@inproceedings{bcf6e7c8819341ed806d4ac2f19c6f04,

title = "Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster",

abstract = "It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically to succeed in finding solutions at low density. This phenomenon has been justified numerically by the existence of subdominant and dense connected regions of solutions, which are accessible by simple learning algorithms. In this paper, we establish formally such a phenomenon for both the symmetric and asymmetric binary perceptrons. We show that at low constraint density (equivalently for overparametrized perceptrons), there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu in STOC'21. In addition, even close to the critical threshold, we show that there exist clusters of linear diameter for the symmetric perceptron, as well as for the asymmetric perceptron under additional assumptions.",

keywords = "binary perceptron, efficient algorithm, neural networks, perceptron model, solution space",

author = "Emmanuel Abbe and Shuangping Li and Allan Sly",

note = "Publisher Copyright: {\textcopyright} 2022 Owner/Author.; 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 ; Conference date: 20-06-2022 Through 24-06-2022",

year = "2022",

month = sep,

day = "6",

doi = "10.1145/3519935.3519975",

language = "English (US)",

series = "Proceedings of the Annual ACM Symposium on Theory of Computing",

publisher = "Association for Computing Machinery",

pages = "860--873",

editor = "Stefano Leonardi and Anupam Gupta",

booktitle = "STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing",

}