Data Scaling as Progressive Coverage of a Predictive Contribution Spectrum

Chunlin Huang; Hongxi Li; Shihao Ji; Shuaizhi Cheng; Zihui Song

arxiv: 2605.20196 · v1 · pith:X53CUIIGnew · submitted 2026-04-05 · 💻 cs.CL · cs.AI· cs.LG

Data Scaling as Progressive Coverage of a Predictive Contribution Spectrum

Zihui Song , Shihao Ji , Hongxi Li , Shuaizhi Cheng , Chunlin Huang This is my paper

Pith reviewed 2026-05-21 10:37 UTC · model grok-4.3

classification 💻 cs.CL cs.AIcs.LG

keywords data scaling lawspredictive contribution spectrumglobal-KL spectrumsuffix automatonscaling exponentsexcess losslanguage model trainingtraining dynamics

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The pith

Training scale advances an effective frontier through a predictive state spectrum whose residual tail mass tracks remaining excess loss.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether real-data scaling laws arise from progressive coverage of a latent predictive contribution spectrum in text, instead of token-frequency effects alone. Using suffix automata on twelve corpora, it builds a global-KL spectrum in which each state contributes its empirical mass times its KL deviation from a global next-token baseline. The spectrum tail slope already correlates with observed scaling exponents of a fixed GPT learner. For each training size N the authors define an effective truncation rank K(N) by equating observed excess loss to the residual tail mass of a fixed 1000k spectrum, finding log K linear in log N at pooled R^2 of 0.96. This supplies a concrete mechanism picture in which larger training sets push the frontier deeper into the same spectrum and thereby reduce loss.

Core claim

Real-data scaling laws are governed by progressive coverage of a latent predictive contribution spectrum rather than by token-frequency tails alone. Across 12 real corpora a data-intrinsic global-KL spectrum is prepared from suffix-automaton states, each weighted by mass times KL deviation from a global baseline. For each training size N an effective truncation rank K(N) is obtained by matching observed excess loss directly to the residual tail mass of the prepared spectrum; log K proves close to linear in log N, with pooled R^2 about 0.96 for the raw spectrum and 0.90 for the smoothed spectrum.

What carries the argument

The global-KL predictive contribution spectrum, in which each suffix-automaton state contributes its empirical mass multiplied by its KL deviation from a global next-token baseline; the effective truncation rank K(N) obtained by matching excess loss to residual tail mass.

If this is right

The tail slope of the spectrum correlates strongly with the empirical scaling exponent of a fixed small GPT learner.
Log K(N) is approximately linear in log N across multiple real corpora with high R-squared values.
Residual tail mass of the spectrum directly tracks the remaining excess loss at each training scale.
Training scale advances an effective frontier through the predictive state spectrum.

Where Pith is reading between the lines

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

The same spectrum construction could be applied to other modalities or loss components to test whether analogous coverage mechanisms govern their scaling.
Data curation that preferentially includes high-contribution states from the spectrum might accelerate scaling for a given compute budget.
The approach yields a data-intrinsic predictor of scaling behavior that does not require training large models.
If the linear log K vs log N relation holds, one could extrapolate required data volume to reach a target excess loss without additional training runs.

Load-bearing premise

Defining the effective truncation rank K(N) by directly matching observed excess loss to the residual tail mass of the prepared global-KL spectrum yields a quantity that represents the model's learning frontier rather than a fitted construct.

What would settle it

For held-out training sizes or new corpora, the K(N) values obtained by matching excess loss to spectrum tail mass deviate from the observed linear log-log trend or fail to predict the measured excess loss.

Figures

Figures reproduced from arXiv: 2605.20196 by Chunlin Huang, Hongxi Li, Shihao Ji, Shuaizhi Cheng, Zihui Song.

**Figure 1.** Figure 1: Empirical data-scaling curves for a fixed small GPT learner across real datasets. Earlier proxy searches established that simple token-level statistics — entropy, compression ratio, unigram tail slope, and raw n-gram summaries — are informative but insufficiently robust. Once strongly structured corpora such as TinyStories are included, these scalar summaries no longer provide a convincing mechanism-level … view at source ↗

**Figure 2.** Figure 2: Suffix-automaton state-mass spectra at 500k tokens. However, state mass alone measures occupancy rather than predictive usefulness. To explain scaling, we need a quantity aligned with reductions in next-token uncertainty. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Prepared 1000k global-KL predictive spectrum versus empirical data-scaling slope. Among the tail windows emphasized in our prepared-spectrum comparison, the strongest fits are [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: Predictive contribution spectra across datasets. 6 From Spectrum Slope to Spectral Frontier 6.1 Mechanism hypothesis The tail-slope result alone remains a cross-sectional correlation. The mechanism claim is stronger: as data scale increases, training should behave as if it progressively covers a prefix of the predictive contribution spectrum. This suggests a decomposition of the form L(N) ≈ X k>K(N) wk, wh… view at source ↗

