TY - JOUR

T1 - Free Energy Wells and Overlap Gap Property in Sparse PCA

AU - Arous, Gérard Ben

AU - Wein, Alexander S.

AU - Zadik, Ilias

N1 - Funding Information:
A.S.W. is partially supported by NSF grant DMS-1712730 and by the Simons Collaboration on Algorithms and Geometry. I.Z. is supported by a CDS Moore-Sloan postdoctoral fellowship. The authors would like to thank David Gamarnik for helpful comments on an earlier draft of this work. We also thank the anonymous reviewers for their helpful comments.
Publisher Copyright:
© 2020 G.B. Arous, A. S. Wein & I. Zadik.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

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M3 - Conference article

AN - SCOPUS:85097971682

SN - 2640-3498

VL - 125

SP - 479

EP - 482

JO - Proceedings of Machine Learning Research

JF - Proceedings of Machine Learning Research

T2 - 33rd Conference on Learning Theory, COLT 2020

Y2 - 9 July 2020 through 12 July 2020

ER -