slim: Sparse Linear Regression using Nonsmooth Loss Functions and L1 Regularization

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

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

The $n$-dimensional response vector.

The $n$ by $d$ design matrix. d$\ge$2 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.rat

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.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 $\sqrt{d}$.

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".

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.

Value

An object with S3 class "slim" is returned:
betaA matrix of regression estimates whose columns correspond to regularization parameters.
interceptThe value of intercepts corresponding to regularization parameters.
YThe value of Y used in the program.
XThe value of X used in the program.
lambdaThe sequence of regularization parameters lambda used in the program.
nlambdaThe number of values used in lambda.
methodThe method from the input.
sparsityThe sparsity levels of the solution path.
iteA 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.
verboseThe verbose from the input.

Details

Standard Lasso $$\min {\frac{1}{2n}}|| Y - X \beta ||_2^2 + \lambda || \beta ||_1$$ Dantzig selector solves the following optimization problem $$\min || \beta ||_1, \quad \textrm{s.t. } || X'(Y - X \beta) ||_{\infty} < \lambda$$ $L_q$ loss Lasso solves the following optimization problem $$\min n^{-\frac{1}{q}}|| Y - X \beta ||_q + \lambda || \beta ||_1$$ where $1

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.

Examples

Run this code

## 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)

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