## Abstract

In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an e-Accurate solution with probability at least 1-p in at most O((n/e) log (1/p)) iterations, where n is the number of blocks. This extends recent results of Nesterov (SIAM J Optim 22(2): 341-362, 2012), which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing efrom the logarithmic term. More importantly, in contrast with the aforementioned work in which the author achieves the results by applying the method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving first true iteration complexity bounds. For strongly convex functions the method converges linearly. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Finally, we demonstrate numerically that the algorithm is able to solve huge-scale l_{1}-regularized least squares problems with a billion variables.

Original language | English (US) |
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Pages (from-to) | 1-38 |

Number of pages | 38 |

Journal | Mathematical Programming |

Volume | 144 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2014 |

## Keywords

- Block coordinate descent
- Composite minimization
- Convex optimization
- Coordinate relaxation
- Gauss-Seidel method
- Gradient descent
- Huge-scale optimization
- Iteration complexity
- LASSO
- Sparse regression

## ASJC Scopus subject areas

- Software
- Mathematics(all)