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Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention

Jing Huang, Daniel Wurgaft, Rachit Bansal, Laura Ruis, Naomi Saphra, David Alvarez-Melis, Andrew Kyle Lampinen, Christopher Potts, Ekdeep Singh Lubana

arXiv:2605.29548v2cs.LG

TL;DR

The paper asks why larger models learn tasks that smaller models fail to learn, even when the tasks are representable. It combines power-law analysis, controlled synthetic mixtures, and OLMo pretraining to study this question. Across these settings, larger models learn rare and complex tasks by reducing resource competition and gradient interference, though the account is not complete and complements expressivity and sample-efficiency explanations.

  • Problem

    The paper asks why larger models can learn tasks that smaller models fail to learn, including cases where smaller models remain unsuccessful despite extensive training.

  • Method

    The paper combines a phenomenological power-law analysis with controlled mixtures of tasks varying in frequency and complexity, then validates the account using OLMo models trained on injected novel tasks.

  • Results

    Larger models learn rare, infrequent, and complex tasks that smaller models fail to learn, while showing greater task-feature retention and less gradient interference.

  • Takeaways & Limitations

    The findings support a data-centric view in which model sizing and training-data mixtures jointly shape which capabilities become learnable.

  • Takeaways & Limitations

    The account is not complete: expressivity and sample efficiency are complementary explanations, and the power-law argument need not hold asymptotically.

Abstract

from arXiv · show

Larger models learn tasks smaller models do not. What drives this phenomenon? We develop a simple phenomenological argument that power-law scaling already suggests that a larger model will be able to learn a part of the data distribution that a smaller model fails to learn, even with infinite training data. To validate this claim and identify its causes, we study the effects of model scaling on a synthetic setup consisting of a mixture of tasks that show monotonic scaling curves. The results point to a data-induced competition over resources (neurons). Specifically, smaller models allocate their neurons to high frequency or low complexity tasks, and so they learn solutions that perform poorly on rare and complex tasks. Moreover, this happens even when solutions capable of expressing the desired task exist. We then assess how a larger model circumvents this data-centric bottleneck, finding that it traces to a reduced interference mechanism: larger models can allocate enough resources to common tasks that the gradient updates for those tasks become weak, which means that they do not overwrite rare-task features as they slowly accumulate. Finally, to further validate these claims, we pretrain OLMo models (4M to 4B parameters) on novel tasks of varying frequency and complexity. The results mirror those from our synthetic data experiments: only the larger OLMo models learn the infrequent and complex tasks, and these larger models embed more task features in their representations and show less gradient interference between tasks. Overall, we offer a data-centric account of why larger models learn tasks that smaller models fail to. This helps explain why larger models are better in practice, and it can inform practical questions concerning model sizing and training data mixtures.

1 Introduction

The paper argues that larger models can learn rare and complex tasks that smaller models fail to learn even with extensive training, because larger models reduce data-induced competition and gradient interference. Synthetic experiments and OLMo pretraining support this data-centric account.

  • Motivation: Power-law scaling suggests smaller models can fail to learn parts of a task distribution even under asymptotic training.This limitation is distinct from ordinary sample inefficiency: additional data may not allow the smaller model to catch up.
  • Method: The synthetic setup controls task frequency and complexity while ensuring the individual tasks remain expressible by the models.This separates learning-dynamics limitations from simple representational incapacity.
  • Results: Larger models learn rare and complex tasks on average, as predicted analytically and verified experimentally.The result appears in the controlled regression-task setting.
  • Mechanism: Reduced resource competition lets larger models retain rare-task memories because common-task updates create less gradient interference.Smaller models repeatedly restart learning rare tasks, whereas larger models build on previously observed rare-task data.
  • Real-model validation: OLMo models from 4M to 4B parameters reproduce the pattern: only larger models learn infrequent and complex tasks, while embedding more task features and showing less gradient interference.These results extend the synthetic findings to pretrained language models.
  • Implications: The paper frames scaling as an interaction between model size, task frequency, task complexity, and gradient-based learning dynamics rather than expressivity alone.It suggests that data-mixture design may also affect which capabilities are learned.

2 A Phenomenological Model Predicts Larger Models Learn More

The phenomenological analysis distinguishes gains obtainable through more data from gains that require more parameters. Power-law scaling motivates the latter regime, where a smaller model cannot reach the larger model’s loss even with infinite data.

