What do Language Models Learn and When? The Implicit Curriculum Hypothesis

Emmy Liu; Kaiser Sun; Millicent Li; Isabelle Lee; Lindia Tjuatja; Jen-tse Huang; Graham Neubig

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What do Language Models Learn and When? The Implicit Curriculum Hypothesis

Emmy Liu, Kaiser Sun, Millicent Li, Isabelle Lee, Lindia Tjuatja, Jen-tse Huang, Graham Neubig

arXiv:2604.08510v1Apr 9, 2026cs.CL

TL;DR

The paper addresses what skills language models acquire during pretraining and when, beyond what aggregate validation-loss scaling laws reveal. It tests the Implicit Curriculum Hypothesis with composable tasks across models and finds stable emergence orderings, compositional sequencing, and predictable trajectories from model representations.

Problem
Scaling laws describe improvement in validation loss with compute but do not identify which skills models acquire or when they acquire them.
Method
The paper tracks emergence across simple composable tasks in 9 models from 4 families spanning 410M–13B parameters and analyzes task representations.
Results
Emergence orderings are consistent across models, composite tasks usually follow prerequisites, and function-vector proximity predicts held-out task trajectories.
Takeaways & Limitations
Pretraining is more structured than loss curves reveal: skill emergence follows a cross-model compositional order readable from model internals.
Takeaways & Limitations
The prerequisite relation reflects the researchers’ task design and does not establish corresponding internal primitives or dependencies in the model.

Abstract

from arXiv · show

Large language models (LLMs) can perform remarkably complex tasks, yet the fine-grained details of how these capabilities emerge during pretraining remain poorly understood. Scaling laws on validation loss tell us how much a model improves with additional compute, but not what skills it acquires in which order. To remedy this, we propose the Implicit Curriculum Hypothesis: pretraining follows a compositional and predictable curriculum across models and data mixtures. We test this by designing a suite of simple, composable tasks spanning retrieval, morphological transformations, coreference, logical reasoning, and mathematics. Using these tasks, we track emergence points across four model families spanning sizes from 410M-13B parameters. We find that emergence orderings of when models reach fixed accuracy thresholds are strikingly consistent ($ρ= .81$ across 45 model pairs), and that composite tasks most often emerge after their component tasks. Furthermore, we find that this structure is encoded in model representations: tasks with similar function vector representations also tend to follow similar trajectories in training. By using the space of representations derived from our task set, we can effectively predict the training trajectories of simple held-out compositional tasks throughout the course of pretraining ($R^2 = .68$-$.84$ across models) without previously evaluating them. Together, these results suggest that pretraining is more structured than loss curves reveal: skills emerge in a compositional order that is consistent across models and readable from their internals.

1 Introduction

The paper proposes that skills emerge during pretraining in a stable, compositional order and tests this across models using simple diagnostic tasks. It finds consistent emergence orderings, delayed emergence of composite tasks, and trajectory predictability from model representations.

The Implicit Curriculum Hypothesis predicts that skills emerge in a stable compositional order consistent across models.
ρ = .81 across 45 model pairs, with emergence orderings consistent across model families under fixed accuracy thresholds.The ordering ranges from ρ = .64 to .93; consistency does not hold for relative thresholds.
Composite tasks generally emerge after their elemental prerequisites, supporting a compositional relationship between skills.
Tasks with nearby function-vector representations follow similar learning trajectories, making internal representations informative about training dynamics.
R^2 = .68–.84 across models for predicting held-out composite-task trajectories without evaluating those tasks during training.Per-task R^2 exceeds .95.
The results suggest that pretraining structure is richer than aggregate loss curves, with skill order, compositional dependencies, and internal representations aligned.

2 Preliminaries

The paper frames pretraining as potentially structured around discrete, compositional skills rather than only aggregate loss, and formalizes hypotheses about their emergence order, cross-model stability, and representational alignment.

