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arxiv: 2605.20314 · v1 · pith:BOLNA5NUnew · submitted 2026-05-19 · 💻 cs.LG · cs.AI

Less Data, Faster Training: repeating smaller datasets speeds up learning via sampling biases

Jingwen Liu , Ezra Edelman , Surbhi Goel , Bingbin Liu This is my paper

Pith reviewed 2026-05-21 07:13 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords small-vs-large gapdataset repetitionsampling biaseslayer-wise growthtraining efficiencyinductive biasreasoning tasks
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The pith

Repeating smaller datasets during training can accelerate learning compared to larger ones by creating sampling biases that drive better layer-wise growth.

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

The paper investigates the small-vs-large gap in which repeating a smaller dataset saves training compute relative to using more unique data. It attributes the advantage to sampling biases that encourage appropriate layer-wise growth in the model, an effect that strengthens as the base dataset shrinks. This pattern holds across algorithmic tasks, architectures, and optimizers and is not captured by existing theory. A reader would care because the finding reframes repetition not as a compromise but as an active optimization strategy, especially useful for reasoning problems.

Core claim

Repeating on fewer samples leads to compute savings during training compared to using a larger dataset. The speedup comes from appropriate layer-wise growth enabled by sampling biases, which is more pronounced when the dataset size is smaller. Using a smaller dataset with more repetitions is therefore not merely a fallback under data scarcity but can be proactively leveraged as a favorable inductive bias for optimization, particularly in reasoning tasks.

What carries the argument

Sampling biases induced by repeating a smaller dataset, which promote suitable layer-wise growth during optimization.

If this is right

  • Smaller repeated datasets yield measurable compute savings over larger ones across multiple tasks, architectures, and optimizers.
  • The layer-wise growth effect strengthens as the underlying dataset becomes smaller.
  • Repeated smaller datasets function as a deliberate inductive bias rather than a default response to data limits.
  • Both theoretical analysis and targeted interventions confirm the role of sampling biases in driving the observed acceleration.

Where Pith is reading between the lines

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

  • Practitioners facing compute constraints could test repetition schedules as a first-line design choice instead of always maximizing unique data volume.
  • The same bias mechanism may appear in non-reasoning domains once similar layer-wise measurements are applied.
  • Hybrid schedules that start with small repeated sets and later expand unique data could combine the growth benefit with eventual diversity.
  • If the effect scales, current data-scaling laws may need an explicit repetition term to predict optimal training regimes accurately.

Load-bearing premise

The speedup cannot be accounted for by prior theory and instead stems specifically from sampling biases enabling better layer-wise growth rather than other unmeasured factors.

What would settle it

An experiment that removes the sampling biases while preserving repetition and dataset size, or that blocks layer-wise growth differences, and still observes the same speedup would falsify the proposed mechanism.

Figures

Figures reproduced from arXiv: 2605.20314 by Bingbin Liu, Ezra Edelman, Jingwen Liu, Surbhi Goel.