  • Scaling setup: Compute-optimal scaling predicts lower loss for larger models, but does not establish whether smaller models are fundamentally unable to match it.The analysis therefore compares resource-constrained and asymptotic-data losses.
  • Scaling setup: Under asymptotic resources, L∞(N) ∝N^-α; when α > γ, increasing model size still improves the achievable loss.The cited Chinchilla-style compute exponent is γ = 0.34, while α is the parameter exponent for asymptotic loss.
  • Data scaling: A smaller model recovers the larger model’s loss via data scaling when its compute-optimal loss exceeds the larger model’s but its asymptotic loss is lower.This formalizes the case where the smaller model is undertrained and can eventually catch up.
  • Model scaling: Model scaling is required when even asymptotic data scaling leaves the smaller model above the larger model’s loss.This identifies a genuine marginal benefit attributable to the larger model’s parameter count.

3 Scaling Allows Learning Rare Tasks by Reducing Gradient Interference

The paper argues that model width enables rare and complex tasks to be learned by ranking features by utility and reducing competition between common- and rare-task representations. Larger models retain rare-task updates because common-task gradients weaken after their high-utility features are learned.

  • 3.1 Scaling Enables Learning Rare and Complex Tasks: The multi-task setup uses K = 32 simultaneous linear regression tasks with power-law frequencies and task spectra, allowing frequency and complexity to be controlled independently.Different tasks occupy orthogonal feature blocks, and slower spectral decay represents greater complexity.
  • 3.1 Scaling Enables Learning Rare and Complex Tasks: The phenomenological power-law argument is limited because power-law scaling need not hold asymptotically.The authors use this argument to motivate subsequent rigorous claims.
  • 3.1 Scaling Enables Learning Rare and Complex Tasks: Larger width preferentially improves low-frequency tasks because it retains lower-utility features that smaller models omit.Figure 2 reports a frequency-dependent retention phase diagram and losses matching the analytic prediction across task-frequency exponents.
  • 3.1 Scaling Enables Learning Rare and Complex Tasks: A width-N encoder retains the N task features with highest utility, so infrequent or complex tasks are learned only when their features enter this ranking.Task frequency weights feature utility, while slowly decaying spectra indicate greater task complexity.
  • 3.2 Scaling Reduces Interference and Allows for Retention of Rare Task Observations: Common-task updates weaken after high-utility common features are explained, leaving spare width for rare-task features.Theorem 4 links aggregate common-task gradients to their unresolved residual signal.
  • 3.2 Scaling Reduces Interference and Allows for Retention of Rare Task Observations: Width scaling lowers common-task competition and can make rare features locally stable rather than replacing them through an update-and-forget dynamic.The paper identifies both reduced aggregate common-task gradients and lower weakest occupied common-task utility as mechanisms that enable rare-task retention.

4 Corroborating Claims with the OLMo Pretraining Pipeline

The OLMo experiments test how model size and injected task frequency affect learning, representations, rare-task retention, and gradient interference. Larger models learn rarer tasks, represent more task features, retain rare-task information better, and show stronger task-aligned gradients with less interference.

  • 4.1 Setup: OLMo models from 4M to 4B parameters are pretrained on Dolma v1.7 with controlled injections of novel tasks at varying frequencies.The pipeline analyzes loss, representation, and gradient behavior across model sizes.
  • 4.1 Setup: The comparison and modular addition tasks use three-token sequences and require geometric structures for generalization beyond memorizing training instances.The tasks contain 10K instances each, split evenly between training and testing.
  • 4.2 Behavioral Evidence: Larger models learn lower-frequency tasks better, and task learning follows frequency order; on TADD, only larger models at higher frequency show grokking.Higher test accuracy distinguishes learning generalizable task structures from merely achieving low training loss.
  • 4.2 Behavioral Evidence: Equivalent global task frequency yields higher task loss when instances are spaced farther apart, indicating that larger gaps impair rare-task retention.The matched-frequency experiment injects N instances every N batches for N = 1, 10, 20, 50, 100.
  • 4.3 Representational Evidence: Larger models and models trained on more frequent task data learn task-relevant features faster, and feature presence is highly correlated with test accuracy.TCMP uses global token ordering features, while TADD uses Fourier-mode features localized across model representations.
  • 4.4 Gradient Evidence: At injection steps, larger models have more task-aligned batch gradients and less interference from non-task gradients than smaller models.The reported full-gradient similarity is 0.08 ± 0.02 for the 1B model and 0.04 ± 0.04 for the 300M model; the 20M model shows unstable similarities and random collisions.

5 Discussion

The paper frames scaling as a data-centric interaction among learning dynamics, task frequency, and task complexity rather than model expressivity alone. It suggests that retention of rare-task information can support abstraction and that changing data mixtures may sometimes be more efficient than increasing model size.

