20,176 research outputs found
Piecewise linear regularized solution paths
We consider the generic regularized optimization problem
. Efron, Hastie,
Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for
the LASSO--that is, if is squared error loss and is
the norm of --the optimal coefficient path is piecewise linear,
that is, is piecewise
constant. We derive a general characterization of the properties of (loss ,
penalty ) pairs which give piecewise linear coefficient paths. Such pairs
allow for efficient generation of the full regularized coefficient paths. We
investigate the nature of efficient path following algorithms which arise. We
use our results to suggest robust versions of the LASSO for regression and
classification, and to develop new, efficient algorithms for existing problems
in the literature, including Mammen and van de Geer's locally adaptive
regression splines.Comment: Published at http://dx.doi.org/10.1214/009053606000001370 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Learning to Approximate a Bregman Divergence
Bregman divergences generalize measures such as the squared Euclidean
distance and the KL divergence, and arise throughout many areas of machine
learning. In this paper, we focus on the problem of approximating an arbitrary
Bregman divergence from supervision, and we provide a well-principled approach
to analyzing such approximations. We develop a formulation and algorithm for
learning arbitrary Bregman divergences based on approximating their underlying
convex generating function via a piecewise linear function. We provide
theoretical approximation bounds using our parameterization and show that the
generalization error for metric learning using our framework
matches the known generalization error in the strictly less general Mahalanobis
metric learning setting. We further demonstrate empirically that our method
performs well in comparison to existing metric learning methods, particularly
for clustering and ranking problems.Comment: 19 pages, 4 figure
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