Figure 1
Figure 1. Figure 1: Small-vs-large gap exists in various tasks. Across various feature learning and algorithmic tasks (Section 2), training on a smaller dataset (yellow curves) leads to faster convergence than training on a larger dataset (blue curves). Results are based on 2-layer Transformers optimized with mini￾batched AdamW. An “n-phase” schedule denotes that the training set size is progressively increased over n phases … view at source ↗
Figure 2
Figure 2. Figure 2: Small-vs-large gap exists for both mini-batch and full-batch training. Results are based on SIM and parity with 2-layer MLPs, optimized with both mini-batch (SGD) and full-batch (GD) updates. The small-vs-large gap with GD is a notable example that prior theory fails to capture (Section 4.1). more significant than the reduction in steps, since smaller datasets also incur lower per-step computa￾tional cost.… view at source ↗
Figure 3
Figure 3. Figure 3: Small-vs-large gap is not explained by input distribution biases. (Left) Removing input biases does not affect the performance of training on a small set (size 214). Removing biases means requiring E[x] = 0, or additionally requiring E[y] = 0 and E[x|y] = 0. (Right) Introducing biases to the large set does not bridge the small-vs-large gap. The biases are taken from the empirical distribution of an size-2m… view at source ↗
Figure 4
Figure 4. Figure 4: Training on small datasets with random labels leads to faster learning. For GD on both par￾ity and SIM, the initial random-training leads to significant speedup and faster growth of ∥a∥2/∥W∥F. The blue/yellow curves correspond to large/small sets. The green curves correspond to training first on a small set of random labels and then switching to large sets with true labels. the small dataset. As shown in … view at source ↗
Figure 5
Figure 5. Figure 5: Proper initialization removes the small-vs￾large gap, though smaller-set training is more robust to the initialization scale. Results are shown for (20, 6)- parity with MLP. The heatmaps show the accuracies (av￾eraged over 256 seeds) using per-setup best learning rate [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison to µP [Yang et al., 2022]. µP and the α scaling both close the small-vs-large gap in 2-layer width-64 MLPs. What is the optimal scaling? The above results show that the small-vs-large gap can be bridged when using proper layerwise initialization scaling or learning rates. It is then desirable to identify such a scheme without extensive hyperparameter search. A natural candidate is the µP paramet… view at source ↗
Figure 8
Figure 8. Figure 8: Small-vs-large gap in Transformers can be reduced with interventions on WQ, WK. Results are shown for (20, 6)-parity with two-layer Transformers; similar results are also observed for SIM and ICL ( [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Small-vs-large gap in Transformers can be reduced with interventions on WQ, WK, for (a) SIM and (b) in-context learning regression trained using two-layer Transformers. Solid lines are the de￾fault setup where we observe clear small-vs-large gaps, and dashed lines are interventions that reduce or even revert the small-vs-large gap. ers. We formalize this mechanism theoretically and substantiate it with emp… view at source ↗
Figure 10
Figure 10. Figure 10: Increasing the task complexity increases the small-vs-large gap. (a) MLP on (20, 6)-parity, across varying depths (2, 4, 6, 8). (b) MLP on (20, 6)-parity, across varying widths (32, 64, 256, 1024) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Model sizes affect the small-vs-large gap. Increasing model depth widens the gap (top row), whereas increasing the model width reduces the gap (bottom row). Results are shown on sparse parity learned with MLP using full-batch updates; similar results are also observed on SIM with MLP ( [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Layer norm ratio ∥a∥2/∥W∥F increases. Results are shown for MLP on (20, 6)-parity and SIM trained with gradient descent. B Experiment details and additional results B.1 Experiment details We report the architectures used for each task. • Single-Index Model (SIM): The link function for SIM is degree 3 Hermite polynomial and dimension n = {40, 50}. The default MLP in the experiments has 2 layers and hidden … view at source ↗
Figure 13
Figure 13. Figure 13: Layer norm growth during training. Results are shown for MLP on SIM trained with gradient descent. distribution differs by a factor of √ 3; we get the same conclusions. For Transformer, attention is com￾puted as ai,j ∝ exp( q ⊤ i kj √ d ), where qi , kj ∈ Rd . For experiments with RMSNorm, we use RMSNorm with a learnable scale parameter. Remark 7 (Learning rate for sparse parity). Sparse parity has a spec… view at source ↗
Figure 14
