Cognitive Training for Language Models: Towards General Capabilities via Cross-Entropy Games

Clément Hongler; Franck Gabriel; Valentin Hartmann; Arthur Renard; Andrew Emil

math.OCMar 23, 2026

Cognitive Training for Language Models: Towards General Capabilities via Cross-Entropy Games

Authors: Clément Hongler, Franck Gabriel, Valentin Hartmann, Arthur Renard, Andrew Emil

Organizations: XENT LABS · EPFL · UNIVERSITÉ LYON 1

Abstract

Defining a constructive process to build general capabilities for language models in an automatic manner is considered an open problem in artificial intelligence. Towards this, we consider the problem of building a curriculum of tasks that grows a model via relevant skill discovery. We provide a concrete framework for this task, using a family of tasks called Cross-Entropy Games, which we postulate is universal in a suitable sense. We show that if it is possible to grow the curriculum for relevant skill discovery by iterating a greedy optimization algorithm, then, under natural assumptions, there is essentially only one meta-objective possible (up to a few hyper-parameters). We call the resulting process cognitive training. We postulate that, given sufficiently capable language models as players and meta-samplers, cognitive training provides a principled way to relevant skill discovery; and hence to the extent general capabilities are achievable via greedy curriculum learning, cognitive training would be a solution.

Figures & tables

Panel of Figure 4 — Figure 4.1. Left: Geometric mean of game scores as a function of the diversity hyper-parameter $\delta$ . Middle: geometric mean of game scores during training. Right: number of distinct games selected during training. All values are averaged over 64 independently sampled toy worlds; shaded bands show standard error across worlds.

LLM Training

Large language model training studies the data, objectives, optimizers, and systems used to build and adapt language models.

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Apr 9, 2026J/K move · Enter open · S save

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

Emmy Liu, Kaiser Sun, Millicent Li, Isabelle Lee+3

Language Technologies Institute, Carnegie Mellon University · Department of Computer Science, Data Science and AI Institute, Johns Hopkins University · Khoury College of Computer Science, Northeastern University+1

Emergent Abilities in Language Models Language Model Pretraining

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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.

Abstract

LLM Training

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