TY - GEN
T1 - Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron
AU - Abbe, Emmanuel
AU - Li, Shuangning
AU - Sly, Allan
N1 - Publisher Copyright:
© 2022 IEEE.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - freezing
KW - interpo-lation
KW - neural networks
KW - partition function
KW - perceptron model
KW - sharp thresholds
KW - solution space
UR - http://www.scopus.com/inward/record.url?scp=85127115705&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85127115705&partnerID=8YFLogxK
U2 - 10.1109/FOCS52979.2021.00041
DO - 10.1109/FOCS52979.2021.00041
M3 - Conference contribution
AN - SCOPUS:85127115705
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 327
EP - 338
BT - Proceedings - 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science, FOCS 2021
PB - IEEE Computer Society
T2 - 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021
Y2 - 7 February 2022 through 10 February 2022
ER -