Optimal scaling laws in learning hierarchical multi-index models
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915778202173440 |
|---|---|
| author | Defilippis, Leonardo Krzakala, Florent Loureiro, Bruno Maillard, Antoine |
| author_facet | Defilippis, Leonardo Krzakala, Florent Loureiro, Bruno Maillard, Antoine |
| contents | In this work, we provide a sharp theory of scaling laws for two-layer neural networks trained on a class of hierarchical multi-index targets, in a genuinely representation-limited regime. We derive exact information-theoretic scaling laws for subspace recovery and prediction error, revealing how the hierarchical features of the target are sequentially learned through a cascade of phase transitions. We further show that these optimal rates are achieved by a simple, target-agnostic spectral estimator, which can be interpreted as the small learning-rate limit of gradient descent on the first-layer weights. Once an adapted representation is identified, the readout can be learned statistically optimally, using an efficient procedure. As a consequence, we provide a unified and rigorous explanation of scaling laws, plateau phenomena, and spectral structure in shallow neural networks trained on such hierarchical targets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05846 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal scaling laws in learning hierarchical multi-index models Defilippis, Leonardo Krzakala, Florent Loureiro, Bruno Maillard, Antoine Machine Learning In this work, we provide a sharp theory of scaling laws for two-layer neural networks trained on a class of hierarchical multi-index targets, in a genuinely representation-limited regime. We derive exact information-theoretic scaling laws for subspace recovery and prediction error, revealing how the hierarchical features of the target are sequentially learned through a cascade of phase transitions. We further show that these optimal rates are achieved by a simple, target-agnostic spectral estimator, which can be interpreted as the small learning-rate limit of gradient descent on the first-layer weights. Once an adapted representation is identified, the readout can be learned statistically optimally, using an efficient procedure. As a consequence, we provide a unified and rigorous explanation of scaling laws, plateau phenomena, and spectral structure in shallow neural networks trained on such hierarchical targets. |
| title | Optimal scaling laws in learning hierarchical multi-index models |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.05846 |