We consider ``one-at-a-time'' coordinate-wise descent algorithms for a class
of convex optimization problems. An algorithm of this kind has been proposed
for the \$L\_1\$-penalized regression (lasso) in the literature, but it seems to
have been largely ignored. Indeed, it seems that coordinate-wise algorithms are
not often used in convex optimization. We show that this algorithm is very
competitive with the well-known LARS (or homotopy) procedure in large lasso
problems, and that it can be applied to related methods such as the garotte and
elastic net. It turns out that coordinate-wise descent does not work in the
``fused lasso,'' however, so we derive a generalized algorithm that yields the
solution in much less time that a standard convex optimizer. Finally, we
generalize the procedure to the two-dimensional fused lasso, and demonstrate
its performance on some image smoothing problems.
%0 Journal Article
%1 friedman2007pathwise
%A Friedman, Jerome
%A Hastie, Trevor
%A Höfling, Holger
%A Tibshirani, Robert
%D 2007
%J The Annals of Applied Statistics
%K coordinate-descent, fused-lasso optimization total_variation
%N 2
%P 302--332
%R 10.1214/07-aoas131
%T Pathwise coordinate optimization
%U http://dx.doi.org/10.1214/07-aoas131
%V 1
%X We consider ``one-at-a-time'' coordinate-wise descent algorithms for a class
of convex optimization problems. An algorithm of this kind has been proposed
for the \$L\_1\$-penalized regression (lasso) in the literature, but it seems to
have been largely ignored. Indeed, it seems that coordinate-wise algorithms are
not often used in convex optimization. We show that this algorithm is very
competitive with the well-known LARS (or homotopy) procedure in large lasso
problems, and that it can be applied to related methods such as the garotte and
elastic net. It turns out that coordinate-wise descent does not work in the
``fused lasso,'' however, so we derive a generalized algorithm that yields the
solution in much less time that a standard convex optimizer. Finally, we
generalize the procedure to the two-dimensional fused lasso, and demonstrate
its performance on some image smoothing problems.
@article{friedman2007pathwise,
abstract = {{We consider ``one-at-a-time'' coordinate-wise descent algorithms for a class
of convex optimization problems. An algorithm of this kind has been proposed
for the \$L\_1\$-penalized regression (lasso) in the literature, but it seems to
have been largely ignored. Indeed, it seems that coordinate-wise algorithms are
not often used in convex optimization. We show that this algorithm is very
competitive with the well-known LARS (or homotopy) procedure in large lasso
problems, and that it can be applied to related methods such as the garotte and
elastic net. It turns out that coordinate-wise descent does not work in the
``fused lasso,'' however, so we derive a generalized algorithm that yields the
solution in much less time that a standard convex optimizer. Finally, we
generalize the procedure to the two-dimensional fused lasso, and demonstrate
its performance on some image smoothing problems.}},
added-at = {2018-12-07T09:10:16.000+0100},
archiveprefix = {arXiv},
author = {Friedman, Jerome and Hastie, Trevor and H\"{o}fling, Holger and Tibshirani, Robert},
biburl = {https://www.bibsonomy.org/bibtex/20d0a9118f986b6e1cbebcc266699ae80/jpvaldes},
citeulike-article-id = {3979777},
citeulike-linkout-0 = {http://arxiv.org/abs/0708.1485},
citeulike-linkout-1 = {http://arxiv.org/pdf/0708.1485},
citeulike-linkout-2 = {http://dx.doi.org/10.1214/07-aoas131},
day = 14,
doi = {10.1214/07-aoas131},
eprint = {0708.1485},
interhash = {846b5d7bbe708c8358773f089da30d19},
intrahash = {0d0a9118f986b6e1cbebcc266699ae80},
issn = {1932-6157},
journal = {The Annals of Applied Statistics},
keywords = {coordinate-descent, fused-lasso optimization total_variation},
month = dec,
number = 2,
pages = {302--332},
posted-at = {2017-11-20 14:09:13},
priority = {2},
timestamp = {2018-12-07T09:40:56.000+0100},
title = {{Pathwise coordinate optimization}},
url = {http://dx.doi.org/10.1214/07-aoas131},
volume = 1,
year = 2007
}