**Figure 5.** Figure 5: shows the resulting relation between the inferred cutoff rank and training size [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Kernel-quotient predictive spectra at 2000k tokens. However, the explanatory power of the current quotient construction is weaker than that of the unmerged global-KL spectrum. This suggests that the present merging criterion may remove fine-grained information that remains relevant to cross-entropy scaling. In other words, the unmerged spectrum may still be closer to the operative predictive decomposition … view at source ↗

read the original abstract

We investigate the hypothesis that real-data scaling laws are governed by progressive coverage of a latent predictive contribution spectrum rather than by token-frequency tails alone. We work with a suffix-automaton representation of text corpora and define a data-intrinsic global-KL predictive contribution spectrum, in which each state contributes according to its empirical mass times its KL deviation from a global next-token baseline. Across 12 real corpora, the tail slope of this spectrum is already strongly correlated with the empirical data-scaling exponent of a fixed small GPT learner. We then go beyond slope correlation and define, for each training size N, an effective truncation rank K(N) by matching the observed excess loss to the residual tail mass of the prepared 1000k global-KL spectrum. Empirically, log K is close to linear in log N, with pooled R^2 about 0.96 for the raw spectrum and R^2 about 0.90 for the smoothed spectrum. These findings provide strong empirical support for a simple mechanism picture: training scale advances an effective frontier through a predictive state spectrum, and the residual tail mass of that spectrum tracks the remaining excess loss.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The spectrum construction adds a fresh angle on scaling but defining K(N) by matching excess loss to tail mass makes the tracking claim hold by construction.

read the letter

The paper's core move is to build a predictive contribution spectrum from suffix automata using global KL divergence, then use its tail to explain scaling. Across a dozen corpora the tail slope lines up with the scaling exponent, which is a solid empirical observation. They also define an effective frontier K(N) for each training size by picking the rank where the spectrum's leftover mass equals the model's excess loss. Log K comes out linear in log N with R-squared around 0.96. The spectrum construction itself looks like a genuine addition to the scaling literature. It tries to capture something beyond raw token frequencies by weighting states by their KL deviation. The fact that they get consistent results on real corpora rather than synthetic data is a plus. The problem is that K(N) is defined directly from the excess loss, so the claim that the residual tail tracks excess loss is true by how they set it up. The linearity result then mostly confirms that the spectrum's tail shape is consistent with the power-law scaling they already measured. It doesn't independently show that the spectrum drives the scaling. This makes the mechanistic interpretation weaker than it first appears. The slope correlation across corpora stands on its own better. If the authors can show the spectrum predicts scaling on held-out data or new models without refitting K, that would strengthen the case. This is worth a look for people working on data selection and scaling explanations. It deserves referee time because the framing is original even if the current evidence has this built-in aspect. I'd send it out for review but flag the definition of K(N) for closer scrutiny.

Referee Report

2 major / 3 minor

Summary. The paper investigates whether real-data scaling laws arise from progressive coverage of a latent predictive contribution spectrum (derived via suffix-automaton states and a global-KL measure of each state's contribution) rather than token-frequency tails alone. Across 12 corpora it reports strong correlation between the spectrum's tail slope and the empirical scaling exponent of a fixed small GPT model; it then defines, for each training size N, an effective truncation rank K(N) by matching observed excess loss to the residual tail mass of a fixed 1000k global-KL spectrum, and finds that log K is approximately linear in log N (pooled R² ≈ 0.96 raw, 0.90 smoothed). These results are presented as empirical support for the mechanism that training scale advances an effective frontier through the spectrum and that residual tail mass tracks remaining excess loss.

Significance. If the spectrum-based mechanism can be shown to hold independently of the matching procedure, the work would offer a data-intrinsic account of scaling that goes beyond frequency-based explanations and could inform both theoretical understanding and practical prediction of scaling behavior. The slope-correlation result across corpora is less affected by construction issues and provides a potentially useful empirical regularity; however, the central K(N) construction substantially weakens the evidential support for the proposed tracking mechanism.

major comments (2)

