Free Energy Wells and Overlap Gap Property in Sparse PCA

Gérard Ben Arous, Alexander S. Wein, Ilias Zadik

Research output: Contribution to journalConference articlepeer-review

Abstract

We study a variant of the sparse PCA (principal component analysis) problem in the “hard” regime, where the inference task is possible yet no polynomial-time algorithm is known to exist. Prior work, based on the low-degree likelihood ratio, has conjectured a precise expression for the best possible (sub-exponential) runtime throughout the hard regime. Following instead a statistical physics inspired point of view, we show bounds on the depth of free energy wells for various Gibbs measures naturally associated to the problem. These free energy wells imply hitting time lower bounds that corroborate the low-degree conjecture: we show that a class of natural MCMC (Markov chain Monte Carlo) methods (with worst-case initialization) cannot solve sparse PCA with less than the conjectured runtime. These lower bounds apply to a wide range of values for two tuning parameters: temperature and sparsity misparametrization. Finally, we prove that the Overlap Gap Property (OGP), a structural property that implies failure of certain local search algorithms, holds in a significant part of the hard regime.

Original languageEnglish (US)
Pages (from-to)479-482
Number of pages4
JournalProceedings of Machine Learning Research
Volume125
StatePublished - 2020
Event33rd Conference on Learning Theory, COLT 2020 - Virtual, Online, Austria
Duration: Jul 9 2020Jul 12 2020

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

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