Class LASSO
The Lasso typically yields a sparse solution, of which the parameter vector β has relatively few nonzero coefficients. In contrast, the solution of L_{2}regularized least squares (i.e. ridge regression) typically has all coefficients nonzero. Because it effectively reduces the number of variables, the Lasso is useful in some contexts.
For overdetermined systems (more instances than variables, commonly in machine learning), we normalize variables with mean 0 and standard deviation 1. For underdetermined systems (fewer instances than variables, e.g. compressed sensing), we assume white noise (i.e. no intercept in the linear model) and do not perform normalization. Note that the solution is not unique in this case.
There is no analytic formula or expression for the optimal solution to the L_{1}regularized least squares problems. Therefore, its solution must be computed numerically. The objective function in the L_{1}regularized least squares is convex but not differentiable, so solving it is more of a computational challenge than solving the L_{2}regularized least squares. The Lasso may be solved using quadratic programming or more general convex optimization methods, as well as by specific algorithms such as the least angle regression algorithm.
References
 R. Tibshirani. Regression shrinkage and selection via the lasso. J. Royal. Statist. Soc B., 58(1):267288, 1996.
 B. Efron, I. Johnstone, T. Hastie, and R. Tibshirani. Least angle regression. Annals of Statistics, 2003
 SeungJean Kim, K. Koh, M. Lustig, Stephen Boyd, and Dimitry Gorinevsky. An InteriorPoint Method for LargeScale L1Regularized Least Squares. IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, 2007.

Constructor Summary

Method Summary
Modifier and TypeMethodDescriptionstatic LinearModel
Fits a L1regularized least squares model.static LinearModel
Fits a L1regularized least squares model.static LinearModel
Fits a L1regularized least squares model.static LinearModel
fit
(Formula formula, DataFrame data, Properties params) Fits a L1regularized least squares model.

Constructor Details

LASSO
public LASSO()


Method Details

fit
Fits a L1regularized least squares model. Parameters:
formula
 a symbolic description of the model to be fitted.data
 the data frame of the explanatory and response variables. NO NEED to include a constant column of 1s for bias. Returns:
 the model.

fit
Fits a L1regularized least squares model. The hyperparameters inprop
includesmile.lasso.lambda
is the shrinkage/regularization parameter. Large lambda means more shrinkage. Choosing an appropriate value of lambda is important, and also difficult.smile.lasso.tolerance
is the tolerance for stopping iterations (relative target duality gap).smile.lasso.iterations
is the maximum number of IPM (Newton) iterations.
 Parameters:
formula
 a symbolic description of the model to be fitted.data
 the data frame of the explanatory and response variables. NO NEED to include a constant column of 1s for bias.params
 the hyperparameters. Returns:
 the model.

fit
Fits a L1regularized least squares model. Parameters:
formula
 a symbolic description of the model to be fitted.data
 the data frame of the explanatory and response variables. NO NEED to include a constant column of 1s for bias.lambda
 the shrinkage/regularization parameter. Returns:
 the model.

fit
public static LinearModel fit(Formula formula, DataFrame data, double lambda, double tol, int maxIter) Fits a L1regularized least squares model. Parameters:
formula
 a symbolic description of the model to be fitted.data
 the data frame of the explanatory and response variables. NO NEED to include a constant column of 1s for bias.lambda
 the shrinkage/regularization parameter.tol
 the tolerance to stop iterations (relative target duality gap).maxIter
 the maximum number of IPM (Newton) iterations. Returns:
 the model.