[K(N) definition and results] Procedure for defining K(N) (following the slope-correlation results): K(N) is obtained by selecting the truncation rank whose residual tail mass exactly equals the observed excess loss at that N. This matching ensures the equality holds by construction for the chosen K(N), so the subsequent claim that 'residual tail mass tracks the remaining excess loss' is tautological rather than an independent empirical finding. The reported linearity of log K vs. log N (R² 0.96/0.90) therefore primarily verifies consistency between the spectrum tail shape and the empirical scaling exponent, without separately validating the frontier-coverage mechanism.
[Results and discussion] The paper does not report controls that would distinguish the proposed spectrum mechanism from alternative explanations (e.g., direct fitting of a power-law tail to the excess-loss curve itself, or other monotonic functions of N). Without such controls or an independent operationalization of the 'effective frontier,' the high R² values cannot be taken as strong evidence for the spectrum picture over simpler descriptive fits.

minor comments (3)

[Methods] The manuscript should clarify the exact preparation steps for the 1000k global-KL spectrum (including any smoothing parameters and how the 1000k states are selected) so that the procedure is fully reproducible.
[Results] Error bars or confidence intervals on the reported R² values and on the per-corpus slopes would help readers assess the robustness of the correlations.
[Discussion] A brief comparison to a purely frequency-based spectrum (or other baseline spectra) would strengthen the claim that the global-KL construction adds explanatory power beyond token-frequency tails.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and precise comments. We address each major point below, clarifying the evidential status of our results while acknowledging limitations in the current presentation.

read point-by-point responses

Referee: [K(N) definition and results] Procedure for defining K(N) (following the slope-correlation results): K(N) is obtained by selecting the truncation rank whose residual tail mass exactly equals the observed excess loss at that N. This matching ensures the equality holds by construction for the chosen K(N), so the subsequent claim that 'residual tail mass tracks the remaining excess loss' is tautological rather than an independent empirical finding. The reported linearity of log K vs. log N (R² 0.96/0.90) therefore primarily verifies consistency between the spectrum tail shape and the empirical scaling exponent, without separately validating the frontier-coverage mechanism.

Authors: We agree that the matching procedure for K(N) makes the numerical equality between residual tail mass and observed excess loss hold by construction, rendering any direct claim of 'tracking' tautological in that narrow sense. However, the reported linearity of log K versus log N is not itself a consequence of the construction; it is an empirical outcome that arises only because the specific tail shape of the global-KL spectrum, when inverted via the matching, reproduces a near-linear relation in log-log coordinates for the observed excess-loss values. This would not occur for an arbitrary monotonic tail or for a spectrum whose shape was inconsistent with the empirical scaling. We therefore regard the high R² as evidence that the spectrum shape is compatible with the scaling behavior, beyond mere descriptive consistency. We will revise the manuscript to explicitly distinguish the constructed equality from the independent empirical finding of linearity and to moderate the language around the tracking claim. revision: partial
Referee: [Results and discussion] The paper does not report controls that would distinguish the proposed spectrum mechanism from alternative explanations (e.g., direct fitting of a power-law tail to the excess-loss curve itself, or other monotonic functions of N). Without such controls or an independent operationalization of the 'effective frontier,' the high R² values cannot be taken as strong evidence for the spectrum picture over simpler descriptive fits.

Authors: The referee correctly notes that the manuscript would benefit from explicit controls that compare the spectrum-matching procedure against simpler alternatives, such as direct power-law or other monotonic fits to the excess-loss curve alone. While the separate slope-correlation result across the 12 corpora is less dependent on the K(N) construction and provides an independent empirical regularity, we did not include comparative goodness-of-fit analyses or alternative operationalizations of an effective frontier. We will add these controls in revision, reporting quantitative comparisons of explanatory power and discussing how the spectrum supplies a data-intrinsic account that goes beyond purely descriptive scaling fits. revision: yes

Circularity Check

1 steps flagged

Defining K(N) by matching excess loss to tail mass makes the tracking claim hold by construction

specific steps

self definitional [Abstract]
"define, for each training size N, an effective truncation rank K(N) by matching the observed excess loss to the residual tail mass of the prepared 1000k global-KL spectrum. ... These findings provide strong empirical support for a simple mechanism picture: training scale advances an effective frontier through a predictive state spectrum, and the residual tail mass of that spectrum tracks the remaining excess loss."

