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The paper argues that a training sample's influence on a model is not a fixed property: it can change sign and spike sharply at developmental phase transitions, so data attribution must be stagewise.

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2026-08-04 09:59 UTC pith:C5VKRPZL

load-bearing objection Real conceptual contribution and a solid toy model, but the BIF experiments do not actually test the paper's phase-transition mechanism—the local sampler cannot see the between-phase term. the 3 major comments →

arxiv 2510.12071 v2 pith:C5VKRPZL submitted 2025-10-14 cs.LG

Influence Dynamics and Stagewise Data Attribution

Jin Hwa Lee , Matthew Smith , Maxwell Adam , Jesse Hoogland This is my paper
classification cs.LG
keywords training data attributioninfluence functionssingular learning theorystagewise developmentphase transitionsBayesian influence functionsusceptibilitylanguage model development
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.

Current training-data attribution treats the influence one sample has on another as a single static score, but neural networks learn in distinct stages and the paper argues this picture is fundamentally incomplete. Drawing on singular learning theory, it predicts that influence is dynamic—non-monotonic, capable of sign flips, and sharply peaking at the phase transitions that define stagewise learning. The paper validates this prediction analytically and empirically in a toy hierarchical model, where influence peaks align with the model's progressive acquisition of semantic distinctions, and in language models, where token-level influence changes align with known developmental events. If correct, the field should move from pointwise to stagewise data attribution: the question is not only which samples matter, but when and why they matter during training.

Core claim

The central claim is that the Bayesian influence function (BIF)—the derivative of a posterior expectation of an observable with respect to a sample's training weight—serves as a generalized susceptibility that registers stagewise phase transitions. Decomposing the BIF by the law of total covariance into within-phase and between-phase contributions shows that the between-phase term, πUπV(μi,U−μi,V)(μj,U−μj,V), peaks when the posterior is evenly split across two phases and is largest for samples on which the phases disagree most. This yields two predictions: influence can flip sign, and influence peaks at transitions. The paper confirms these predictions with an analytical derivation in a deep

What carries the argument

The Bayesian Influence Function (BIF), defined as the negative posterior covariance between a sample's loss and an observable, replaces the Hessian-inverse classical influence function and stays well-defined even on degenerate loss landscapes. Its role is to make influence a distributional, checkpoint-local quantity, and its decomposition via conditioning on the posterior's phase yields the between-phase covariance term—the mechanism that produces sign flips and sharp peaks. It is estimated in practice with a stochastic-gradient Markov-chain Monte Carlo sampler localized to each checkpoint.

Load-bearing premise

The load-bearing premise is that SGD-trained neural networks can be modeled as a Bayesian learning process whose posterior undergoes the phase transitions predicted by singular learning theory—a correspondence the paper's own limitations section describes as the primary theoretical gap.

What would settle it

Train a neural network in a regime with no degeneracy and no expected phase transition (for example, a well-conditioned model where the Bernstein–von Mises theorem should hold) and measure BIF over training: if influence still shows sharp peaks and sign flips, the phase-transition mechanism is not what drives them; if influence stays smooth and monotonic, the mechanism is supported.

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

If this is right

  • Static end-of-training attribution scores can be actively misleading: a sample that helps early may harm later, and vice versa.
  • Sharp peaks in Bayesian influence localize phase transitions and single out the samples that characterize them.
  • Token-level influence in language models is observable and non-monotonic, with timing that lines up with known developmental milestones such as induction-circuit formation.
  • Ablation timing matters for interveners: retraining experiments show that briefly removing a sample at its peak-influence stage produces the largest loss change.
  • The framework calls for stagewise data attribution as a new default object of study, with influence as a trajectory rather than a point estimate.

Where Pith is reading between the lines

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

  • If the picture holds, attribution benchmarks should stop reporting single final scores and instead score methods on whether they reproduce influence trajectories, sign flips, and transition timing.
  • The between-phase term offers a way to cluster samples by which phase they favor, turning influence spike patterns into a data-structure discovery tool instead of just a debugger.
  • Peaks in influence could be used as an online, training-time detector of phase transitions—and, if transitions are steerable, as a hook for intervening in model development.
  • The framework gives a mechanistic gloss on implicit curricula: rather than being imposed, the curriculum emerges because different data become maximally influential at different stages, which might explain why explicit curricula only sometimes help.

