Scalable sparse covariance estimation via self-concordance

TL;DR

This paper introduces a scalable optimization method for sparse covariance estimation that leverages self-concordant functions, providing improved convergence guarantees and efficiency in high-dimensional settings.

Contribution

It develops a proximal Newton algorithm tailored for self-concordant functions with inexact evaluations, advancing optimization techniques for covariance estimation.

Findings

The proposed method achieves faster convergence rates.

It demonstrates superior recovery performance in sparse covariance estimation.

The algorithm reduces computational complexity in high-dimensional problems.

Abstract

We consider the class of convex minimization problems, composed of a self-concordant function, such as the $\log\det$ metric, a convex data fidelity term $h(\cdot)$ and, a regularizing -- possibly non-smooth -- function $g(\cdot)$. This type of problems have recently attracted a great deal of interest, mainly due to their omnipresence in top-notch applications. Under this \emph{locally} Lipschitz continuous gradient setting, we analyze the convergence behavior of proximal Newton schemes with the added twist of a probable presence of inexact evaluations. We prove attractive convergence rate guarantees and enhance state-of-the-art optimization schemes to accommodate such developments. Experimental results on sparse covariance estimation show the merits of our algorithm, both in terms of recovery efficiency and complexity.