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We show that weight matrix updates in learning algorithms follow Dyson Brownian motion, linking stochasticity to learning rate and minibatch size. This reveals universal features from random matrix theory in machine learning models.

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Area of Science:

  • Machine Learning
  • Statistical Physics
  • Random Matrix Theory

Background:

  • Weight matrix updates are central to machine learning algorithms.
  • Understanding the dynamics of these updates is crucial for algorithm performance and generalization.
  • Existing theories often lack a unifying framework to explain the stochastic nature of these updates.

Purpose of the Study:

  • To demonstrate that weight matrix updates can be modeled using Dyson Brownian motion.
  • To establish a connection between the stochasticity of learning and the ratio of learning rate to minibatch size.
  • To identify universal and nonuniversal features in the resulting distributions and relate them to established random matrix theory concepts.

Main Methods:

  • Applying the framework of Dyson Brownian motion to analyze weight matrix updates.
  • Relating the learning rate and minibatch size ratio to the level of stochasticity.
  • Analyzing the Coulomb gas distribution of eigenvalues.
  • Identifying specific random matrix theory distributions (Wigner surmise, semicircle) in models.

Main Results:

  • Weight matrix updates are shown to be describable by Dyson Brownian motion.
  • A scaling relationship between stochasticity and the learning rate/minibatch size ratio is confirmed.
  • Universal and nonuniversal features of the Coulomb gas distribution are discussed.
  • The Wigner surmise and semicircle distributions are explicitly identified in teacher-student and Gaussian restricted Boltzmann machine models.

Conclusions:

  • Dyson Brownian motion provides a powerful framework for understanding learning algorithm dynamics.
  • The findings offer robust evidence for conjectured scaling relationships in machine learning.
  • This work bridges concepts from random matrix theory and machine learning, offering new insights into model behavior.