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 / 5 minor

Summary. The paper argues that training data influence is not static but evolves during learning, and proposes a stagewise data attribution framework grounded in singular learning theory (SLT). The theoretical core (Section 2.3) models a first-order phase transition as a two-phase posterior mixture, decomposes the Bayesian influence function (BIF) into within-phase and between-phase covariance terms, and predicts that influence can change sign and peak sharply when the posterior mass is split evenly between phases. Empirical support is sought in two settings: a deep linear network on a hierarchical dataset (Section 3), where BIF trajectories are compared with an analytic influence computation and leave-one-out retraining; and Pythia language models (Section 4), where per-token BIFs across structural token classes are claimed to show non-monotonic dynamics, including sign flips and an induction-related peak. The paper concludes that data attribution methods should track influence as a function of training time rather than returning a single static score.

Significance. If the central claim holds, the paper challenges a foundational assumption of most TDA methods and offers a new diagnostic—BIF as a generalized susceptibility—for detecting developmental transitions in neural networks. The strongest part of the paper is the toy model: the analytic derivation in Section C.6 and the leave-one-out retraining experiments are independent of the BIF estimator and agree qualitatively with the BIF trajectory, providing credible evidence that influence can vary non-monotonically and that peaks align with hierarchy-learning stages. The paper is also transparent about its main gap, the SGD-to-Bayes correspondence, in the Limitations. However, the specific theoretical mechanism advanced in Section 2.3—that peaks arise from the between-phase covariance of a bimodal posterior—is not actually probed by the local BIF experiments, because the estimator is deliberately localized around a single checkpoint. The LM results are suggestive but share this limitation and additionally rely on a normalized BIF whose sign interpretation differs from the theoretical covariance. The empirical phenomenon is therefore more secure than the proposed causal explanation.

major comments (3)
  1. [§2.3, §B (Eq. 15, Algorithm 1)] The central prediction in Section 2.3 attributes sign flips and peaks to the between-phase term πUπV(μi,U−μi,V)(μj,U−μj,V), which requires the posterior p(w|D) to be bimodal with πU≈πV≈0.5. But the local BIF estimator in Eq. (15) and Algorithm 1 is an RMSProp-SGLD sampler initialized at checkpoint w* with a strong localization drift γ(w−w*). Such a sampler stays in a single basin and estimates a unimodal posterior around w*; it structurally cannot estimate the between-phase covariance. Consequently, the BIF peaks in Figures 3–5 and 15 measure only within-phase covariances and do not test the Section 2.3 mechanism. The independent analytic IF and LOO results support non-monotonic influence, but not the specific claim that peaks are susceptibility divergences at phase transitions. Please either revise the causal claims to what the local BIF can actually measure, or add an experiment that c
  2. [§3, §C.2] The toy-model validation is weakened by the hyperparameter selection procedure. Section C.2 states that the BIF hyperparameters shown in Figure 3 (β=1000, ε=1e-3, γ=5e+3) were chosen as the point with the highest correlation between the BIF trace and the LOO loss-difference trace. This makes the reported BIF–LOO agreement in Figure 3 an in-sample fit, not a blind confirmation. The grid in Figure 6 shows that the correlation varies smoothly and substantially with the hyperparameters. Please report the BIF–LOO agreement for a hyperparameter set chosen by an independent criterion (e.g., the LLC-calibration procedure described in C.2) or show the distribution of BIF–LOO correlations over the whole grid, and justify that the chosen point is not cherry-picked.
  3. [§B, §4, Fig. 5] The language-model experiments use the normalized BIF, defined as the Pearson correlation between losses (Section B), whereas the theoretical predictions in Section 2.3 concern the covariance in Eq. (2). A sign flip in Pearson correlation is not equivalent to a sign flip in covariance, since changes in the marginal variances of the losses can induce sign changes in the correlation even when the covariance does not change sign. Claims in Section 4 about “sign flips” (e.g., the left/right delimiter relationship) and “negative influence” are therefore not directly supported by the theory. Please either report the unnormalized BIF for the sign-flip claims, or explicitly analyze how standardization affects the signs and peaks, or rephrase the claims in terms of normalized BIF and state that the theoretical sign-flip prediction is not being tested at LM scale.
minor comments (5)
  1. [§A.1, Eqs. (7)–(8)] There is a sign inconsistency. For Fβn = −log∫e^{−β·ℓ}φ dμ, one has ∂F/∂βi = −E[ℓi], so E[ℓi] = −∂F/∂βi and BIF = ∂E[ℓj]/∂βi = −∂²F/∂βi∂βj. Equations (7)–(8) omit this minus sign and therefore define the BIF with the opposite sign, contradicting Eq. (2). Please correct the derivation or the sign convention.
  2. [§A.2, Eq. (14)] The text says the log-sum-exp is a smooth approximation for the max function, but Eq. (14) shows Fn ≈ mini Fn(Wi). It should say min, not max.
  3. [§B] The sentence “The full algorithm is described in Section B” is self-referential; it should reference Algorithm 1, where the RMSPropSGLD procedure is defined.
  4. [§C.1] Minor typo: “N∼(0, σ2)” should read “N(0, σ²)” or “𝒩(0, σ²)”. Also, the main text does not state which values of C and T from Table 1 were used for the BIF plots in Figure 3; please report them.
  5. [§C.2 and Fig. 3 caption] The Figure 3 caption lists the BIF hyperparameters but does not say that they were selected to maximize BIF–LOO correlation. That fact is in Section C.2 but should be stated prominently wherever the agreement is presented as validation.

