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

Abstract: 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.

• The paper demonstrates that LLMs acquire distinct skills in a stable, compositional order, validated by consistent behavioral thresholds and function vector analysis.
• It employs 91 diagnostic tasks across multiple model families to determine the precise timing and sequence of skill emergence.
• The findings imply that tracking emergence order can guide pretraining interventions and improve interpretability of model internals.

The Implicit Curriculum Hypothesis: Structure and Compositionality in LLM Skill Emergence

Introduction and Motivation

The pretraining of LLMs yields a continuous reduction in validation loss, but this aggregate metric fails to elucidate the fine-grained acquisition and ordering of distinct cognitive skills during training. Empirically, qualitative transitions—sometimes abrupt—occur in the emergence of specific behaviors and task competencies, a phenomenon not fully captured by scaling law analyses. Recent theoretical advances posit that models acquire discrete, compositional units of computation (referred to as “quanta” or skills), but practical evidence delineating the order and internal structure of these acquisitions across model families has been limited.

This paper posits the Implicit Curriculum Hypothesis: skills are acquired in a consistent, compositional order across diverse LLMs, and this structure is not only observable in external task performance, but also encoded in the representational geometry of model internals. The hypothesis is formalized behaviorally (emergence of composites after their constituents, consistency of emergence order across models) and representationally (similar function vectors for tasks with similar emergence profiles).

Methodology

A suite of 91 diagnostic tasks, including both elemental (e.g., string manipulation, translation, coreference, arithmetic, logic) and systematically constructed compositional variants, forms the empirical testbed. Composite tasks enforce explicit dependencies by construction, allowing direct tests of the hypothesized curriculum structure. Intermediate checkpoints from four LLM families (OLMo2/3, Pythia, LLM360/Amber/Crystal), spanning 410M–13B parameters, are scored on all tasks to resolve the timing and sequence of emergence events.

To evaluate representational alignment, task-specific function vectors (FVs) are extracted from the residual stream or dominant causal indirect effect heads in each model. Task similarity is quantified via cosine similarity in function vector space, and trajectory prediction leverages kernel ridge regression to assess whether learning curves of unseen compositional tasks can be inferred from this manifold.

Emergence Order Regularities

Analysis reveals that, under thresholded accuracy criteria, the order in which skills emerge remains highly stable across architectures, depths, data mixtures, and parameter counts. Spearman correlations of emergence orderings range from 0.64 to 0.93 across 45 model pairs (mean $\rho=0.81$), emphasizing that emergent curriculum structure is robust to hardware, software, and data distribution variations.

Figure 1: Emergence order across model families and sizes is regular, even as absolute emergence times shift.

A consensus sequence is visible: copying and basic string operations emerge first, followed by morphological transformation, simple retrieval, coreference, and straightforward logic; only then do complex arithmetic, reasoning, and compositional tasks activate. Critically, composite tasks overwhelmingly emerge later than their atomic constituents, consistent with the hypothesis that skill acquisition follows the compositional graph imposed by the task structure.

Figure 2: Emergence order heatmap corroborates stable emergence across models and task types, with composites delayed relative to their prerequisites.

The stability is specific to the use of fixed absolute performance thresholds ($\theta_{\text{abs}}$). Relative thresholds (fraction of maximum performance for a given task, $\alpha \cdot \text{max}$) yield substantially reduced ordering consistency across models, confirming that emergence as milestone competency, not relative gain, aligns with the acquisition of functional circuits in models.

Representational Geometry Predicts Skill Learning

Task representations as function vectors resolve whether internal model geometry is predictive of behavioral emergence. Learning trajectories of composite tasks are robustly inferred from their neighbors in function vector space, with leave-one-out prediction $R^2$ in the range 0.68 to 0.84 and MAE on held-out tasks as low as 0.07. In the majority of cases, the predicted trajectory captures both the inflection points and growth rates of the true skill learning curve, without requiring external performance measurement on the held-out task.

Figure 3: Example composite task learning curve prediction for OLMo2-7B via kernel regression on function vectors.

Exclusion of compositional bases in the regression (“simple tasks only”) yields uniform degradation in fit ($\Delta$MAE $= +0.135$), establishing that composite skills are not mere linear combinations of atomic skills, but manifest distinct, structured mechanisms in representation space.

Learning Trajectory and Scaling Analysis

Full learning trajectories for each model (Figures 4–12) detail the temporal development of every elemental and composite task through pretraining, capturing both smooth progressions and instabilities (particularly in larger models, e.g., Pythia-12B, OLMo2-13B). Across models, the synchronization of skill phase transitions further underscores the regularity of the implicit curriculum.

Discussion and Implications

Behavioral and Representational Evidence for Stable Curricula: The results provide strong support for the existence of a compositional, model-independent order in LLM skill emergence, as posited by the Implicit Curriculum Hypothesis. This predictive structure is observable both in externally measured task learning and in the abstract, high-dimensional representation geometry of the residual stream.

Inductive Bias and Curriculum Structure: The observed order suggests an interaction between data distribution, model architecture, and stochastic gradient dynamics that enforces a simplicity bias and compositional skill dependency—even with large, unstructured web-scale pretraining corpora. Tasks that depend on chaining or fusion of lower-level skills are reliably acquired only after the underlying substrate has stabilized, a property with direct implications for curriculum learning, data mixture design, and model diagnostics.

Monitoring and Steering Pretraining: Since the order and timing of skill emergence are highly consistent across training runs and families, the diagnostic task suite introduced could serve as a real-time monitoring tool for pretraining progress, flagging anticipated or anomalous deviations in skill development. This could enable data or architecture interventions to steer learning toward targeted capacities or to prevent premature overfitting to complex composites.

Emergence, Quanta, and Model Theory: The findings provide empirical grounding and refinement for quantized skill acquisition theories [michaud2023the], indicating that quanta in LLMs are not fully independent but are intricately compositional. The representational alignment between skill emergence and function vectors invites further mechanistic interpretability work to map and potentially manipulate skill circuits via activation engineering.

Conclusion

Skill acquisition during LLM pretraining is not a stochastic or black-box process but follows a reproducible, compositional sequence that is consistent across architectures and learning setups. This stable implicit curriculum is directly testable with diagnostic tasks and decodable from model internals, providing new levers for interpretability, pretraining monitoring, and principled curriculum design. Understanding and leveraging this curriculum may inform both the engineering and theory of next-generation LLMs.

Reference:

"What do LLMs Learn and When? The Implicit Curriculum Hypothesis" (2604.08510)