## Abstract

The cyclic block coordinate descent-type (CBCD-type) methods, which perform iterative updates for a few coordinates (a block) simultaneously throughout the procedure, have shown remarkable computational performance for solving strongly convex minimization problems. Typical applications include many popular statistical machine learning methods such as elastic-net regression, ridge penalized logistic regression, and sparse additive regression. Existing optimization literature has shown that for strongly convex minimization, the CBCD-type methods attain iteration complexity of O(p log(1/)), where is a pre-specified accuracy of the objective value, and p is the number of blocks. However, such iteration complexity explicitly depends on p, and therefore is at least p times worse than the complexity O(log(1/)) of gradient descent (GD) methods. To bridge this theoretical gap, we propose an improved convergence analysis for the CBCD-type methods. In particular, we first show that for a family of quadratic minimization problems, the iteration complexity O(log^{2}(p) · log(1/)) of the CBCD-type methods matches that of the GD methods in term of dependency on p, up to a log^{2} p factor. Thus our complexity bounds are sharper than the existing bounds by at least a factor of p/log^{2}(p). We also provide a lower bound to confirm that our improved complexity bounds are tight (up to a log^{2}(p) factor), under the assumption that the largest and smallest eigenvalues of the Hessian matrix do not scale with p. Finally, we generalize our analysis to other strongly convex minimization problems beyond quadratic ones.

Original language | English (US) |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Journal of Machine Learning Research |

Volume | 18 |

State | Published - Apr 1 2018 |

### Bibliographical note

Funding Information:This research is supported by NSF DMS1454377-CAREER; NSF IIS 1546482-BIGDATA; NIH R01MH102339; NSF IIS1408910; NSF IIS1332109; NIH R01GM083084; NSF CMMI1727757.

Publisher Copyright:

© 2018 Xingguo Li, Tuo Zhao, Raman Arora, Han Liu and Mingyi Hong.

## Keywords

- Cyclic block coordinate descent
- Gradient descent
- Improved iteration complexity
- Quadratic minimization
- Strongly convex minimization