Affiliation:
1. School of Computer Science Georgia Institute of Technology Atlanta Georgia USA
2. Center of Mathematical Sciences and Applications Harvard University Cambridge Massachusetts USA
Abstract
AbstractWe prove, under an assumption on the critical points of a real‐valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1‐RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1‐RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.
Subject
Applied Mathematics,Computer Graphics and Computer-Aided Design,General Mathematics,Software
Reference48 articles.
1. Community detection and stochastic block models: Recent developments;Abbe E.;J Mach Learn Res,2017
2. D.AchlioptasandA.Coja‐Oghlan.Algorithmic barriers from phase transitions. Paper presented at: 2008 49th annual IEEE symposium on foundations of computer science IEEE.2008793–802.
3. On the solution-space geometry of random constraint satisfaction problems
4. E.Abbe S.Li andA.Sly.Proof of the contiguity conjecture and lognormal limit for the symmetric perceptron. Paper presented at: 2021 IEEE 62nd annual symposium on foundations of computer science (FOCS) IEEE.2022327–338.
5. E.Abbe S.Li andA.Sly.Binary perceptron: Efficient algorithms can find solutions in a rare well‐connected cluster Proceedings of the 54th annual ACM SIGACT symposium on theory of computing.2022860–873.