Circularity Check

1 steps flagged

Theoretical peak/sign-flip predictions follow algebraically from the assumed bimodal posterior, but independent toy/LM measurements keep the central phenomenon grounded.

specific steps
  1. self definitional [Section 2.3, 'Decomposing influence with the law of total covariance' and 'Predicting stagewise changes in influence']
    "A first-order phase transition is characterized by the posterior distribution p(w| D) having significant mass in two distinct neighborhoods, which we label U and V. ... At the peak of a phase transition, πU ≈π V ≈0.5. ... The between-phase influence term is maximized when the posterior mass is evenly split (πU ≈π V ≈0.5 ), causing a sharp peak in total influence at the critical point of a transition."

    The 'prediction' is obtained by inserting the paper's own definition of a phase-transition peak (posterior mass evenly split, πU≈πV≈0.5) into the law of total covariance applied to the BIF definition (Eq. 2). The between-phase term πUπV(µi,U−µi,V)(µj,U−µj,V) is maximized at πU=πV by construction, so 'influence peaks at transitions' is a property of the assumed mixture model rather than an independent constraint. Sign flips are likewise a direct algebraic consequence of phase-dependent within-phase covariances. This is partial built-in-ness of the theoretical core, not a fitting of the empirical data.

full rationale

Section 2.3's central prediction is a formal consequence of the model setup: once the posterior is written as a two-phase mixture with πU+πV=1, the law of total covariance forces the between-phase contribution πUπV(µi,U−µi,V)(µj,U−µj,V), and this quadratic term is maximal at πU=πV. Because the paper defines the peak of a phase transition as πU≈πV≈0.5, the predicted 'sharp peak in total influence at the critical point' is largely built into the definitions. The same holds for sign flips if within-phase covariances differ. This is the one genuinely definitional element. It does not make the whole paper circular: the toy-model phenomenon is also established by leave-one-out retraining (Eq. 3) and by an analytic perturbation calculation in Section C.6 that does not use the mixture decomposition; the language-model results are compared to external developmental markers (induction-head timing from Olsson et al. 2022; Tigges et al. 2024). A separate validity concern, not a circularity, is that the implemented local BIF (Eq. 15, Algorithm 1) samples a posterior localized to w* and therefore cannot cross free-energy barriers, so the between-phase covariance term is not directly measured by the BIF experiments; this limits how strongly Figures 3–5 and 15 test the Section 2.3 mechanism. In addition, the main toy BIF plot used hyperparameters selected for the highest BIF–LOO correlation (Section C.2), so that particular agreement is partly selection-optimized; the analytic IF and LOO remain independent checks. Self-citations (Kreer et al. 2025, Hoogland et al. 2024/2025, Baker et al. 2025, Lehalleur et al. 2025) supply methods and framing, but the load-bearing SLT results are cited to Watanabe (2009, 2018), not to a self-citation chain, and no uniqueness theorem is imported. Overall, partial definitional circularity in the theoretical core, with independent empirical grounding, warrants a 4.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The theoretical framework relies on standard statistical identities and on SLT results from the literature, plus the paper-specific assumption of a two-phase posterior. The toy model adds a domain assumption about alignment of singular vectors. No new physical entities are introduced; the main free parameters are BIF hyperparameters that were tuned against the validation signal.