Scaling Laws: Scaling laws describe validation-loss improvement with compute but do not specify which skills models acquire or when they acquire them.Aggregate loss also does not directly correlate with downstream performance.
Quantization Hypothesis: The Quantization Hypothesis treats smooth scaling curves as the result of models acquiring discrete skills, or quanta, but typically leaves their structural dependencies unexplored.The paper distinguishes this independent, additive framing from its focus on compositional skill structure.
Simplicity Bias and Compositional Skill Structure: Prior work on simplicity bias and skill dependency structures motivates asking whether large-scale pretraining follows a compositional ordering of skill acquisition.These threads suggest simpler functions may precede complex ones and that skills may have dependencies.
The Implicit Curriculum Hypothesis: The Implicit Curriculum Hypothesis proposes that tasks constructed with compositional dependencies emerge no later than the tasks that depend on them.The dependency relation reflects the researchers’ task design rather than a claim about the model’s internal primitives.
The Implicit Curriculum Hypothesis: The hypothesis also predicts that emergence orderings form a partial order consistent across models.Cross-model stability is evaluated through rank correlations between emergence orderings.
Representational Alignment: Tasks with nearby internal representations are hypothesized to follow similar learning trajectories, enabling prediction of unseen-task trajectories from representational geometry.The paper defines each task’s learning trajectory as its performance over training and uses a distance over trajectories.

3 Methodology

The methodology evaluates emergence order and representational structure across open-weight models using composable exact-match tasks, threshold-based emergence times, and function-vector trajectory prediction.

Model and checkpoint selection: The study compares nine open-weight models from four families, spanning 410M to 13B parameters, using checkpoints through the first 1T training tokens.Approximately 20 checkpoints per model provide roughly 20B-token granularity.
Task design: The benchmark contains 91 elemental and composite tasks with intuitive compositional relationships, evaluated using exact-match accuracy.Synthetic chains compose elemental operations in sequence, yielding 38 composite tasks with known prerequisites.
Emergence measurement: Emergence time is defined as the first checkpoint exceeding either a fixed accuracy threshold or a fraction of the model’s best task performance.These absolute and relative definitions focus on ordering rather than the sharpness of emergence.
Representational analysis: Task representations are extracted as function vectors from correctly answered in-context prompts at the final non-padding token, using either causal attention-head or hidden-state extraction.Task similarity is measured from these representations, and an RBF kernel plus kernel ridge regression predicts held-out composite trajectories.
Behavioral analyses: The study tests prerequisite ordering by checking whether each composite’s prerequisites emerge no later than the composite and reports the resulting violation rate.Cross-model stability is measured with Spearman correlations between full task emergence orderings under both threshold definitions.
Trajectory prediction: A leave-one-out protocol predicts each held-out composite trajectory from nearby function-vector tasks, comparing all-task and elemental-only bases with smoothed trajectories.Prediction quality uses per-task Pearson r2 and MAE; reduced quality with elemental-only bases tests for a composition bottleneck.

4 Emergence Order Results

Emergence orderings are highly consistent across models under absolute accuracy thresholds, and composite tasks generally follow their component skills, with some inversions.

ρ = .64–.93 across model pairs, with all 45 correlations significant under an absolute 80% threshold.Within OLMo-2, correlations ranged from .72 to .93; cross-family correlations remained substantial.
54/76 composite tasks emerged no earlier than their parent tasks.There were also 19 weak inversions and 3 strong inversions; all strong inversions involved the first-letter component task.
Emergence order is broadly stable across model families, from copying and simple string operations through knowledge-dependent and complex compositional tasks.The ordering proceeds from early copying and coreference to transformations, translation, and a long tail of difficult tasks.
Relative-threshold emergence produces substantially lower cross-model correlations than absolute-threshold emergence.Absolute thresholds are set above chance for all tasks, whereas relative thresholds depend on each model’s maximum performance.

5 Representational Similarity and Prediction Results

Task representations are linked to learning dynamics: function-vector similarity supports prediction of held-out composite-task trajectories, although accuracy varies by task and representation set.

R2 = .67–.838 for leave-one-out prediction of composite-task trajectories using function-vector space.Mean absolute error ranged from .068 to .195 on a 0–1 accuracy scale.
Restricting prediction to elemental tasks increased MAE by mean ΔMAE = +.135 for every model.The comparison indicates that composite-task representations contribute to trajectory prediction.
For OLMo2-7B, predicted trajectories closely tracked ground truth for fr eng upper and plural lower.The reported fits were R2 = .99, MAE = .017 for fr eng upper and R2 = .89, MAE = .028 for plural lower.
Prediction was weaker for eng fr upper, with R2 = .51 and MAE = .068.Its trajectory was less well approximated by the nearest neighbors in representation space.

