Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

Emmanuel Auguste Abbe, Shuangning Li, Allan Sly

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

Abstract

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15. We establish that the partition function of this model, normalized by its expected value, converges to a log-normal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mézard '89 in the asymmetric case. In a recent work of Perkins-Xu '21, the last two conjectures were also established by proving that the partition function concentrates on an exponential scale, under an analytical assumption on a real-valued function. This left open the contiguity conjecture and the lognor-mal limit characterization, which are established here unconditionally, with the analytical assumption verified. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

Original languageEnglish (US)
Title of host publicationProceedings - 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science, FOCS 2021
PublisherIEEE Computer Society
Pages327-338
Number of pages12
ISBN (Electronic)9781665420556
DOIs
StatePublished - 2022
Event62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 - Virtual, Online, United States
Duration: Feb 7 2022Feb 10 2022

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2022-February
ISSN (Print)0272-5428

Conference

Conference62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021
Country/TerritoryUnited States
CityVirtual, Online
Period2/7/222/10/22

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

Keywords

  • freezing
  • interpo-lation
  • neural networks
  • partition function
  • perceptron model
  • sharp thresholds
  • solution space

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