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R/linreg_ADMM_spGLASSO.R
In fdaSP: Sparse Functional Data Analysis Methods
Defines functions
linreg_ADMM_spGLASSO
# Authors: Mauro Bernardi
# Department Statistical Sciences
# University of Padova
# Via Cesare Battisti, 241
# 35121 PADOVA, Italy
# E-mail: mauro.bernardi@unipd.it
# Last change: March 16, 2023
#' Sparse Group Least Absolute Shrinkage and Selection Operator
#'
#' Sparse group-LASSO, or sparse group \eqn{L_2}-regularized regression, solves the following optimization problem
#' \deqn{\textrm{min}_\beta ~ \frac{1}{2}\|y-X\beta-Z\gamma\|_2^2 + \lambda[(1-\alpha) \sum_{g=1}^G \|S_g T\beta\|_2+\alpha\Vert T_1\beta\Vert_1]}
#' to obtain a sparse coefficient vector \eqn{\beta\in\mathbb{R}^p} and a coefficient vector \eqn{\gamma\in\mathbb{R}^q}. For each group, each row of
#' the matrix \eqn{S_g\in\mathbb{R}^{n_g\times p}} has non-zero entries only for those variables belonging
#' to that group. These values are provided by the arguments \code{groups} and \code{group_weights} (see below).
#' Each variable must belong to only one group. The diagonal matrix \eqn{T\in\mathbb{R}^{p\times p}} contains the variable-specific weights. These values are
#' provided by the argument \code{var_weights} (see below). The diagonal matrix \eqn{T_1\in\mathbb{R}^{p\times p}} contains
#' the variable-specific \eqn{L_1} weights. These values are provided by the argument \code{var_weights_L1} (see below).
#' The regularization path is computed for the sparse group-LASSO penalty at a grid of values for the regularization
#' parameter \eqn{\lambda} using the alternating direction method of multipliers (ADMM). See Boyd et al. (2011)
#' for details on the ADMM method. The elastic net regularization is a combination of \eqn{\ell_2} stability and
#' \eqn{\ell_1} sparsity constraint simultenously and can be obtained by
#' specifying \eqn{G=1}, the matrix \eqn{T} and \eqn{T_1} as the identity matrix of dimension \eqn{p}, and the selection matrix
#' \eqn{S_1} as a row vector of ones of dimension \eqn{p}. The adaptive LASSO can be obtained as a special
#' case of the sparse group-LASSO penalty by setting \eqn{\alpha=1}, the matrix \eqn{T} as the identity matrix of dimension \eqn{p} and the
#' matrix \eqn{T_1} as the diagonal matrix containing the \eqn{L_1} variable-specific weights.
#'
#' @param X an \eqn{(n\times p)} matrix of penalized predictors.
#' @param Z an \eqn{(n\times q)} full column rank matrix of predictors that are not penalized.
#' @param y a length-\eqn{n} response vector.
#' @param groups either a vector of consecutive integers describing the grouping of the coefficients,
#' or a list with two elements: the first element is a vector containing the variables belonging to each group,
#' while the second element is a vector containing the group lenghts, (see example below).
#' @param group_weights a vector of length \eqn{G} containing group-specific weights. The default is square root of the group cardinality, see Yuan and Lin (2006).
#' @param var_weights a vector of length \eqn{p} containing variable-specific weights. The default is a vector of ones.
#' @param var_weights_L1 a vector of length \eqn{p} containing variable-specific weights for the \eqn{L_1} penalty. The default is a vector of ones.
#' @param standardize.data logical. Should data be standardized?
#' @param lambda either a regularization parameter or a vector of regularization parameters. In this latter case the routine computes the whole path. If it is NULL the path values for lambda are provided by the routine.
#' @param alpha the sparse group-LASSO mixing parameter, with \eqn{0\leq\alpha\leq1}. The penalty is defined as
#' \eqn{\alpha = 1} is the LASSO penalty, and \eqn{\alpha = 0} the group-LASSO penalty.
#' @param lambda.min.ratio smallest value for lambda, as a fraction of the maximum lambda value, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero).
#' The default depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.0001,
#' close to zero. If nobs < nvars, the default is 0.01. A very small value of lambda.min.ratio will lead to a saturated fit
#' in the nobs < nvars case.
#' @param nlambda the number of lambda values - default is 30.
#' @param intercept logical. If it is TRUE, a column of ones is added to the design matrix.
#' @param control a list of control parameters. See ‘Details’.
