Pathwise coordinate optimization. (English) Zbl 1378.90064

Summary: 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.


90C25 Convex programming
62J07 Ridge regression; shrinkage estimators (Lasso)


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