Physics of Skill Learning

TL;DR

Physics of Skill Learning introduces three progressively abstract models—the Geometry, Resource, and Domino models—to explain the Domino effect in neural-network skill acquisition. The Geometry model represents skills as linear directions in parameter space, with a loss that aggregates task-specific losses; the Resource model recasts this as competition for a finite learning resource, and the Domino model emerges under strong hierarchy and no waste. The work demonstrates how optimization (notably SignGD) and task structure yield sequential learning, derives conserved quantities that predict learning-time scaling, and connects these insights to neural scaling laws, modularity, and optimizer design. Collectively, the framework offers practical guidance for speeding training (via weighting schemes and modularity) and a conceptual lens for interpreting how complex, compositional tasks unfold in high-dimensional learning systems.

Abstract

We aim to understand physics of skill learning, i.e., how skills are learned in neural networks during training. We start by observing the Domino effect, i.e., skills are learned sequentially, and notably, some skills kick off learning right after others complete learning, similar to the sequential fall of domino cards. To understand the Domino effect and relevant behaviors of skill learning, we take physicists' approach of abstraction and simplification. We propose three models with varying complexities -- the Geometry model, the Resource model, and the Domino model, trading between reality and simplicity. The Domino effect can be reproduced in the Geometry model, whose resource interpretation inspires the Resource model, which can be further simplified to the Domino model. These models present different levels of abstraction and simplification; each is useful to study some aspects of skill learning. The Geometry model provides interesting insights into neural scaling laws and optimizers; the Resource model sheds light on the learning dynamics of compositional tasks; the Domino model reveals the benefits of modularity. These models are not only conceptually interesting -- e.g., we show how Chinchilla scaling laws can emerge from the Geometry model, but also are useful in practice by inspiring algorithmic development -- e.g., we show how simple algorithmic changes, motivated by these toy models, can speed up the training of deep learning models.

Figures (21)

• Figure 1: Physicists are famous for making up (sometimes overly) simplistic models. We take the same spirit to understand skill learning. We propose three models trading off between reality and simplicity: the Geometry model, the Resource model, and the Domino model.
• Figure 2: The Domino effect (sequential learning of tasks) occurs for sparse parity learning. Top: The Domino effect can be attributed to task imbalance (left) or compositional dependency (right). Bottom: how task imbalance influences learning dynamics.
• Figure 3: Organization. Physics-like theories (models) are inspired by experiments and contribute to guiding the design of new experiments.
• Figure 4: The Geometry model + SignGD leads to robust Domino effects (the second task starts to rapidly learn only after the first task completes learning), while SGD and Adam do not. (a) skill dynamics with different optimizers and different probabilities. (b) With SignGD, the learning time ratio saturates to 2 as two tasks become more unbalanced, while SGD grows linearly and Adam grows sub-linearly.
• Figure 5: Results of the Geometry model. Top: Skills show sequential learning dynamics under different power law distributions $\alpha=\{1,2,4\}$ and $n_{\rm task}=\{2,3,4,5,10\}$. Bottom: the number of gradient-aligned dimensions $n_{\rm align}$ (interpreted as resources) also shows a sequential order.