#'
#' @return A named list containing \describe{
#' \item{sp.coefficients}{a length-\eqn{p} solution vector for the parameters \eqn{\beta}, which corresponds to the minimum in-sample MSE.}
#' \item{coefficients}{a length-\eqn{q} solution vector for the parameters \eqn{\gamma}, which corresponds to the minimum in-sample MSE.
#' It is provided only when either the matrix \eqn{Z} in input is not NULL or the intercept is set to TRUE.}
#' \item{sp.coef.path}{an \eqn{(nlambda\times p)} matrix of estimated \eqn{\beta} coefficients for each lambda of the provided sequence.}
#' \item{coef.path}{an \eqn{(nlambda\times q)} matrix of estimated \eqn{\gamma} coefficients for each lambda of the provided sequence.
#' It is provided only when either the matrix \eqn{Z} in input is not NULL or the intercept is set to TRUE.}
#' \item{lambda}{sequence of lambda.}
#' \item{lambda.min}{value of lambda that attains the minimum in-sample MSE.}
#' \item{mse}{in-sample mean squared error.}
#' \item{min.mse}{minimum value of the in-sample MSE for the sequence of lambdas.}
#' \item{convergence}{logical. 1 denotes achieved convergence.}
#' \item{elapsedTime}{elapsed time in seconds.}
#' \item{iternum}{number of iterations.}
#' }
#'
#' When you run the algorithm, output returns not only the solution, but also the iteration history recording
#' following fields over iterates:
#' \describe{
#' \item{objval}{objective function value}
#' \item{r_norm}{norm of primal residual}
#' \item{s_norm}{norm of dual residual}
#' \item{eps_pri}{feasibility tolerance for primal feasibility condition}
#' \item{eps_dual}{feasibility tolerance for dual feasibility condition.}
#' }
#' Iteration stops when both \code{r_norm} and \code{s_norm} values
#' become smaller than \code{eps_pri} and \code{eps_dual}, respectively.
#' @section Details: The control argument is a list that can supply any of the following components:\describe{
#' (To understand exactly what these do see the source code: higher levels give more detail.).
#' \item{adaptation}{logical. If it is TRUE, ADMM with adaptation is performed. The default value is TRUE.}
#' \item{rho}{an augmented Lagrangian parameter. The default value is 1.}
#' \item{tau.ada}{an adaptation parameter in [1,2]. Only needed if adaptation = TRUE. The default value is 2. See Boyd et al. (2011) for details.}
#' \item{mu.ada}{an adaptation parameter. Only needed if adaptation = TRUE. The default value is 10. See Boyd et al. (2011) for details.}
#' \item{abstol}{absolute tolerance stopping criterion. The default value is sqrt(sqrt(.Machine$double.eps)).}
#' \item{reltol}{relative tolerance stopping criterion. The default value is sqrt(.Machine$double.eps).}
#' \item{maxit}{maximum number of iterations. The default value is 100.}
#' \item{print.out}{logical. If it is TRUE, a message about the procedure is printed. The default value is TRUE.}
#' }
#'
#' @examples
#' \donttest{
#'
#' ### generate sample data
#' set.seed(2023)
#' n <- 50
#' p <- 30
#' X <- matrix(rnorm(n*p), n, p)
#' beta <- c(rep(4, 12), rep(0, p - 13), -2)
#' y <- X %*% beta + rnorm(n, sd = sqrt(crossprod(X %*% beta)) / 20)
#'
#' ### define groups of dimension 3 each
#' group1 <- rep(1:10, each = 3)
#'
#' ### set regularization parameter grid
#' lam <- 10^seq(1, -2, length.out = 30)
#'
#' ### set the alpha parameter
#' alpha <- 0.5
#'
#' ### set the hyper-parameters of the ADMM algorithm
#' maxit <- 1000
#' adaptation <- TRUE
#' rho <- 1
#' reltol <- 1e-5
#' abstol <- 1e-5
#'
#' ### run example
#' mod <- fdaSP:::linreg_ADMM_spGLASSO(X = X, y = y, groups = group1, standardize.data = FALSE, intercept = FALSE,
#' lambda = lam, alpha = alpha, control = list("adaptation" = adaptation,
#' "rho" = rho, "maxit" = maxit,
#' "abstol" = abstol, "reltol" = reltol,
#' "print.out" = FALSE))
#'
#' ### graphical presentation
#' matplot(log(lam), mod$sp.coef.path, type = "l", main = "Sparse Group Lasso solution path",
#' bty = "n", xlab = TeX("$\\log(\\lambda)$"), ylab = "")
#' }
#'
#' @references
#' \insertRef{simon_etal.2013}{fdaSP}
#'
#' \insertRef{yuan_lin.2006}{fdaSP}
#'
#' \insertRef{zou_hastie.2005}{fdaSP}
#'
#' \insertRef{zou.2006}{fdaSP}
#'
#' \insertRef{hastie_etal.2015}{fdaSP}
#'
#' \insertRef{boyd_etal.2011}{fdaSP}
#'
#' \insertRef{lin_etal.2022}{fdaSP}
#'
#' @noRd
linreg_ADMM_spGLASSO <- function(X, Z = NULL, y, groups, group_weights = NULL, var_weights = NULL, var_weights_L1 = NULL,
standardize.data = TRUE, lambda = NULL, alpha = NULL, lambda.min.ratio = NULL, nlambda = 30,
intercept = FALSE, control = list()) {
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# add constant
if (standardize.data == TRUE) {
if (intercept == TRUE){
intercept = FALSE
warning("* linreg_ADMM_spGLASSO : intercept should be set to FALSE is data are standardized!\n")
}
}
if (intercept == TRUE) {
if (!is.null(Z)) {
Z <- cbind(matrix(data = 1, nrow = length(y), ncol = 1), Z)
} else {
Z <- matrix(data = 1, nrow = length(y), ncol = 1)
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: data validity
if (!check_data_matrix(X)) {
stop("* linreg_ADMM_spGLASSO : input 'X' is invalid data matrix.")
