Sparse Linear Regression using Nonsmooth Loss Functions and L1 Regularization

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Description

The function "slim" implements a family of Lasso variants for estimating high dimensional sparse linear models including Dantzig Selector, LAD Lasso, SQRT Lasso, Lq Lasso for estimating high dimensional sparse linear model. We adopt the alternating direction method of multipliers (ADMM) and convert the original optimization problem into a sequential L1-penalized least square minimization problem, which can be efficiently solved by combining the linearization and multi-stage screening of varialbes. Missing values can be tolerated for Dantzig selector in the design matrix and response vector.

Usage

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slim(X, Y, lambda = NULL, nlambda = NULL, 
     lambda.min.value = NULL,lambda.min.ratio = NULL, 
     rho = 1, method="lq", q = 2, res.sd = FALSE, 
     prec = 1e-5, max.ite = 1e5, verbose = TRUE)

Arguments

Y

The n-dimensional response vector.

X

The n by d design matrix. d2 is required.

lambda

A sequence of decresing positive numbers to control the regularization. Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Users can also specify a sequence to override this. Default value is from lambda.max to lambda.min.ratio*lambda.max. For Lq regression, the default value of lambda.max is π√{\log(d)/n}. For Dantzig selector, the default value of lambda.max is the minimum regularization parameter, which yields an all-zero estiamtes.

nlambda

The number of values used in lambda. Default value is 5.

lambda.min.value

The smallest value for lambda, as a fraction of the uppperbound (lambda.max) of the regularization parameter. The program can automatically generate lambda as a sequence of length = nlambda starting from lambda.max to lambda.min.ratio*lambda.max in log scale. The default value is \log(d)/n for for Dantzig selector 0.3*lambda.max for Lq Lasso.

lambda.min.ratio

The smallest ratio of the value for lambda. The default value is 0.3 for Lq Lasso and 0.5 for Dantzig selector.

rho

The penalty parameter used in ADMM. The default value is √{d}.

method

Dantzig selector is applied if method = "dantzig" and L_q Lasso is applied if method = "lq". Standard Lasso is provided if method = "lasso". The default value is "lq".

q

The loss function used in Lq Lasso. It is only applicable when method = "lq" and must be in [1,2]. The default value is 2.

res.sd

Flag of whether the response varialbles are standardized. The default value is FALSE.

prec

Stopping criterion. The default value is 1e-5.

max.ite

The iteration limit. The default value is 1e5.

verbose

Tracing information printing is disabled if verbose = FALSE. The default value is TRUE.

Details

Standard Lasso

\min {\frac{1}{2n}}|| Y - X β ||_2^2 + λ || β ||_1


Dantzig selector solves the following optimization problem

\min || β ||_1, \quad \textrm{s.t. } || X'(Y - X β) ||_{∞} < λ


L_q loss Lasso solves the following optimization problem

\min n^{-\frac{1}{q}}|| Y - X β ||_q + λ || β ||_1


where 1<= q <=2. Lq Lasso is equivalent to LAD Lasso and SQR Lasso when q=1 and q=2 respectively.

Value

An object with S3 class "slim" is returned:

beta

A matrix of regression estimates whose columns correspond to regularization parameters.

intercept

The value of intercepts corresponding to regularization parameters.

Y

The value of Y used in the program.

X

The value of X used in the program.

lambda

The sequence of regularization parameters lambda used in the program.

nlambda

The number of values used in lambda.

method

The method from the input.

sparsity

The sparsity levels of the solution path.

ite

A list of vectors where ite[[1]] is the number of external iteration and ite[[2]] is the number of internal iteration with the i-th entry corresponding to the i-th regularization parameter.

verbose

The verbose from the input.

Author(s)

Xingguo Li, Tuo Zhao, Lie Wang, Xiaoming Yuan and Han Liu
Maintainer: Xingguo Li <xingguo.leo@gmail.com>

References

1. E. Candes and T. Tao. The Dantzig selector: Statistical estimation when p is much larger than n. Annals of Statistics, 2007.
2. A. Belloni, V. Chernozhukov and L. Wang. Pivotal recovery of sparse signals via conic programming. Biometrika, 2012.
3. L. Wang. L1 penalized LAD estimator for high dimensional linear regression. Journal of Multivariate Analysis, 2012.
4. J. Liu and J. Ye. Efficient L1/Lq Norm Regularization. Technical Report, 2010. 5. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends in Machine Learning, 2011. 6. B. He and X. Yuan. On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers. Technical Report, 2012.

See Also

flare-package, print.slim, plot.slim, coef.slim and predict.slim.

Examples

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## load library
library(flare)
## generate data
n = 50
d = 100
X = matrix(rnorm(n*d), n, d)
beta = c(3,2,0,1.5,rep(0,d-4))
eps = rnorm(n)
Y = X%*%beta + eps
nlamb = 5
ratio = 0.3

## Regression with "dantzig", general "lq" and "lasso" respectively
out1 = slim(X=X,Y=Y,nlambda=nlamb,lambda.min.ratio=ratio,method="dantzig")
out2 = slim(X=X,Y=Y,nlambda=nlamb,lambda.min.ratio=ratio,method="lq",q=1)
out3 = slim(X=X,Y=Y,nlambda=nlamb,lambda.min.ratio=ratio,method="lq",q=1.5)
out4 = slim(X=X,Y=Y,nlambda=nlamb,lambda.min.ratio=ratio,method="lq",q=2)
out5 = slim(X=X,Y=Y,nlambda=nlamb,lambda.min.ratio=ratio,method="lasso")

## Display results
print(out4)
plot(out4)
coef(out4)