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Spin Glass Concepts in Computer Science, Statistics, and Learning

Andrea Montanari

TL;DR

This article will try to explain how ideas from spin glass theory have seeded recent developments in computer science, statistics, and machine learning.

Abstract

Spin glass theory studies the structure of sublevel sets and minima (or near-minima) of certain classes of random functions in high dimension. Near-minima of random functions also play an important role in high-dimensional statistics and statistical learning, where minimizing the empirical risk (which is a random function of the model parameters) is the method of choice for learning a statistical model from noisy data. Finally, near-minima of random functions are obviously central to average-case analysis of optimization algorithms. Computer science, statistics, and machine learning naturally lead to questions that are traditionally not addressed within physics and mathematical physics. I will try to explain how ideas from spin glass theory have seeded recent developments in these fields. (This article was written on the occasion of the 2024 Abel Prize to Michel Talagrand.)

Spin Glass Concepts in Computer Science, Statistics, and Learning

TL;DR

This article will try to explain how ideas from spin glass theory have seeded recent developments in computer science, statistics, and machine learning.

Abstract

Spin glass theory studies the structure of sublevel sets and minima (or near-minima) of certain classes of random functions in high dimension. Near-minima of random functions also play an important role in high-dimensional statistics and statistical learning, where minimizing the empirical risk (which is a random function of the model parameters) is the method of choice for learning a statistical model from noisy data. Finally, near-minima of random functions are obviously central to average-case analysis of optimization algorithms. Computer science, statistics, and machine learning naturally lead to questions that are traditionally not addressed within physics and mathematical physics. I will try to explain how ideas from spin glass theory have seeded recent developments in these fields. (This article was written on the occasion of the 2024 Abel Prize to Michel Talagrand.)
Paper Structure (20 sections, 7 theorems, 77 equations, 1 figure)

This paper contains 20 sections, 7 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivations and questions
  3. Statistical physics.
  4. Computer science.
  5. High-dimensional statistics.
  6. Sampling.
  7. Parisi's formula
  8. Computer science approaches
  9. Message passing, belief propagation, and the cavity method
  10. Approximate Message Passing and high-dimensional statistics
  11. A general algorithm and its characterization
  12. Low-rank matrix estimation via AMP
  13. Optimal low-rank matrix estimation and statistical computation gaps
  14. Algorithimc thresholds for spin glasses
  15. Optimizing spin glass Hamiltonians
  16. ...and 5 more sections

Key Result

Theorem 1

Let ${\sf P}_{\mathscrsfs{U}}^*(\xi):=\inf_{\gamma\in\mathscrsfs{U}} {\sf P}(\gamma;\xi)$. Then, for $H_{\hbox{\tiny \rm SK}}({\boldsymbol{\sigma}})$ the Hamiltonian of Eq. eq:Hsigma with $\boldsymbol{A}\sim{\sf GOE}(n)$, $\xi_{\hbox{\tiny \rm SK}}(t)=t^2/2$, $\Sigma_n=\{+1,-1\}^n$, we have (almost

Figures (1)

  • Figure 1:

Theorems & Definitions (21)

  • Theorem 1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 5.1: AMP iteration; Matrix data case
  • Definition 5.2: State Evolution process and recursion
  • Theorem 2
  • Remark 5.1
  • Remark 5.2
  • Proposition 5.3
  • ...and 11 more