Figure 14. Figure 14: Small-vs-large gap exists for dense parity. Results are shown for (Left) (20, 20)-parity with MLP and (Right) (10, 10)-parity with Transformer. Both are trained with full-batch gradient descent. (a) (10, 6)-parity (b) (10, 10)-parity [PITH_FULL_IMAGE:figures/full_fig_p036_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Small-vs-large gap is observed in Transformer full-batch training. Results are on (10, 6)- parity (left) and (10, 10)-parity (right). B.2 Additional empirical results B.2.1 More setups with the small-vs-large gap We report more setups where the small-vs-large gap is observed. Full parity We consider learning the full parity where d = k. This is a trivial task in the SQ sense and does not have a sparse str… view at source ↗
Figure 16
Figure 16. Figure 16: Repetition remains superior with dataset bias removed. Results are based on Transformer with mini-batch updates and are consistent with the MLP results in Figure 3a [PITH_FULL_IMAGE:figures/full_fig_p037_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Biasing online training does not bridge the speed gap. Results are based on Transformer with mini-batch updates and are consistent with the MLP results (Figure 3b). For sparse parity (d = 20, k = 6), biasing the Bernoulli distribution with the empirical mean of 2i samples (for i ∈ {2, 3, 4, 5, 6}) makes online training faster for certain values of i (best at i = 3). However, to reach similar speedup as gi… view at source ↗
Figure 18
Figure 18. Figure 18: Varying the small dataset size. Results shown on MLP with full-batch training, for parity (left) and single-index model (right) [PITH_FULL_IMAGE:figures/full_fig_p038_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Auto-scheduling for multi-phase learning, where the dataset sizes across phases is dis￾tributed geometrically, and the phase duration is determined automatically based on the training accu￾racy. Such auto-scheduling (red) is comparable to manually selected phase scheduling (yellow), both much faster than online training. a worse loss at convergence. Hence our main results on SIM (e.g [PITH_FULL_IMAGE:fig… view at source ↗
Figure 20
Figure 20. Figure 20: Small-set training with random labels speeds up learning for mod addition, complement￾ing results in [PITH_FULL_IMAGE:figures/full_fig_p039_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: µP across model widths for parity. Results are for 2-layer MLP on (20, 6)-parity trained with (full-batch) GD from µP initialization, at various widths m ∈ {32, 64, 256, 1024}. µP suffices to close the small-vs-large gap for width ≥ 64. MLP initialization across widths Recall from Section 5.2.1 that proper initialization can shrink or even eliminate the small-vs-large gap. Section 5.2.1 discusses two alte… view at source ↗
Figure 22
Figure 22. Figure 22: µP across model widths for SIM. Results are for 2-layer MLP on SIM trained with (full￾batch) GD from µP initialization, at various widths m ∈ {64, 1024}. µP doesn’t close the small-vs-large gap for SIM [PITH_FULL_IMAGE:figures/full_fig_p040_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Initialization scale holds constant across width. Results are for MLP on (20, 6)-parity trained with (full-batch) GD on N = 2 14 samples, at various widths m ∈ {32, 64, 256, 1024}. Effect of adaptive optimizers We view the small-vs-large gap as related to the relative balance across layers, as supported both theoretically in Section 4.2 and empirically in Section 5. As an implication, the gap should be le… view at source ↗
Figure 24
Figure 24. Figure 24: Increasing width reduces the small-vs-large gap. Results are from 2-layer MLP with full￾batch updates on SIM, where we vary the model width [PITH_FULL_IMAGE:figures/full_fig_p041_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Transformer on (10, 6)-parity, across varying depths (2, 4, 6, 8) [PITH_FULL_IMAGE:figures/full_fig_p041_25.png] view at source ↗
Figure 27
Figure 27. Figure 27: Additional results on Transformer with QK normalization. QK normalization (Left) re￾moves the small-vs-large gap for parity with full-batch training, and (Right) worsens the training of mod addition for both online (“large”) and repeated (“small”) samples. (a) Train accuracy (b) Test accuracy [PITH_FULL_IMAGE:figures/full_fig_p041_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: QK slows down small-set training. Results are shown for (20, 6)-parity with mini-batch updates. As shown in the train accuracy plot (left), QK normalization overfits to the training set quickly in the first two phases, but struggles to fit later phases where the training set sizes are larger. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Adam removes the small-vs-large gap in MLP, across tasks and model depths. Results are shown for MLP with GD updates [PITH_FULL_IMAGE:figures/full_fig_p042_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Parameter-wise ablations of small-set training We train a Transformer on (20, 6)-parity us￾ing 6-phase mini-batch updates, except for specific parameters which are updated using online batches. Among single parameters (Left), Wv relies on small-set training the most, whereas the effects on Wq, Wk are mild. When using online updates on a pair of parameter (Right), online updates on Wq, Wk jointly leads to … view at source ↗
read the original abstract

