## Abstract

In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex function. The theoretical speedup, as compared to the serial method, and referring to the number of iterations needed to approximately solve the problem with high probability, is a simple expression depending on the number of parallel processors and a natural and easily computable measure of separability of the smooth component of the objective function. In the worst case, when no degree of separability is present, there may be no speedup; in the best case, when the problem is separable, the speedup is equal to the number of processors. Our analysis also works in the mode when the number of blocks being updated at each iteration is random, which allows for modeling situations with busy or unreliable processors. We show that our algorithm is able to solve a LASSO problem involving a matrix with 20 billion nonzeros in 2 h on a large memory node with 24 cores.

Original language | English (US) |
---|---|

Pages (from-to) | 433-484 |

Number of pages | 52 |

Journal | Mathematical Programming |

Volume | 156 |

Issue number | 1-2 |

DOIs | |

State | Published - Mar 1 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015, The Author(s).

## Keywords

- Big data optimization
- Composite objective
- Convex optimization
- Expected separable over-approximation
- Huge-scale optimization
- Iteration complexity
- LASSO
- Parallel coordinate descent
- Partial separability

## ASJC Scopus subject areas

- Software
- General Mathematics