A Multilevel Framework for Sparse Optimization with Application to Inverse Covariance Estimation and Logistic Regression

Eran Treister, Javier S. Turek, Irad Yavneh

Abstract: Solving $l_1$ regularized optimization problems is common in the fields of computational biology, signal processing, and machine learning. Such $l_1$ regularization is utilized to find sparse minimizers of convex functions. A well-known example is the least absolute shrinkage and selection operator (LASSO) problem, where the $l_1$ norm regularizes a quadratic function. A multilevel framework is presented for solving such $l_1$ regularized sparse optimization problems efficiently. We take advantage of the expected sparseness of the solution, and create a hierarchy of problems of similar type, which is traversed in order to accelerate the optimization process. This framework is applied for solving two problems: (1) the sparse inverse covariance estimation problem, and (2) $l_1$ regularized logistic regression. In the first problem, the inverse of an unknown covariance matrix of a multivariate normal distribution is estimated, under the assumption that it is sparse. To this end, an $l_1$ regularized log-determinant optimization problem needs to be solved. This task is challenging especially for large-scale datasets, due to time and memory limitations. In the second problem, the $l_1$ regularization is added to the logistic regression classification objective to reduce overfitting to the data and obtain a sparse model. Numerical experiments demonstrate the efficiency of the multilevel framework in accelerating existing iterative solvers for both of these problems.