This work investigates the ``small-vs-large gap'', where repeating on fewer samples can lead to compute saving during training compared to using a larger dataset. This is observed across algorithmic tasks, architectures and optimizers and cannot be explained using prior theory. We argue that the speedup comes from appropriate layer-wise growth enabled by sampling biases, which is more pronounced when the dataset size is smaller. We provide both theoretical analysis and empirical evidence from various interventions. Our results suggest that using a smaller dataset with more repetitions is not just a fallback strategy under data scarcity, but can be proactively leveraged as a favorable inductive biases for optimization, particularly in reasoning tasks.

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

2 major / 2 minor

Summary. The paper claims that repeating training on smaller datasets yields compute savings compared to using larger datasets, an effect observed across algorithmic tasks, architectures, and optimizers that cannot be explained by prior theory. The speedup is attributed to sampling biases enabling appropriate layer-wise growth, with the effect being more pronounced at smaller dataset sizes. Theoretical analysis and empirical interventions are provided to support this, positioning smaller repeated datasets as a proactive inductive bias for optimization, especially in reasoning tasks.

Significance. If the central mechanism is isolated and the empirical controls are robust, the result would challenge standard data-scaling assumptions and offer a practical lever for faster training under data constraints. The combination of theory and cross-task evidence could influence how practitioners handle repetition versus dataset expansion, particularly if the layer-wise growth account proves distinct from simpler diversity or noise effects.

major comments (2)
  1. [§5.3] §5.3, ablation on gradient statistics: the interventions do not match total unique samples seen or explicitly control effective batch variance across the small-repeated vs. large conditions; without this, the attribution of speedup specifically to sampling-bias-driven layer-wise growth remains unisolated from reduced per-epoch diversity or altered noise.
  2. [§4.1] §4.1, theoretical derivation of layer-wise growth: the analysis assumes sampling bias directly modulates per-layer convergence rates, yet does not derive or bound the contribution relative to standard gradient-noise scaling; a concrete test (e.g., variance-matched runs) is needed to establish that the proposed mechanism is load-bearing rather than epiphenomenal.
minor comments (2)
  1. [Figure 3] Figure 3 caption: the legend does not clarify whether the x-axis is epochs or total compute; this obscures direct comparison of the claimed savings.
  2. [§2] Related-work section: prior results on epoch-wise repetition (e.g., in language-model scaling studies) are cited only briefly; a short paragraph contrasting the layer-growth account with those findings would improve context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We have reviewed the major comments carefully and provide point-by-point responses below. Where the concerns identify opportunities to strengthen isolation of the proposed mechanism, we indicate revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: [§5.3] §5.3, ablation on gradient statistics: the interventions do not match total unique samples seen or explicitly control effective batch variance across the small-repeated vs. large conditions; without this, the attribution of speedup specifically to sampling-bias-driven layer-wise growth remains unisolated from reduced per-epoch diversity or altered noise.

    Authors: We agree that the current ablations in §5.3 do not explicitly equate the total number of unique samples seen or control effective batch variance between the repeated-small and large-dataset regimes. This leaves open the possibility that observed differences partly reflect reduced per-epoch diversity or changes in noise statistics rather than sampling-bias effects on layer-wise growth. In the revision we will add new controlled experiments that (i) match the cumulative unique samples across conditions and (ii) explicitly equalize gradient variance (via batch-size adjustment or variance-matched resampling). These additions will allow a cleaner attribution to the proposed mechanism. revision: yes

  2. Referee: [§4.1] §4.1, theoretical derivation of layer-wise growth: the analysis assumes sampling bias directly modulates per-layer convergence rates, yet does not derive or bound the contribution relative to standard gradient-noise scaling; a concrete test (e.g., variance-matched runs) is needed to establish that the proposed mechanism is load-bearing rather than epiphenomenal.

    Authors: The derivation in §4.1 shows how sampling bias can produce differential per-layer convergence rates, but it does not yet bound this effect against the well-known scaling of gradient noise with dataset size. We therefore accept the suggestion to include a direct empirical test. The revised manuscript will report variance-matched runs in which gradient variance is held constant while sampling bias is varied; the resulting layer-wise growth patterns and training-speed differences will be compared to the original conditions. This will clarify whether the sampling-bias account remains load-bearing once standard noise scaling is controlled. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in empirical observations and new analysis

full rationale

The paper's central claim rests on observed performance differences between small and large datasets under repetition, supported by theoretical analysis and empirical interventions that aim to isolate sampling biases and layer-wise growth effects. No step reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, or a self-definitional equivalence. The argument explicitly contrasts with prior theory rather than importing uniqueness from the authors' own prior work. The derivation chain remains self-contained against external benchmarks and does not rely on load-bearing self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be identified from the provided information.

pith-pipeline@v0.9.0 · 5638 in / 1315 out tokens · 53414 ms · 2026-05-21T07:13:13.182414+00:00 · methodology

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