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arxiv: 1712.00409 · v1 · submitted 2017-12-01 · 💻 cs.LG · stat.ML

Recognition: 2 theorem links

Deep Learning Scaling is Predictable, Empirically

Joel Hestness , Sharan Narang , Newsha Ardalani , Gregory Diamos , Heewoo Jun , Hassan Kianinejad , Md. Mostofa Ali Patwary , Yang Yang
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Yanqi Zhou
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:56 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords deep learning scalingpower lawgeneralization errortraining data sizemodel sizeempirical studymachine translationlanguage modeling
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The pith

Deep learning generalization error decreases as a power law of training set size across multiple domains.

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

The paper measures how error rates change when training data grows much larger, testing four domains: machine translation, language modeling, image processing, and speech recognition. It finds that error follows a power-law relationship with data size, so that each doubling of data produces a predictable fractional drop in error. Model changes or other improvements move the entire curve up or down but leave the scaling exponent unchanged. Optimal model size grows slower than linearly with data volume. These patterns let researchers forecast returns from adding data or compute without running every experiment at full scale.

Core claim

Empirical tests show that generalization error scales as a power law with training set size in each of the four domains examined. The exponent that sets the rate of improvement stays the same when architectures or other model improvements are introduced; those changes only shift the absolute error level. Model size needed for best performance scales sublinearly with data size. The measurements cover a wide range of data volumes and produce consistent scaling behavior within the tested regimes.

What carries the argument

Power-law scaling of generalization error with training set size

If this is right

  • Accuracy targets can be set by extrapolating the measured power law rather than by exhaustive trial runs.
  • Decisions on whether to collect more data can be guided by the expected error reduction per additional example.
  • Model architecture work can be assessed by how far it shifts the error curve rather than by any change in the scaling rate.
  • Hardware and systems planning can use the sublinear model-size relation to estimate compute needs as datasets grow.
  • Continued scaling of data and compute is expected to deliver steady, predictable gains within the domains studied.

Where Pith is reading between the lines

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

  • If the scaling persists, theoretical explanations should target why the observed exponents take their particular values rather than only proving existence of some scaling.
  • The invariance of the exponent under model changes suggests that data volume may dominate long-term progress more than incremental architectural tweaks.
  • Sublinear growth of model size with data implies that the computational cost per example falls as datasets enlarge, improving efficiency at scale.
  • A break in the power law at extreme sizes would signal a new regime, such as exhaustion of useful information in the data source.

Load-bearing premise

The power-law relationships seen in the tested range of data and model sizes will continue without breaks when both are made much larger.

What would settle it

A new experiment at ten times the largest data volume tested here that shows error deviating from the fitted power-law curve by more than the observed variation.

read the original abstract

Deep learning (DL) creates impactful advances following a virtuous recipe: model architecture search, creating large training data sets, and scaling computation. It is widely believed that growing training sets and models should improve accuracy and result in better products. As DL application domains grow, we would like a deeper understanding of the relationships between training set size, computational scale, and model accuracy improvements to advance the state-of-the-art. This paper presents a large scale empirical characterization of generalization error and model size growth as training sets grow. We introduce a methodology for this measurement and test four machine learning domains: machine translation, language modeling, image processing, and speech recognition. Our empirical results show power-law generalization error scaling across a breadth of factors, resulting in power-law exponents---the "steepness" of the learning curve---yet to be explained by theoretical work. Further, model improvements only shift the error but do not appear to affect the power-law exponent. We also show that model size scales sublinearly with data size. These scaling relationships have significant implications on deep learning research, practice, and systems. They can assist model debugging, setting accuracy targets, and decisions about data set growth. They can also guide computing system design and underscore the importance of continued computational scaling.

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.

Referee Report

3 major / 3 minor

Summary. The manuscript presents a large-scale empirical study of scaling in deep learning across four domains (machine translation, language modeling, image processing, and speech recognition). It claims that generalization error follows a power-law dependence on training set size, that the power-law exponent is invariant to model architecture improvements (which only shift the prefactor), and that optimal model size grows sublinearly with data size. These relationships are positioned as making DL scaling predictable, with implications for research, practice, and systems design.