free parameters (6)
  • BIF inverse temperature β (toy) = 1000
    Selected as the value maximizing BIF-LOO correlation in Figure 6 (Section C.2).
  • BIF localization strength γ (toy) = 5e+3
    Selected as the value maximizing BIF-LOO correlation in Figure 6 (Section C.2).
  • BIF step size ε (toy) = 1e-3
    Selected as the value maximizing BIF-LOO correlation in Figure 6 (Section C.2).
  • Perturbation strength ε (analytic toy) = -0.1
    Used to approximate the ablation effect in the analytic influence derivation (Section C.6).
  • BIF localization strength γ (LM) = 500 or 1000
    Selected based on KNN class recall in Section D.1.2.
  • BIF inverse temperature β (LM) = 256 or 1024
    Selected based on KNN class recall in Section D.1.2.
axioms (6)
  • standard math Law of total covariance
    Used in Section 2.3 to decompose the BIF into within-phase and between-phase terms.
  • standard math Watanabe's singular learning theory free energy formula
    Used in Section 2.1 and Appendix A to justify phase transitions via competition between loss and complexity.
  • ad hoc to paper Two-phase mixture posterior at phase transitions
    Section 2.3 assumes the posterior is p(w|D)=πU p(w|U)+πV p(w|V); the peak prediction follows from this modeling assumption.
  • domain assumption Deep linear network singular value dynamics with balanced initialization
    Appendix C.6 assumes W=U G V^T with closed-form G(t), which holds under small initialization and small learning rate.
  • domain assumption SGD trajectory approximated by a Bayesian posterior
    Section 2.1 follows the Developmental Interpretability recipe; the paper acknowledges this is an open gap in the Limitations.
  • domain assumption RMSPropSGLD samples approximate the true local posterior
    Section B relies on SGLD estimates of the BIF as covariance over the posterior.

pith-pipeline@v1.3.0-alltime-deepseek · 20697 in / 11817 out tokens · 97891 ms · 2026-08-04T09:59:46.057321+00:00 · methodology

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read the original abstract

Current training data attribution (TDA) methods treat the influence one sample has on another as static, but neural networks learn in distinct stages that exhibit changing patterns of influence. In this work, we introduce a framework for stagewise data attribution grounded in singular learning theory. We predict that influence can change non-monotonically, including sign flips and sharp peaks at developmental transitions. We first validate these predictions analytically and empirically in a toy model, showing that dynamic shifts in influence directly map to the model's progressive learning of a semantic hierarchy. Finally, we demonstrate these phenomena at scale in language models, where token-level influence changes align with known developmental stages.

Figures

Figures reproduced from arXiv: 2510.12071 by Jesse Hoogland, Jin Hwa Lee, Matthew Smith, Maxwell Adam.

