Statistical physics of deep learning: Optimal learning of a multi-layer perceptron near interpolation

2025-10-28 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, short

Summary

Jean Barbier and colleagues, in their paper revised on July 23, 2026, apply statistical physics to analyze deep learning models, specifically a multi-layer perceptron (MLP). This research extends the framework beyond narrow networks or kernel methods to capture rich feature learning effects. They investigate an MLP where its width scales with the input dimension, making it more capable of feature learning than ultra-wide or narrow networks. The study focuses on the challenging interpolation regime, where trainable parameters and data are comparable, forcing the model to adapt. Using a matched teacher-student setting, the authors establish fundamental limits for learning random deep neural network targets. Their findings reveal a complex phenomenology with various learning transitions, showing that optimal performance, achieved through model "specialisation," can be difficult for training algorithms to reach due to attraction to sub-optimal solutions. Specialisation propagates inhomogeneously from shallow to deep layers and across neurons, with deeper targets proving harder to learn.

Key takeaway

For AI Scientists optimizing deep learning models, this research highlights critical dynamics in the interpolation regime. You should recognize that achieving optimal performance requires navigating complex "specialisation" pathways, as training algorithms can converge on sub-optimal solutions. Consider the inhomogeneous nature of learning across layers and neurons, and anticipate increased difficulty when tackling deeper target functions. Your training strategies might benefit from methods that explicitly encourage or monitor layer-specific specialization to avoid local minima.

Key insights

Statistical physics reveals deep learning's feature learning limits and optimal specialization in the interpolation regime.

Principles

• Optimal learning involves model "specialisation."
• Deeper neural network targets are harder to learn.
• Specialisation propagates inhomogeneously across layers.

Method

The study uses a matched teacher-student setting to analyze supervised learning of a multi-layer perceptron in the interpolation regime, where width scales with input dimension.

In practice

• Be aware of sub-optimal solutions during training.
• Consider layer-wise specialization in deep networks.
• Deeper targets increase learning difficulty.

Topics

• Statistical Physics
• Deep Learning
• Multi-layer Perceptron
• Interpolation Regime
• Feature Learning
• Learning Dynamics

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.