is non-circular: Section 5 calls ASTRA 'an unbiased estimator of the iHVP' while Appendix G (Eqs. 33–35) shows finite truncation adds implicit damping 1/(αJ), so for J=200–300 the estimator is biased; this is a correctness gap (no exact-iHVP reference comparison), not a circularity. Overall circularity is negligible.

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

The central claim rests on the standard G about H approximation and on solver hyperparameters that are tuned per dataset. No new entities are invented. The low-curvature analysis additionally relies on the EKFAC eigenbasis as a proxy for true curvature.

free parameters (5)
  • GGN damping lambda for ASTRA-IF and ASTRA-SOURCE = per dataset: 0.0017, 0.00091, 0.0052, 0.014, 0.011, 0.10
    Table 3: a different lambda is used for each dataset; for IF it is the SOURCE-implied damping averaged over segments, for SOURCE it is 1/(eta_l K_l).
  • ASTRA learning rate alpha = 0.1 lambda, 0.01 lambda, or lambda depending on dataset
    Table 3 shows alpha = 0.1 lambda for most settings, 0.01 lambda for CIFAR-10, alpha = lambda for WikiText-2; Figure 3 uses a sweep and selects 1e-2 as best.
  • Preconditioner damping lambda_tilde = equal to lambda in all settings
    Table 3 sets lambda_tilde = lambda; this choice is not derived, though it is a natural heuristic.
  • Mini-batch size for ASTRA updates = 256, 128, 16, 8
    Table 3 reports different batch sizes per dataset; this influences stochastic noise in the iHVP estimate.
  • Number of iterations J = 200 (most), 300 (WikiText-2)
    Table 3 fixes the truncation point; the paper claims convergence within this budget, but does not verify against an exact iHVP.
assumptions (5)
  • domain assumption The Generalized Gauss-Newton Hessian (GGN) approximates the true Hessian: G about H.
    Used in Section 2.2 to define influence functions with (G+lambda I)^{-1} in place of H^{-1} (Eq 2). Standard in TDA literature.
  • domain assumption For the regression and classification tasks considered, the Fisher Information Matrix equals the GGN: F = G.
    Appendix B.3 derives this for exponential family outputs; required so that Hessian-vector products can be estimated with the GGN.
  • standard math Neumann series convergence requires alpha < 1/(sigma_max(G)+lambda) for the deterministic update; stochastic mini-batch updates inherit the same fixed point.
    Section 2.1 states the condition; ASTRA depends on the iteration in Eq 5 having the same fixed point as the target iHVP.
  • domain assumption A truncated Neumann series behaves approximately as increased damping, adding roughly 1/(alpha J) to lambda.
    Derived in Section G with approximations; used to motivate the low-curvature analysis in Section 6.
  • domain assumption The EKFAC eigendecomposition is a reliable proxy for the true curvature eigenspaces when binning eigenvalues and projecting influence scores.
    Section 6 uses EKFAC eigenvalues to define bins S1..S5 and projections; the true eigendecomposition is intractable for the reported models.

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Pith. "Pith review of Better Training Data Attribution via Better Inverse Hessian-Vector Products." pith.science (2026). https://pith.science/paper/DHPN2GYZ

@misc{pith2026250714740,
  author       = {Pith},
  title        = {Pith review of: Better Training Data Attribution via Better Inverse Hessian-Vector Products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHPN2GYZ}},
  note         = {Machine review of arXiv:2507.14740}
}
read the original abstract

Training data attribution (TDA) provides insights into which training data is responsible for a learned model behavior. Gradient-based TDA methods such as influence functions and unrolled differentiation both involve a computation that resembles an inverse Hessian-vector product (iHVP), which is difficult to approximate efficiently. We introduce an algorithm (ASTRA) which uses the EKFAC-preconditioner on Neumann series iterations to arrive at an accurate iHVP approximation for TDA. ASTRA is easy to tune, requires fewer iterations than Neumann series iterations, and is more accurate than EKFAC-based approximations. Using ASTRA, we show that improving the accuracy of the iHVP approximation can significantly improve TDA performance.

Figures

Figures reproduced from arXiv: 2507.14740 by the authors.

Figure 1
Figure 1. The objective is to compute the damped iHVP [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. TDA Performance. Single model ASTRA-IF and ASTRA-SOURCE beat EKFAC-based counterparts in most settings, as well as other TDA methods such as TracIn [83] and TRAK [22] when measured by average LDS over the query set Dquery. ASTRA also enjoys a larger performance boost from ensembling. Improvement is particularly large for convolution architectures such as ResNet-9. Error bars (where available) indicate 1 standard err… view at source ↗
Figure 3
Figure 3. Training Curves. Loss and LDS curves for SNI and ASTRA measured over 10 seeds (shaded region = 1 standard error) on an arbitrary query point zq, using influence functions as the TDA method. SNI makes slower progress compared to ASTRA as measured by LDS. Evaluating Training Data Attribution Performance We evaluate the performance of our TDA algorithms on a popular evaluation metric called Linear Datamodeling Score (L… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Training Curves in Various Eigenspaces. Top: The values h Si fzq as ASTRA and SNI train on the objective in Equation 4 for an arbitrary zq. The subspaces represented by S1, . . . , S5 are spanned by eigenvectors with eigenvalues σ > 10−1 , . . . , σ > 10−5 respectively…

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Forward citations

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.