Sparse PCA: Algorithms, Adversarial Perturbations and Certificates

TL;DR

This paper develops robust polynomial-time algorithms for Sparse PCA that withstand adversarial perturbations, using certificates of sparse eigenvalues, and establishes lower bounds showing the inherent cost of robustness.

Contribution

It introduces the first polynomial-time algorithms for robust Sparse PCA based on certificates, and proves lower bounds indicating the fundamental trade-off between robustness and computational efficiency.

Findings

Algorithms based on certificates are robust against small adversarial perturbations.

New efficient certificates for sparse eigenvalues enable resilient algorithms.

Lower bounds demonstrate the inherent cost of robustness in Sparse PCA.

Abstract

We study efficient algorithms for Sparse PCA in standard statistical models (spiked covariance in its Wishart form). Our goal is to achieve optimal recovery guarantees while being resilient to small perturbations. Despite a long history of prior works, including explicit studies of perturbation resilience, the best known algorithmic guarantees for Sparse PCA are fragile and break down under small adversarial perturbations. We observe a basic connection between perturbation resilience and \emph{certifying algorithms} that are based on certificates of upper bounds on sparse eigenvalues of random matrices. In contrast to other techniques, such certifying algorithms, including the brute-force maximum likelihood estimator, are automatically robust against small adversarial perturbation. We use this connection to obtain the first polynomial-time algorithms for this problem that are resilient against additive adversarial perturbations by obtaining new efficient certificates for upper bounds on sparse eigenvalues of random matrices. Our algorithms are based either on basic semidefinite programming or on its low-degree sum-of-squares strengthening depending on the parameter regimes. Their guarantees either match or approach the best known guarantees of \emph{fragile} algorithms in terms of sparsity of the unknown vector, number of samples and the ambient dimension. To complement our algorithmic results, we prove rigorous lower bounds matching the gap between fragile and robust polynomial-time algorithms in a natural computational model based on low-degree polynomials (closely related to the pseudo-calibration technique for sum-of-squares lower bounds) that is known to capture the best known guarantees for related statistical estimation problems. The combination of these results provides formal evidence of an inherent price to pay to achieve robustness.