## 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.

Original language | English |
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Pages (from-to) | S566-S592 |

Journal | SIAM Journal on Scientific Computing |

Volume | 38 |

Issue number | 5 |

DOIs | |

State | Published - 1 Jan 2016 |

Externally published | Yes |

## Keywords

- Block coordinate descent
- Covariance selection
- Multilevel methods
- Proximal Newton
- Sparse inverse covariance estimation
- Sparse optimization
- l regularized logistic regression

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics