Model identification and local linear convergence of coordinate descent

Quentin Klopfenstein (1) , Quentin Bertrand (2) , Alexandre Gramfort (2) , Joseph Salmon (3) , Samuel Vaiter (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
3 IMAG - Institut Montpelliérain Alexander Grothendieck

For composite nonsmooth optimization problems, Forward-Backward algorithm achieves model identification (e.g., support identification for the Lasso) after a finite number of iterations, provided the objective function is regular enough. Results concerning coordinate descent are scarcer and model identification has only been shown for specific estimators, the support-vector machine for instance. In this work, we show that cyclic coordinate descent achieves model identification in finite time for a wide class of functions. In addition, we prove explicit local linear convergence rates for coordinate descent. Extensive experiments on various estimators and on real datasets demonstrate that these rates match well empirical results.

Domaines: Statistiques [math.ST]; Optimisation et contrôle [math.OC]

Quentin Klopfenstein, Quentin Bertrand, Alexandre Gramfort, Joseph Salmon, Samuel Vaiter. Model identification and local linear convergence of coordinate descent. 2020. ⟨hal-03019711⟩