  • Discussion: Larger models learn rare tasks through resource competition, memory retention, task frequency, and task complexity, highlighting learning dynamics beyond expressivity.The account is presented as a data-centric explanation of why smaller models can fail on tasks they could express.
  • Discussion: Increasing a target task’s frequency may provide a more efficient way to learn it than increasing model size.This is offered as an example of more intentional design of training data mixtures.
  • Discussion: Retaining task instances longer can let models accumulate signals across batches and learn more generalizable task structures.The paper therefore presents memorization as potentially beneficial for abstraction in rare-task settings.

Limitations

The paper’s account emphasizes learning dynamics, task frequency, and task complexity rather than offering a complete explanation of scaling. Its realistic pretraining validation supports the theory but leaves analytic and empirical gaps.

  • Scope of the account: The account is complementary to expressivity and sample-efficiency explanations, rather than a complete account of scaling.Expressivity constrains representation, sample efficiency shapes data use, and this account emphasizes learning dynamics interacting with task frequency and complexity.
  • Empirical scope: The OLMo validation strongly matches theoretical predictions, but realistic pretraining experiments may leave analytic gaps.The authors specifically note that larger-scale language models and over-trained language models were not empirically verified.
  • Synthetic setting: The synthetic setup uses orthogonal task blocks, fixed ambient dimension D = 1024, and K ∈{16, 32} tasks, usually K = 32.Within each task block, spectra follow power laws; unless otherwise stated, tasks share α = 1.
  • Synthetic setting: The student model is a shared width-N encoder with task-specific linear decoders jointly optimized with the encoder.The relevant encoder object is the projector PW, which is invariant to the encoder’s right-multiplicative gauge.
  • Experimental procedure: Experiments use AdamW, inverse-square-root scheduling, gradient clipping at norm 1.0, and fresh batches without replay.Batch size is B = 1 024 for phase-diagram and rank-1 sweeps and B = 512 for retention sweeps.
  • Metrics: Evaluation tracks per-task loss, subspace alignment, and residual common-task signal on freshly sampled batches separate from training.The frequent task set is defined as the smallest top-prior set with cumulative mass at least 0.8.

B.4 Localizing and Measuring Task Features in Sec. 4.3

The paper localizes task-relevant representations and measures their presence using task-specific probes. Synthetic theory further characterizes the width-dependent eigenspace and feature-utility ordering underlying learning.

  • Comparison task: DAS identifies a 1-D first-layer residual-stream subspace causally encoding global token order for comparison.Interchange interventions achieve a 96% success rate, supporting both localization and causal use for prediction.
  • Modular addition: For modular addition, Fourier analysis measures the presence of frequency modes in residual-stream representations across layers.The analysis groups residual vectors by output and computes variance captured by each frequency.
  • Causal localization: Null interventions on MLP layers identify which layers have the largest causal effects on model output.The result is consistent with first-layer localization as the earliest point where global token order is causally encoded.
  • Synthetic theory: For fixed encoder width, minimizing mixture loss is equivalent to maximizing Tr(U^⊤MU), so an optimum spans the top-N eigenspace of M.The optimal representation therefore retains directions ranked by eigenvalues of the weighted task covariance.
  • Synthetic theory: Task features are ranked by utility πkλk,j, and a width-N optimum keeps the N largest utilities.A task’s residual loss depends on how many of its coordinates are retained.
  • Synthetic theory: A rare feature first invades through the weakest occupied common direction when its utility exceeds that direction’s utility.The perturbation toward rare feature br decreases loss iff πrλr > µF_i.

C.4 Microscopic competition in a one-neuron, two-task model

In the one-neuron, two-task model, frequent and rare orthogonal tasks compete for a shared direction. Rare-task alignment decays across gaps dominated by common-task updates.

  • Competition dynamics: With one neuron, the frequent task dominates; with two neurons, separate specialization lets rare-task alignment approach and sustain values near 1.The tasks are orthogonal and sampled with probabilities 0.9 and 0.1 in the illustrated softmax-gated model.
  • Model: The width-1 encoder is a unit vector u = cos θ a + sin θ b spanning the two orthogonal task directions.Task-specific losses depend on the squared alignment with a and b.
  • Gradient drift: Expected gradient drift drives the shared neuron toward the more common task when its probability exceeds the rare task’s probability.For task probabilities p and q < p, E[∆θ | θ] = η(q −p) sin(2θ) + O(η^2).
  • Retention: θG ≈e−2ηGθ0, so rare-task alignment decays exponentially across the gap between rare observations.This follows from linearizing the dynamics near alignment with the common task.

