genDWD {DWDLargeR}

Solve the generalized distance weighted discrimination (DWD) model.

Description

Solve the generalized DWD model by using a symmetric Gauss-Seidel based alternating direction method of multipliers (ADMM) method.

Usage

genDWD(X,y,C,expon, tol = 1e-5, maxIter = 2000, method = 1, printDetails = 0,
             rmzeroFea = 1, scaleFea = 1)

Arguments

X

A d x n matrix of n training samples with d features.

y

A vector of length n of training labels. The element of y is either -1 or 1.

C

A number representing the penalty parameter for the generalized DWD model.

expon

A positive number representing the exponent q of the residual r_i in the generalized DWD model. Common choices are expon = 1,2,4.

tol

The stopping tolerance for the algorithm. (Default = 1e-5)

maxIter

Maximum iteration allowed for the algorithm. (Default = 2000)

method

Method for solving generalized DWD model. The default is set to be 1 for the highly efficient sGS-ADMM algorithm. User can also select method = 2 for the directly extended ADMM solver.

printDetails

Switch for printing details of the algorithm. Default is set to be 0 (not printing).

rmzeroFea

Switch for removing zero features in the data matrix. Default is set to be 1 (removing zero features).

scaleFea

Switch for scaling features in the data matrix. This is to make the features having roughly similar magnitude. Default is set to be 1 (scaling features).

Details

This is a symmetric Gauss-Seidel based alternating method of multipliers (sGS-ADMM) algorithm for solving the generalized DWD model of the following formulation:

\min \sum_i \theta_q (r_i) + C e^T x_i

subject to the constraints

Z^T w + \beta y + \xi - r = 0, ||w||<=1, \xi>=0,

where Z = X diag(y), e is a given positive vector such that ||e||_\infty = 1, and \theta_q is a function defined by \theta_q(t) = 1/t^q if t>0 and \theta_q(t)=\infty if t<=0.

Value

A list consists of the result from the algorithm.

w

The unit normal of hyperplane that distinguishes the two classes.

beta

The distance of the hyperplane to the origin (\beta in the above formulation).

xi

A slack variable of length n for the possibility that the two classes may not be separated cleanly by the hyperplane (\xi in the above formulation).

r

The residual r:= Z^T w + \beta y + \xi.

alpha

Dual variable of the generalized DWD model.

info

A list consists of the information from the algorithm.

runhist

A list consists of the run history throughout the iterations.

Author(s)

Xin-Yee Lam, J.S. Marron, Defeng Sun, and Kim-Chuan Toh

References

Lam, X.Y., Marron, J.S., Sun, D.F., and Toh, K.C. (2018) “Fast algorithms for large scale generalized distance weighted discrimination", Journal of Computational and Graphical Statistics, forthcoming.
https://arxiv.org/abs/1604.05473

Examples

# load the data
data("mushrooms")
# calculate the best penalty parameter
C = penaltyParameter(mushrooms$X,mushrooms$y,expon=1)
# solve the generalized DWD model
result = genDWD(mushrooms$X,mushrooms$y,C,expon=1)