Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster
This addresses the challenge of finding solutions in binary perceptrons, which is relevant for machine learning theory and algorithm design, though it is incremental as it builds on prior empirical observations.
The paper proves that, despite most solutions being isolated, there exists a well-connected cluster of solutions in binary perceptrons at low constraint density, and demonstrates that an efficient algorithm can find solutions in this cluster with high probability, settling an open problem from prior work.
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 '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.