D Further Experimental Results: Complexity Sweeps

Complexity sweeps show that learning order depends jointly on task frequency and complexity. Larger width improves retention, but rare tasks still struggle while substantial frequent-task residual remains.

  • Complexity construction: Task complexity is varied by changing power-law exponents αk, with slower spectral decay representing more complex tasks.The most frequent task is assigned αmax and is simplest; the rarest receives αmin and is most complex.
  • Complexity construction: Varying complexity changes the number of directions needed to cover 90% of task signal to 4–7 directions.The experiments use power-law task spectra with different α ranges across tasks.
  • Learning order: Under wider complexity ranges, learning order is not monotonically predicted by frequency alone.The complexity gap increases emphasis on the top two modes’ learning and can alter which tasks are learned first.
  • Learning order: Observed loss continues to align with predictions based on learning the top N utility features, confirming that frequency and complexity jointly determine learning dynamics.The utility-based prediction remains effective in non-uniform complexity settings.
  • Retention: Larger models capture tasks across frequencies more completely, while rare tasks struggle when frequent-task residual signal remains high.Even the largest models show difficulty when substantial common-task signal is still left to explain.
  • Retention: Matched-frequency injections show that larger models retain rare-task signal more effectively than smaller models.The experiment withholds the lowest-utility task and reintroduces it while keeping overall frequency matched across settings.
  • Experimental design: Additional sweeps vary β ∈{0.5, 1.0, 1.5, 2.0} over widths N ∈{8, 16, 32, 64, 128}.These settings reproduce the feature-utility ordering analysis at lower staircase resolution.

E.1.2 Simplified Case: Rank-1 Tasks

In the rank-1 setting, theory predicts that width determines which tasks are retained, with the retention boundary following task utility or prior ordering. Experiments closely match the predicted staircase and preserve the predicted two-phase dynamics across prior skews.

  • Prediction: Theorem 3 predicts that a width-N minimizer retains the N task-features with largest utility u_kj = π_k λ_kj.In the rank-1 specialization, setting d_T = 1 and α_k = 1 reduces utility ordering to prior ordering.
  • Prediction: The rank-1 prediction produces a perfectly linear staircase in task index for the critical width required to retain task k.
  • Verification: Empirical transition widths N_emp(k) show almost perfect alignment with the theoretical staircase, with minor disparities attributed to the sampled width grid.
  • Prior-skew controls: Across β ∈ {0.5, 1.0, 1.5, 2.0}, the two-phase dynamic persists and the threshold shifts as predicted by theory.
  • Residual controls: Rare-task signal drops to zero when frequent-task residual δ_F exceeds the analytic threshold δ*, then rises steeply toward near-unity below it.As β grows, the threshold shifts left because the rare task’s leading utility becomes smaller.

E.3 Per-gap dynamics: Reproducing retention results across different injection gaps and widths

Retention results remain consistent across injection gaps and model widths: larger models preserve rare-task signal, while smaller models require sufficiently short gaps for learning. Extending training to 1M steps supports the stability of the reported results.

  • Per-gap dynamics: The per-gap experiments vary injection gaps G ∈ {64, 128, 256, 512, 1024, 1280} and widths N ∈ {32, 96, 128, 192, 256}.
  • Per-gap dynamics: Larger models retain and preserve the learned rare-task signal across injection gaps and widths, while smaller models require sufficiently small gaps for learning to occur.
  • Extended training: Training extensions to 1M steps across widths N = {8, 16, 32, 64, 128, 256} show that the reported results remain stable beyond the 100K-step budget.

F.1 Task Loss vs. General Language Modeling Loss

At matched general language-modeling loss, model size matters most for very low-frequency injected tasks. Larger models achieve lower task-specific loss there, whereas higher-frequency tasks follow similar trajectories across sizes.

  • Low-frequency tasks: At task frequency 2.4 × 10^-7, larger models achieve lower task loss given the same language modeling loss.
  • Higher-frequency tasks: For frequencies from 2.4 × 10^-6 to 2.4 × 10^-5, loss curves across model sizes follow roughly the same trajectory.In this range, model size improves sample efficiency while training dynamics remain similar.

F.2 Compute-optimal Comparison

Under equal compute budgets, larger models are more compute-efficient at learning low-frequency tasks. The comparison uses a compute estimate based on model parameters and training tokens.

  • Compute-optimal comparison: When task frequency is one instance per batch, 2.4 × 10^-7, larger models achieve lower task loss under the same compute budget.Compute is estimated as 6× the number of model parameters × the number of training tokens following Chinchilla scaling laws.
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