5,172 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
A Path Algorithm for Constrained Estimation
Many least squares problems involve affine equality and inequality
constraints. Although there are variety of methods for solving such problems,
most statisticians find constrained estimation challenging. The current paper
proposes a new path following algorithm for quadratic programming based on
exact penalization. Similar penalties arise in regularization in model
selection. Classical penalty methods solve a sequence of unconstrained problems
that put greater and greater stress on meeting the constraints. In the limit as
the penalty constant tends to , one recovers the constrained solution.
In the exact penalty method, squared penalties are replaced by absolute value
penalties, and the solution is recovered for a finite value of the penalty
constant. The exact path following method starts at the unconstrained solution
and follows the solution path as the penalty constant increases. In the
process, the solution path hits, slides along, and exits from the various
constraints. Path following in lasso penalized regression, in contrast, starts
with a large value of the penalty constant and works its way downward. In both
settings, inspection of the entire solution path is revealing. Just as with the
lasso and generalized lasso, it is possible to plot the effective degrees of
freedom along the solution path. For a strictly convex quadratic program, the
exact penalty algorithm can be framed entirely in terms of the sweep operator
of regression analysis. A few well chosen examples illustrate the mechanics and
potential of path following.Comment: 26 pages, 5 figure
Localized Lasso for High-Dimensional Regression
We introduce the localized Lasso, which is suited for learning models that
are both interpretable and have a high predictive power in problems with high
dimensionality and small sample size . More specifically, we consider a
function defined by local sparse models, one at each data point. We introduce
sample-wise network regularization to borrow strength across the models, and
sample-wise exclusive group sparsity (a.k.a., norm) to introduce
diversity into the choice of feature sets in the local models. The local models
are interpretable in terms of similarity of their sparsity patterns. The cost
function is convex, and thus has a globally optimal solution. Moreover, we
propose a simple yet efficient iterative least-squares based optimization
procedure for the localized Lasso, which does not need a tuning parameter, and
is guaranteed to converge to a globally optimal solution. The solution is
empirically shown to outperform alternatives for both simulated and genomic
personalized medicine data
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