Figure 1
Figure 1. Figure 1: Stagewise learning requires stagewise data attribution. (a) In regular models, development is a uniform, monotonic process of posterior concentration around a single solution (Bernstein–von Mises). (b) In singular models, development is a stagewise process where the posterior undergoes phase transitions (Watanabe’s singular learning theory). This stagewise development means the influence one sample has on … view at source ↗
Figure 2
Figure 2. Figure 2: A toy model of hierarchical semantic knowledge acquisition. (a) A toy dataset adopted from Saxe et al. (2019a). Each object maps to a feature vector that describes the hierarchical structure of semantic knowledge, e.g., ‘penguin’ and cat are all living animals, but ‘penguin’ is a bird and cat is a mammal. (b) The correlation matrix of the feature output shows a hierarchical structure. (c) The hierarchical … view at source ↗
Figure 3
Figure 3. Figure 3: Influence over time on a hierarchical semantic dataset. We measure the influence of dog sample on other query samples j with the following: (Left) BIF (β = 1000, ϵ = 1e-3, γ = 5e + 3). (Center) Analytical IF (see full derivation in Section C.6). (Right) Loss difference from Leave-One-Out (LOO) retraining experiment. All three measures agree that the influence one sample has on another can vary non-monotoni… view at source ↗
Figure 4
Figure 4. Figure 4: BIF captures developmental influence. (a) The peak positive influence from dog to different data points is noted with ⃝1 , ⃝2 , and ⃝3 . (b) MDS of the hidden representations in the network over the course of learning. The peaks in influence match the branching points of the MDS trajectory, where each hierarchical category develops (black points). losses that represent the standard for LLM pretraining: ℓi(… view at source ↗
Figure 5
Figure 5. Figure 5: Token–class relationships. BIF between structural classes through training. We observe structural rela￾tionships between tokens reflected in influence patterns between classes, including strong intra-class relationships, development of induction, and relationships between word elements and corresponding delimiter token classes. Dashed lines indicate major inflection points in the BIF. Opacity captures stro… view at source ↗
Figure 6
Figure 6. Figure 6: BIF–LOO trace correlation with varying inverse temperature β, localization strength γ and step size ϵ. On a single grid point between two hyperparameters (e.g. γ, ϵ in the leftmost), multiple points refer to different values of the remaining hyperparameter (β in the leftmost). In general, high γ and low ϵ gave the highest correlation with the gold standard LOO experiment, and β was less significant in the … view at source ↗
Figure 7
Figure 7. Figure 7: BIF for perturbing each data point. The subtitle indicates the perturbed data point. Analytic Loss Difference query pine sparrow penguin cat dog oak rose lily [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Leave-One-Out (LOO) loss difference for perturbing each data point. The title indicates the ablated data point. we take the approximation methods as in Koh & Liang (2020); Grosse et al. (2023). First is to add a constant dampening 19 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Influence with damped Hessian approximation. We measure the classical influence of the dog sample with damped Hessian approximation H˜ with varying dampening constant γ. C.5 Time-Specific Ablation Experiment Our observation that influence is a time-dependent function proposes another perspective - perturbing the same data point but at different training times would lead to non-identical influence. That is,… view at source ↗
Figure 10
Figure 10. Figure 10: Influence with damped Gauss-Newton-Hessian approximation. We measure the classical influence of the dog sample with damped Gauss-Newton-Hessian approximation H˜ with varying dampening constant γ. Integrated Ablation Ablation [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Brief ablation of dog. For each experiment, we ablate dog for duration D = 100 epochs starting from timepoint t. Starting time t is sampled uniformly with an interval of 200 epochs. For each experiment, we integrate the loss difference to the baseline during D. We show the integrated loss of query (a) cat and (b) sparrow over different ablation windows. The most critical period of influence from the dog o… view at source ↗
Figure 12
Figure 12. Figure 12: Analytical loss difference from perturbing each data point. The title indicates the perturbed data point (ε = −0.1). D.1 Structural Token Classification Here, we present additional details and further experiments conducted to investigate the development of influence with respect to how tokens structure text. These experiments were conducted using the 14 million parameter Pythia model. Tokens are generated… view at source ↗
Figure 13
Figure 13. Figure 13: Examples of structural classification of tokens. Bold outline indicates class membership. The classes are as follows: 25 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: KNN BIF predictions change over time. Using the BIF in order to select the top 30 highest influence tokens and taking the majority class from among them (adjusted for class rates), we plot the recall minus the class rate for each class. This means that 0 represents the random baseline. We see that the BIF selects for in-class tokens above random for all categories by the end of training, indicating that i… view at source ↗
Figure 15
Figure 15. Figure 15: Influence dynamics during induction head formation. We trained a small transformer from scratch with high-frequency checkpointing to capture the formation of induction heads. (Top) The normalized BIF between corresponding tokens in repeated sequences (blue) shows a sharp increase and peak, while correlations between non-repeated samples (red) or between non-repeated and repeated segments (green) remain lo… view at source ↗

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