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We introduce a solvable model for deep neural network learning. The ratio of network depth and width determines distinct learning regimes, impacting feature learning capabilities.

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Area of Science:

  • Machine Learning
  • Statistical Physics
  • Deep Learning

Background:

  • Analyzing Bayesian posteriors in deep neural networks is complex.
  • Existing models often require strong assumptions on initialization or data.

Purpose of the Study:

  • To develop a perturbatively solvable model for deep multilayer perceptrons.
  • To explore learning regimes in large neural networks without restrictive assumptions.

Main Methods:

  • Diagrammatic approach analyzing Gibbs measures (Bayesian posteriors).
  • Analysis of deep shaped multilayer perceptrons at arbitrary temperature.
  • Study of limits where input dimension, depth, width, and samples all approach infinity.

Main Results:

  • The limits N₀, N, L, P → ∞ do not commute, yielding a rich phase diagram.
  • The ratio LP/N defines critical depth for feature learning.
  • For LP/N → 0, posteriors match kernel methods; for LP/N → λ > 0, data-dependent kernel deformations occur.

Conclusions:

  • The developed model offers insights into feature learning in deep neural networks.
  • The critical ratio LP/N governs the transition between kernel and data-dependent learning regimes.
  • Explicit formulas for learned features are derived to first order in 1/N.