genDWD: Solve the generalized distance weighted discrimination (DWD)...

In DWDLargeR: Fast Algorithms for Large Scale Generalized Distance Weighted Discrimination

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 ∑_i θ_q (r_i) + C e^T x_i

subject to the constraints

Z^T w + β y + ξ - r = 0, ||w||<=1, ξ>=0,

where Z = X diag(y), e is a given positive vector such that ||e||_∞ = 1, and θ_q is a function defined by θ_q(t) = 1/t^q if t>0 and θ_q(t)=∞ 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 (β 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 (ξ in the above formulation).
r	The residual r:= Z^T w + β y + ξ.
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)

DWDLargeR documentation built on May 2, 2019, 7:27 a.m.