Machine Learning

, Volume 93, Issue 1, pp 31-52

First online:

Block coordinate descent algorithms for large-scale sparse multiclass classification

  • Mathieu BlondelAffiliated withGraduate School of System Informatics, Kobe University Email author 
  • , Kazuhiro SekiAffiliated withGraduate School of System Informatics, Kobe University
  • , Kuniaki UeharaAffiliated withGraduate School of System Informatics, Kobe University

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Over the past decade, 1 regularization has emerged as a powerful way to learn classifiers with implicit feature selection. More recently, mixed-norm (e.g., 1/ 2) regularization has been utilized as a way to select entire groups of features. In this paper, we propose a novel direct multiclass formulation specifically designed for large-scale and high-dimensional problems such as document classification. Based on a multiclass extension of the squared hinge loss, our formulation employs 1/ 2 regularization so as to force weights corresponding to the same features to be zero across all classes, resulting in compact and fast-to-evaluate multiclass models. For optimization, we employ two globally-convergent variants of block coordinate descent, one with line search (Tseng and Yun in Math. Program. 117:387–423, 2009) and the other without (Richtárik and Takáč in Math. Program. 1–38, 2012a; Tech. Rep. arXiv:​1212.​0873, 2012b). We present the two variants in a unified manner and develop the core components needed to efficiently solve our formulation. The end result is a couple of block coordinate descent algorithms specifically tailored to our multiclass formulation. Experimentally, we show that block coordinate descent performs favorably compared to other solvers such as FOBOS, FISTA and SpaRSA. Furthermore, we show that our formulation obtains very compact multiclass models and outperforms 1/ 2-regularized multiclass logistic regression in terms of training speed, while achieving comparable test accuracy.


Multiclass classification Group sparsity Block coordinate descent