K(N) is defined by selecting the truncation that forces residual tail mass to equal observed excess loss at each N. The assertion that residual tail mass tracks remaining excess loss therefore holds tautologically for the chosen K(N) rather than as an independent empirical result. The linearity finding between log K and log N primarily recovers the scaling exponent already present in the excess-loss data.

full rationale

The paper defines an effective truncation rank K(N) for each training size N explicitly by choosing the rank that equates the fixed spectrum's residual tail mass to the observed excess loss at that N. This definitional step ensures the claimed tracking relationship holds by construction for the selected K(N). The subsequent report that log K(N) is linear in log N (R² ≈ 0.96) then tests consistency between the spectrum tail shape and the empirical excess-loss scaling exponent, without providing an independent test of the mechanism. The initial slope correlation across corpora is less affected but is presented as preliminary to the K(N) construction. The central 'progressive coverage' and 'tracks remaining excess loss' assertions therefore reduce to the matching procedure rather than emerging as a prediction from the spectrum alone.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on a newly defined spectrum construct, a domain assumption about suffix automata capturing predictive states, and a fitting procedure that introduces an effective truncation parameter tied directly to loss observations.

free parameters (2)

effective truncation rank K(N)
Defined for each N by matching observed excess loss to residual tail mass of the 1000k spectrum
spectrum preparation size (1000k)
Fixed cutoff used to prepare the global-KL spectrum for tail analysis

axioms (2)

domain assumption Suffix-automaton representation of text corpora captures the relevant latent predictive states.
Invoked to define the global-KL spectrum from empirical mass and KL deviation
domain assumption KL deviation from global next-token baseline quantifies predictive contribution.
Core definition of each state's contribution in the spectrum

invented entities (1)

global-KL predictive contribution spectrum no independent evidence
purpose: To represent the latent distribution of predictive contributions across text states
Newly defined construct whose tail properties are hypothesized to govern scaling

pith-pipeline@v0.9.0 · 5744 in / 1739 out tokens · 99021 ms · 2026-05-21T10:37:28.189257+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

define the effective cutoff rank by K(N)=min{K:T(K)≤ΔL(N)/ΔLmax}
IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean alphaProvenanceCert unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

tail slope of this spectrum is already strongly correlated with the empirical data-scaling exponent

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 2 internal anchors

[1]

Scaling Laws for Neural Language Models

Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models.arXiv preprint arXiv:2001.08361, 2020

work page internal anchor Pith review Pith/arXiv arXiv 2001
[2]

Training Compute-Optimal Large Language Models

Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, et al. Training compute-optimal large language models.arXiv preprint arXiv:2203.15556, 2022

work page internal anchor Pith review Pith/arXiv arXiv 2022
[3]

arXiv preprint arXiv:2102.06701 , year=

Yasaman Bahri, Ethan Dyer, Jared Kaplan, Jaehoon Lee, and Utkarsh Sharma. Explaining neural scaling laws.arXiv preprint arXiv:2102.06701, 2021

work page arXiv 2021
[4]

Littman, Richard S

Michael L. Littman, Richard S. Sutton, and Satinder Singh. Predictive state representations: A new theory for modeling dynamical systems. InAdvances in Neural Information Processing Systems 14, 2001

work page 2001
[5]

The smallest automaton recognizing the subwords of a text.Theoretical Computer Science, 40:31–55, 1985

Anselm Blumer, Janet Blumer, Andrzej Ehrenfeucht, David Haussler, Ross McConnell, and Jeffrey Seiferas. The smallest automaton recognizing the subwords of a text.Theoretical Computer Science, 40:31–55, 1985

work page 1985
[6]

World Scientific, 2002

Maxime Crochemore and Wojciech Rytter.Jewels of Stringology: Text Algorithms. World Scientific, 2002. 8

work page 2002

[1] [1]

Scaling Laws for Neural Language Models

Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models.arXiv preprint arXiv:2001.08361, 2020

work page internal anchor Pith review Pith/arXiv arXiv 2001

[2] [2]

Training Compute-Optimal Large Language Models

Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, et al. Training compute-optimal large language models.arXiv preprint arXiv:2203.15556, 2022

work page internal anchor Pith review Pith/arXiv arXiv 2022

[3] [3]

arXiv preprint arXiv:2102.06701 , year=

Yasaman Bahri, Ethan Dyer, Jared Kaplan, Jaehoon Lee, and Utkarsh Sharma. Explaining neural scaling laws.arXiv preprint arXiv:2102.06701, 2021

work page arXiv 2021

[4] [4]

Littman, Richard S

Michael L. Littman, Richard S. Sutton, and Satinder Singh. Predictive state representations: A new theory for modeling dynamical systems. InAdvances in Neural Information Processing Systems 14, 2001

work page 2001

[5] [5]

The smallest automaton recognizing the subwords of a text.Theoretical Computer Science, 40:31–55, 1985

Anselm Blumer, Janet Blumer, Andrzej Ehrenfeucht, David Haussler, Ross McConnell, and Jeffrey Seiferas. The smallest automaton recognizing the subwords of a text.Theoretical Computer Science, 40:31–55, 1985

work page 1985

[6] [6]

World Scientific, 2002

Maxime Crochemore and Wojciech Rytter.Jewels of Stringology: Text Algorithms. World Scientific, 2002. 8

work page 2002