Significance. If the reported power-law relationships hold, the work provides a valuable empirical foundation for quantifying the benefits of scaling data and compute in deep learning. The cross-domain consistency and the observation that architecture changes primarily affect the constant term rather than the exponent are particularly useful for guiding practical decisions on data collection and model sizing. The study also highlights open theoretical questions about the origin of the exponents.

major comments (3)
  1. [§3 (Experimental Methodology)] §3 (Experimental Methodology): The description of how training subsets of varying sizes were constructed lacks detail on sampling method (e.g., random vs. contiguous) and any controls to ensure distributional equivalence across scales; without this, it is difficult to rule out selection effects that could artifactually produce or alter the observed power-law exponents.
  2. [Results sections (e.g., §4.1–4.4)] Results sections (e.g., §4.1–4.4): No error bars, confidence intervals, or goodness-of-fit statistics (such as R² or residual analysis) are reported for the fitted power-law exponents, and there is no sensitivity analysis to the choice of fitting range; this weakens the ability to assess the robustness of the central scaling claims.
  3. [§5 (Discussion)] §5 (Discussion): The claim that scaling is 'predictable' rests on the power-law form and exponents persisting beyond the tested regimes, yet the manuscript contains no analysis or discussion of possible breaks, saturation, or changes in effective exponent at substantially larger data volumes or model capacities.
minor comments (3)
  1. [Abstract and §1] The abstract and introduction would benefit from explicitly stating the numerical values of the observed exponents and the precise functional form used for the power-law fits.
  2. [Figures] Figures showing learning curves should overlay the fitted power-law curves and report the fitted parameters for direct visual assessment of fit quality.
  3. [§2 (Related Work)] A brief comparison to prior empirical scaling observations (e.g., in speech or vision) would help situate the novelty of the cross-domain results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful for the referee's positive assessment and constructive suggestions for improving the manuscript. We address each of the major comments below.

read point-by-point responses

  1. Referee: §3 (Experimental Methodology): The description of how training subsets of varying sizes were constructed lacks detail on sampling method (e.g., random vs. contiguous) and any controls to ensure distributional equivalence across scales; without this, it is difficult to rule out selection effects that could artifactually produce or alter the observed power-law exponents.

    Authors: We agree that more detail on subset construction is needed for reproducibility. The training subsets were constructed via random sampling (without replacement) from the full training set to maintain distributional properties. We will revise §3 to include a clear description of this sampling method and any verification steps for distributional equivalence. revision: yes

  2. Referee: Results sections (e.g., §4.1–4.4): No error bars, confidence intervals, or goodness-of-fit statistics (such as R² or residual analysis) are reported for the fitted power-law exponents, and there is no sensitivity analysis to the choice of fitting range; this weakens the ability to assess the robustness of the central scaling claims.

    Authors: We acknowledge the value of these statistical measures. Although the fits were consistent across domains and visually robust, we will add error bars (from repeated trials where feasible), report R² and other fit statistics, and perform sensitivity analysis on the fitting range in the revised results sections. revision: yes

  3. Referee: §5 (Discussion): The claim that scaling is 'predictable' rests on the power-law form and exponents persisting beyond the tested regimes, yet the manuscript contains no analysis or discussion of possible breaks, saturation, or changes in effective exponent at substantially larger data volumes or model capacities.

    Authors: The claims are grounded in the empirical observations within the tested regimes. We will expand the Discussion section to address potential limitations at larger scales, including possible saturation or exponent changes, based on trends at the upper limits of our experiments and related literature. However, empirical analysis at substantially larger scales is beyond the scope of this work due to resource constraints. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical scaling laws are direct observations, not reductions to fitted inputs

full rationale

The manuscript reports measured power-law relationships between generalization error and factors such as training-set size, model size, and compute across four domains. These relationships are obtained by fitting functional forms to experimental data points collected within the tested regimes; the paper does not derive the power-law exponents from prior equations, self-citations, or uniqueness theorems that would make the reported scaling equivalent to its own inputs by construction. Model-size sublinearity is likewise an observed trend, not a prediction forced by the fitting procedure itself. Because the central claims rest on reproducible empirical measurements rather than any self-referential derivation chain, the analysis is self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claims rest on empirical observation and power-law fitting to measured error curves; no new theoretical entities or derivations are introduced beyond standard statistical fitting assumptions.

free parameters (1)
  • power-law exponent
    Fitted separately per domain to describe the rate of error reduction with data size.
axioms (1)
  • domain assumption Power-law functional form adequately captures the scaling relationship over the measured range
    Invoked to summarize observed curves; no derivation from first principles is provided.

pith-pipeline@v0.9.0 · 5557 in / 1230 out tokens · 63530 ms · 2026-05-12T03:56:18.492891+00:00 · methodology

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    Relation between the paper passage and the cited Recognition theorem.

    Our empirical results show power-law generalization error scaling across a breadth of factors, resulting in power-law exponents—the 'steepness' of the learning curve—yet to be explained by theoretical work. Further, model improvements only shift the error but do not appear to affect the power-law exponent. We also show that model size scales sublinearly with data size.

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