Newton-like methods for sparse inverse covariance estimation

Abstract

We propose two classes of second-order optimization methods for solving the sparse inverse covariance estimation problem. The first approach, which we call the Newton-LASSO method, minimizes a piecewise quadratic model of the objective function at every iteration to generate a step. We employ the fast iterative shrinkage thresholding algorithm (FISTA) to solve this subproblem. The second approach, which we call the Orthant-Based Newton method, is a two-phase algorithm that first identifies an orthant face and then minimizes a smooth quadratic approximation of the objective function using the conjugate gradient method. These methods exploit the structure of the Hessian to efficiently compute the search direction and to avoid explicitly storing the Hessian. We also propose a limited memory BFGS variant of the orthant-based Newton method. Numerical results, including comparisons with the method implemented in the QUIC software [1], suggest that all the techniques described in this paper constitute useful tools for the solution of the sparse inverse covariance estimation problem.