6 Conclusion

The paper argues that pretraining has a stable, compositional developmental structure across models and that this structure is readable from internal representations. It also identifies possible uses for monitoring and data-mixture decisions while situating the findings within prior work.

Across 410M–13B models and several families, absolute-threshold emergence orders are consistent, and function-vector similarity predicts held-out compositional-task trajectories.The conclusion describes both behavioral consistency and representational predictability as supporting the Implicit Curriculum Hypothesis.
Stable emergence orders could support pretraining monitoring for capabilities developing ahead of or behind schedule.The authors present this as a practical application rather than an established deployment result.
The paper connects its hypothesis to theories proposing that complex behaviors arise from elementary skills or discrete quanta.The cited related work includes scaling-law and quantization accounts of skill emergence.
Prior evaluation-time studies identify latent skills and compositional ability, while other work examines training phases and task representations.These threads provide context for the paper’s focus on emergence order and internal task structure.

C Full list of Elemental and Composite Tasks

The appendix provides the complete task list and categorizes elemental and composite tasks, drawing tasks from psychometrics and interpretability research.

The full task list is organized into reasoning types across Tables 4 and 5.TextFRCT tasks come from the psychometrics literature, while other tasks were studied or inspired by interpretability work.

D Full list of Elemental and Composite Tasks

The evaluation suite lists elemental and compositional tasks with representative examples spanning retrieval, language transformations, comprehension, reasoning, and coreference.

Task sources: TextFRCT tasks draw from psychometrics, while other tasks were studied in or inspired by interpretability research.The task list is categorized into reasoning types and includes associative-memory, verbal-comprehension, and induction examples.
Elemental tasks: The suite includes elemental tasks covering fact extraction, coreference, simple transformations, associative memory, verbal comprehension, and inductive reasoning.Examples include extracting entities, numbers, or locations; resolving pronouns; transforming case or word forms; retrieving memorized pairs; defining words; and identifying patterns.
Compositional tasks: Compositional tasks combine operations such as translation, capitalization, reversal, pluralization, and morphological transformation.Examples include translating between English, French, and Spanish while also changing case or reversing the output, alongside combined gerund and plural transformations.
Compositional tasks: Representative compositional examples include uppercasing or reversing gerunds and plural forms, such as run → RUNNING and child → NERDLIHC.The suite also includes language-translation compositions across English, French, and Spanish, with first-letter, last-letter, lowercase, uppercase, and reverse variants.

F Emergence Order Agreement Under Alternate Definitions

Emergence-order agreement depends on the threshold definition: absolute thresholds produce substantially stronger cross-model consistency than relative thresholds.

Threshold comparison: Absolute thresholds yield substantially higher cross-model correlations than relative thresholds.The comparison is summarized across alternate emergence-order definitions.
Absolute thresholds: Mean ρ = 0.860 for pairwise emergence-order correlations at absolute threshold θ = 0.5.This is the strongest reported agreement among the threshold definitions listed here.

G Function vector hyperparameters

Function-vector representations are selected per model through a rank-sum search balancing consistency, discriminability, and compositional structure.

Selection criteria: Representation hyperparameters are chosen using within-task consistency, inter-task discriminability, and compositional structure.The search varies representation type, layers, and number of heads over candidate configurations.
Selection procedure: Final selection uses a rank-sum policy across the three criteria, with ties broken by raw metric values.Only correct examples are used to construct the final function vectors.

H All Held-out trajectory predictions

Held-out compositional-task trajectories are predicted using leave-one-out setups across multiple model families and scales.

Leave-one-out setup: Figures 13–21 report leave-one-out predictions for each held-out compositional task.The prediction figures cover Pythia, OLMo, CrystalCoder, and Amber models.
Smaller models: The held-out prediction evaluations include Pythia-410M, OLMo2-1B, and Pythia-1.4B.Each model has a corresponding figure of predicted compositional-task trajectories.
Larger models: The evaluations also include OLMo2-7B, OLMo3-7B, Amber, CrystalCoder, Pythia-12B, and OLMo2-13B.These figures extend the held-out trajectory analysis to larger models and multiple model families.
Prediction configurations: The prediction setup uses model-specific function-vector hyperparameters, including selected representation configurations and ridge-regression parameters.Table 11 summarizes the chosen configurations per model.