admm.genlasso {ADMM}

Generalized LASSO

Description

Generalized LASSO is solving the following equation,

\textrm{min}_x ~ \frac{1}{2}\|Ax-b\|_2^2 + \lambda \|Dx\|_1

where the choice of regularization matrix D leads to different problem formulations.

Usage

admm.genlasso(
  A,
  b,
  D = diag(length(b)),
  lambda = 1,
  rho = 1,
  alpha = 1,
  abstol = 1e-04,
  reltol = 0.01,
  maxiter = 1000
)

Arguments

A

an (m\times n) regressor matrix

b

a length-m response vector

D

a regularization matrix of n columns

lambda

a regularization parameter

rho

an augmented Lagrangian parameter

alpha

an overrelaxation parameter in [1,2]

abstol

absolute tolerance stopping criterion

reltol

relative tolerance stopping criterion

maxiter

maximum number of iterations

Value

a named list containing

x

a length-n solution vector

history

dataframe recording iteration numerics. See the section for more details.

Iteration History

When you run the algorithm, output returns not only the solution, but also the iteration history recording following fields over iterates,

objval

object (cost) function value

r_norm

norm of primal residual

s_norm

norm of dual residual

eps_pri

feasibility tolerance for primal feasibility condition

eps_dual

feasibility tolerance for dual feasibility condition

In accordance with the paper, iteration stops when both r_norm and s_norm values become smaller than eps_pri and eps_dual, respectively.

Author(s)

Xiaozhi Zhu

References

Tibshirani RJ, Taylor J (2011). “The solution path of the generalized lasso.” The Annals of Statistics, 39(3), 1335–1371. ISSN 0090-5364, doi: 10.1214/11-AOS878.

Zhu Y (2017). “An Augmented ADMM Algorithm With Application to the Generalized Lasso Problem.” Journal of Computational and Graphical Statistics, 26(1), 195–204. ISSN 1061-8600, 1537-2715, doi: 10.1080/10618600.2015.1114491.

Examples

## generate sample data
m = 100
n = 200
p = 0.1   # percentange of non-zero elements

x0 = matrix(Matrix::rsparsematrix(n,1,p))
A  = matrix(rnorm(m*n),nrow=m)
for (i in 1:ncol(A)){
  A[,i] = A[,i]/sqrt(sum(A[,i]*A[,i]))
}
b = A%*%x0 + sqrt(0.001)*matrix(rnorm(m))
D = diag(n);

## set regularization lambda value
regval = 0.1*Matrix::norm(t(A)%*%b, 'I')

## solve LASSO via reducing from Generalized LASSO
output  = admm.genlasso(A,b,D,lambda=regval) # set D as identity matrix
niter   = length(output$history$s_norm)
history = output$history

## report convergence plot
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(1:niter, history$objval, "b", main="cost function")
plot(1:niter, history$r_norm, "b", main="primal residual")
plot(1:niter, history$s_norm, "b", main="dual residual")
par(opar)