3,594 research outputs found
A Generic Path Algorithm for Regularized Statistical Estimation
Regularization is widely used in statistics and machine learning to prevent
overfitting and gear solution towards prior information. In general, a
regularized estimation problem minimizes the sum of a loss function and a
penalty term. The penalty term is usually weighted by a tuning parameter and
encourages certain constraints on the parameters to be estimated. Particular
choices of constraints lead to the popular lasso, fused-lasso, and other
generalized penalized regression methods. Although there has been a lot
of research in this area, developing efficient optimization methods for many
nonseparable penalties remains a challenge. In this article we propose an exact
path solver based on ordinary differential equations (EPSODE) that works for
any convex loss function and can deal with generalized penalties as well
as more complicated regularization such as inequality constraints encountered
in shape-restricted regressions and nonparametric density estimation. In the
path following process, the solution path hits, exits, and slides along the
various constraints and vividly illustrates the tradeoffs between goodness of
fit and model parsimony. In practice, the EPSODE can be coupled with AIC, BIC,
or cross-validation to select an optimal tuning parameter. Our
applications to generalized regularized generalized linear models,
shape-restricted regressions, Gaussian graphical models, and nonparametric
density estimation showcase the potential of the EPSODE algorithm.Comment: 28 pages, 5 figure
Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping
We consider the problem of estimating a sparse multi-response regression
function, with an application to expression quantitative trait locus (eQTL)
mapping, where the goal is to discover genetic variations that influence
gene-expression levels. In particular, we investigate a shrinkage technique
capable of capturing a given hierarchical structure over the responses, such as
a hierarchical clustering tree with leaf nodes for responses and internal nodes
for clusters of related responses at multiple granularity, and we seek to
leverage this structure to recover covariates relevant to each
hierarchically-defined cluster of responses. We propose a tree-guided group
lasso, or tree lasso, for estimating such structured sparsity under
multi-response regression by employing a novel penalty function constructed
from the tree. We describe a systematic weighting scheme for the overlapping
groups in the tree-penalty such that each regression coefficient is penalized
in a balanced manner despite the inhomogeneous multiplicity of group
memberships of the regression coefficients due to overlaps among groups. For
efficient optimization, we employ a smoothing proximal gradient method that was
originally developed for a general class of structured-sparsity-inducing
penalties. Using simulated and yeast data sets, we demonstrate that our method
shows a superior performance in terms of both prediction errors and recovery of
true sparsity patterns, compared to other methods for learning a
multivariate-response regression.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS549 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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