elastic.net {quadrupen}

Fit a linear model with elastic-net regularization

Description

Adjust a linear model with elastic-net regularization, mixing a (possibly weighted) \ell_1-norm (LASSO) and a (possibly structured) \ell_2-norm (ridge-like). The solution path is computed at a grid of values for the \ell_1-penalty, fixing the amount of \ell_2 regularization. See details for the criterion optimized.

Usage

elastic.net(
  x,
  y,
  lambda1 = NULL,
  lambda2 = 0.01,
  penscale = rep(1, p),
  struct = NULL,
  intercept = TRUE,
  normalize = TRUE,
  naive = FALSE,
  nlambda1 = ifelse(is.null(lambda1), 100, length(lambda1)),
  min.ratio = ifelse(n <= p, 0.01, 1e-04),
  max.feat = ifelse(lambda2 < 0.01, min(n, p), min(4 * n, p)),
  beta0 = NULL,
  control = list(),
  checkargs = TRUE
)

Arguments

x

matrix of features, possibly sparsely encoded (experimental). Do NOT include intercept. When normalized os TRUE, coefficients will then be rescaled to the original scale.

y

response vector.

lambda1

sequence of decreasing \ell_1-penalty levels. If NULL (the default), a vector is generated with nlambda1 entries, starting from a guessed level lambda1.max where only the intercept is included, then shrunken to min.ratio*lambda1.max.

lambda2

real scalar; tunes the \ell_2 penalty in the Elastic-net. Default is 0.01. Set to 0 to recover the Lasso.

penscale

vector with real positive values that weight the \ell_1-penalty of each feature. Default set all weights to 1.

struct

matrix structuring the coefficients (preferably sparse). Must be at least positive semidefinite (this is checked internally if the checkarg argument is TRUE). The default uses the identity matrix. See details below.

intercept

logical; indicates if an intercept should be included in the model. Default is TRUE.

normalize

logical; indicates if variables should be normalized to have unit L2 norm before fitting. Default is TRUE.

naive

logical; Compute either 'naive' of classic elastic-net as defined in Zou and Hastie (2006): the vector of parameters is rescaled by a coefficient (1+lambda2) when naive equals FALSE. No rescaling otherwise. Default is FALSE.

nlambda1

integer that indicates the number of values to put in the lambda1 vector. Ignored if lambda1 is provided.

min.ratio

minimal value of \ell_1-part of the penalty that will be tried, as a fraction of the maximal lambda1 value. A too small value might lead to unstability at the end of the solution path corresponding to small lambda1 combined with \lambda_2=0. The default value tries to avoid this, adapting to the 'n<p' context. Ignored if lambda1 is provided.

max.feat

integer; limits the number of features ever to enter the model; i.e., non-zero coefficients for the Elastic-net: the algorithm stops if this number is exceeded and lambda1 is cut at the corresponding level. Default is min(nrow(x),ncol(x)) for small lambda2 (<0.01) and min(4*nrow(x),ncol(x)) otherwise. Use with care, as it considerably changes the computation time.

beta0

a starting point for the vector of parameter. When NULL (the default), will be initialized at zero. May save time in some situation.

control

list of argument controlling low level options of the algorithm –use with care and at your own risk– :

verbose:

integer; activate verbose mode –this one is not too much risky!– set to 0 for no output; 1 for warnings only, and 2 for tracing the whole progression. Default is 1. Automatically set to 0 when the method is embedded within cross-validation or stability selection.

timer:

logical; use to record the timing of the algorithm. Default is FALSE.

max.iter:

the maximal number of iteration used to solve the problem for a given value of lambda1. Default is 500.

method:

a string for the underlying solver used. Either "quadra", "pathwise" or "fista". Default is "quadra".

threshold:

a threshold for convergence. The algorithm stops when the optimality conditions are fulfill up to this threshold. Default is 1e-7 for "quadra" and 1e-2 for the first order methods.

monitor:

indicates if a monitoring of the convergence should be recorded, by computing a lower bound between the current solution and the optimum: when '0' (the default), no monitoring is provided; when '1', the bound derived in Grandvalet et al. is computed; when '>1', the Fenchel duality gap is computed along the algorithm.

checkargs

logical; should arguments be checked to (hopefully) avoid internal crashes? Default is TRUE. Automatically set to FALSE when calls are made from cross-validation or stability selection procedures.

Details

The optimized criterion is the following: β^hat_λ1,λ2 = argmin_β 1/2 RSS(β) + λ₁ | D β |₁ + λ/2 ₂ β^T S β, where D is a diagonal matrix, whose diagonal terms are provided as a vector by the penscale argument. The \ell_2 structuring matrix S is provided via the struct argument, a positive semidefinite matrix (possibly of class Matrix).

Value

an object with class quadrupen, see the documentation page quadrupen for details.

See Also

See also quadrupen, plot,quadrupen-method and crossval.

Examples

## Simulating multivariate Gaussian with blockwise correlation
## and piecewise constant vector of parameters
beta <- rep(c(0,1,0,-1,0), c(25,10,25,10,25))
cor <- 0.75
Soo <- toeplitz(cor^(0:(25-1))) ## Toeplitz correlation for irrelevant variables
Sww  <- matrix(cor,10,10) ## bloc correlation between active variables
Sigma <- bdiag(Soo,Sww,Soo,Sww,Soo)
diag(Sigma) <- 1
n <- 50
x <- as.matrix(matrix(rnorm(95*n),n,95) %*% chol(Sigma))
y <- 10 + x %*% beta + rnorm(n,0,10)

labels <- rep("irrelevant", length(beta))
labels[beta != 0] <- "relevant"
## Comparing the solution path of the LASSO and the Elastic-net
plot(elastic.net(x,y,lambda2=0), label=labels) ## a mess
plot(elastic.net(x,y,lambda2=10), label=labels) ## a lot better