}
if (!check_data_vector(y)) {
stop("* linreg_ADMM_spGLASSO : input 'y' is invalid data vector.")
}
y <- as.vector(y)
# data size
if (dim(X)[1] != length(y)) {
stop("* linreg_ADMM_spGLASSO : two inputs 'X' and 'Y' have non-matching dimension.")
}
if (!is.null(Z)) {
if (dim(Z)[1] != length(y)) {
stop("* linreg_ADMM_spGLASSO : two inputs 'Z' and 'Y' have non-matching dimension.")
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: lambda parameter (in case of a single lambda)
if (!is.null(lambda)) {
if (length(lambda) == 1) {
meps <- (.Machine$double.eps)
negsmall <- -meps
if (!check_param_constant(lambda, negsmall)) {
stop("* linreg_ADMM_spGLASSO : reg. parameter 'lambda' is invalid.")
}
if (lambda < meps){
message("* linreg_ADMM_spGLASSO : since 'lambda' is effectively zero, a least-squares solution is returned.")
xsol <- as.vector(aux_pinv(X) %*% matrix(y))
output <- list()
output$x <- xsol
return(output)
}
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# get data dimensions
n <- dim(X)[1]
p <- dim(X)[2]
if (!is.null(Z)) {
q <- dim(Z)[2]
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: alpha parameter
if (is.null(alpha)) {
alpha <- 0.5
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Standardise response and design matrix
if (standardize.data == TRUE) {
res <- standardizemat(X, y)
X.std <- res$X.std
y.std <- res$y.std
mU <- res$mU
mV <- res$mV
} else {
X.std <- X
y.std <- y
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# check for inputs
npar <- dim(X.std)[2]
con <- list(maxit = 100L,
abstol = sqrt(.Machine$double.eps),
reltol = sqrt(sqrt(.Machine$double.eps)),
adaptation = TRUE,
rho = 1,
tau.ada = 2,
mu.ada = 10,
print.out = TRUE)
nmsC <- names(con)
con[(namc <- names(control))] <- control
if (length(noNms <- namc[!namc %in% nmsC])) {
warning("unknown names in control: ", paste(noNms, collapse = ", "))
}
# set default values (for inputs)
maxit <- con$maxit
abstol <- con$abstol
reltol <- con$reltol
adaptation <- con$adaptation
rho <- con$rho
tau.ada <- con$tau.ada
mu.ada <- con$mu.ada
print.out <- con$print.out
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: other parameters
if (!check_param_constant_multiple(c(abstol, reltol))) {
stop("* linreg_ADMM_spGLASSO : tolerance level is invalid.")
}
if (!check_param_integer(maxit, 0.0)) {
stop("* linreg_ADMM_spGLASSO : 'maxiter' should be a positive integer.")
}
maxit = as.integer(maxit)
if (!check_param_constant(rho, 0.0)) {
stop("* linreg_ADMM_spGLASSO : 'rho' should be a positive real number.")
}
if (!check_param_constant(mu.ada, 0.0)) {
stop("* linreg_ADMM_spGLASSO : 'mu.ada' should be a positive real number.")
}
if ((tau.ada < 1.0) || (tau.ada > 2.0)) {
warning("* linreg_spADMM_GLASSO : 'tau.ada' value is suggested to be in [1,2].")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: variable weights
check_weights(x = var_weights, n = dim(X)[2], algname = "linreg_ADMM_spGLASSO", funname = "var_weights")
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: variable weights (L1)
check_weights(x = var_weights_L1, n = dim(X)[2], algname = "linreg_ADMM_spGLASSO", funname = "var_weights_L1")
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage groups
if (is.list(groups) && !is.null(groups$Glen)) {
nG <- groups$Glen
groups_ <- groups$groups
GRmat <- groups2mat_GLASSO(groups = groups_, Glen = nG)
G <- dim(GRmat)[1]
} else {
groups_ <- groups
GRmat <- groups2mat_GLASSO(groups = groups_, Glen = NULL)
G <- dim(GRmat)[1]
nG <- rowSums(GRmat)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Check overlapping groups
res.method <- check_group_overlaps(GRmat)
if (res.method == "ovglasso") {
stop("* linreg_ADMM_spGLASSO : this routine is for fitting the linear regression model with sparse group-LASSO penalty.
Use routine 'linreg_ADMM_OVGLASSO' for fitting the linear regression model with overlap group-LASSO penalty!")
}
if (res.method == "lasso") {
warning("* linreg_spADMM_GLASSO : this routine is for fitting the linear regression model with sparse group-LASSO penalty.
Use (faster) routine 'linreg_ADMM_LASSO' instead or check the provided groups!")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage group weights
if (is.null(group_weights)) {
# Define the weights as in Yuan and Lin 2006, JRSSB
group_weights <- sqrt(diag(tcrossprod(GRmat)))
} else {
check_weights(x = group_weights, n = G, algname = "linreg_ADMM_spGLASSO", funname = "group_weights")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage variable weights
if (is.null(var_weights)) {
var_weights <- 1.0 / colSums(GRmat)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage variable weights: adaptive Lasso
if (is.null(var_weights_L1)) {
var_weights_L1 <- rep(1, p)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get the sequence of lambdas
if (!is.null(nlambda) && is.null(lambda)) {
# get the smallest value of such that the regression
# coefficients estimated by the lasso are all equal to zero
lambda <- lm_lambdamax_spGLASSO(y = y.std, X = X.std, Z = Z,
groups = groups, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda.min.ratio = lambda.min.ratio, maxl = nlambda, alpha = alpha)$lambda.seq
}
if (is.null(nlambda) && is.null(lambda)) {
# get the smallest value of such that the regression
# coefficients estimated by the lasso are all equal to zero
lambda <- lm_lambdamax_spGLASSO(y = y.std, X = X.std, Z = Z,
groups = groups, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda.min.ratio = lambda.min.ratio, maxl = 30, alpha = alpha)$lambda.seq
}
if (!is.null(lambda)) {
nlambda <- length(lambda)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Run lasso
if (is.null(Z)) {
ret <- .admm_spglasso_fast(A = X.std, b = y.std, groups = GRmat,
group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_spglasso_fast",
# A = X.std, b = y.std, groups = GRmat,
# group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
# lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# get estimated coefficients and path
mSpRegP <- t(ret$coef.path)
vSpRegP <- ret$coefficients
} else {
ret <- .admm_spglasso_cov_fast(W = X.std, Z = Z, y = y.std,
groups = GRmat, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_spglasso_cov_fast",
# W = X.std, Z = Z, y = y.std,
# groups = GRmat, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
# lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# get estimated coefficients and path
mSpRegP <- t(ret$sp.coef.path)
mRegP <- t(ret$coef.path)
vSpRegP <- ret$sp.coefficients
vRegP <- ret$coefficients
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get the path and retrieve the scaled estimates
if (standardize.data == TRUE) {
# if (alpha == 0) {
# dim_ <- sum(nG)
# } else if (alpha == 1) {
# dim_ <- p
# } else {
# # dim_ <- sum(nG) + p
# dim_ <- p
# }
dim_ <- p
sp.path <- matrix(nlambda, dim_, data = t(apply(mSpRegP, 2, function(x) solve(mU) %*% x %*% mV)))
vSpRegP <- solve(mU) %*% vSpRegP %*% mV
if (!is.null(Z)) {
path <- matrix(nlambda, q, data = t(apply(mRegP, 2, function(x) x %*% mV)))
vRegP <- vRegP %*% mV
}
} else {
vSpRegP <- matrix(vSpRegP, length(vSpRegP), 1)
sp.path <- t(mSpRegP)
if (!is.null(Z)) {
path <- t(mRegP)
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Print to screen
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
if (print.out == TRUE) {
cat("\n\n\n")
cat("Linear regression model with sparse group-LASSO penalty\n")
cat("Alternating direction method of multipliers\n")
cat("running time (for ", reltol, " relative error):",
sum(ret$elapsedTime), "seconds \n\n\n")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get output
res.names <- c("sp.coefficients",
"sp.coef.path",
"coefficients",
"coef.path",
"lambda.min",
"lambda",
"mse",
"min.mse",
"convergence",
"elapsedTime",
"iternum",
"objfun",
"r_norm",
"s_norm",
"err_pri",
"err_dual",
"rho")
res <- vector(mode = "list", length = length(res.names))
names(res) <- res.names
if (!is.null(Z)) {
res$coefficients <- vRegP
res$coef.path <- path
}
res$sp.coefficients <- vSpRegP
res$sp.coef.path <- sp.path
res$lambda.min <- ret$lambda.min
res$lambda <- ret$lambda
res$mse <- ret$mse
res$min.mse <- ret$min.mse
res$convergence <- ret$convergence
res$elapsedTime <- ret$elapsedTime
res$iternum <- ret$iternum
res$objfun <- ret$objfun
res$r_norm <- ret$r_norm
res$s_norm <- ret$s_norm
res$err_pri <- ret$err_pri
res$err_dual <- ret$err_dual
res$rho <- ret$rho
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Return output
return(res)
}
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Defines functions linreg_ADMM_spGLASSO
# Authors: Mauro Bernardi
# Department Statistical Sciences
# University of Padova
# Via Cesare Battisti, 241
# 35121 PADOVA, Italy
# E-mail: mauro.bernardi@unipd.it
# Last change: March 16, 2023
#' Sparse Group Least Absolute Shrinkage and Selection Operator
#'
#' Sparse group-LASSO, or sparse group \eqn{L_2}-regularized regression, solves the following optimization problem
#' \deqn{\textrm{min}_\beta ~ \frac{1}{2}\|y-X\beta-Z\gamma\|_2^2 + \lambda[(1-\alpha) \sum_{g=1}^G \|S_g T\beta\|_2+\alpha\Vert T_1\beta\Vert_1]}
#' to obtain a sparse coefficient vector \eqn{\beta\in\mathbb{R}^p} and a coefficient vector \eqn{\gamma\in\mathbb{R}^q}. For each group, each row of
#' the matrix \eqn{S_g\in\mathbb{R}^{n_g\times p}} has non-zero entries only for those variables belonging
#' to that group. These values are provided by the arguments \code{groups} and \code{group_weights} (see below).
#' Each variable must belong to only one group. The diagonal matrix \eqn{T\in\mathbb{R}^{p\times p}} contains the variable-specific weights. These values are
#' provided by the argument \code{var_weights} (see below). The diagonal matrix \eqn{T_1\in\mathbb{R}^{p\times p}} contains
#' the variable-specific \eqn{L_1} weights. These values are provided by the argument \code{var_weights_L1} (see below).
#' The regularization path is computed for the sparse group-LASSO penalty at a grid of values for the regularization
#' parameter \eqn{\lambda} using the alternating direction method of multipliers (ADMM). See Boyd et al. (2011)
#' for details on the ADMM method. The elastic net regularization is a combination of \eqn{\ell_2} stability and
#' \eqn{\ell_1} sparsity constraint simultenously and can be obtained by
#' specifying \eqn{G=1}, the matrix \eqn{T} and \eqn{T_1} as the identity matrix of dimension \eqn{p}, and the selection matrix
#' \eqn{S_1} as a row vector of ones of dimension \eqn{p}. The adaptive LASSO can be obtained as a special
#' case of the sparse group-LASSO penalty by setting \eqn{\alpha=1}, the matrix \eqn{T} as the identity matrix of dimension \eqn{p} and the
#' matrix \eqn{T_1} as the diagonal matrix containing the \eqn{L_1} variable-specific weights.
#'
#' @param X an \eqn{(n\times p)} matrix of penalized predictors.
#' @param Z an \eqn{(n\times q)} full column rank matrix of predictors that are not penalized.
#' @param y a length-\eqn{n} response vector.
#' @param groups either a vector of consecutive integers describing the grouping of the coefficients,
#' or a list with two elements: the first element is a vector containing the variables belonging to each group,
#' while the second element is a vector containing the group lenghts, (see example below).
#' @param group_weights a vector of length \eqn{G} containing group-specific weights. The default is square root of the group cardinality, see Yuan and Lin (2006).
#' @param var_weights a vector of length \eqn{p} containing variable-specific weights. The default is a vector of ones.
#' @param var_weights_L1 a vector of length \eqn{p} containing variable-specific weights for the \eqn{L_1} penalty. The default is a vector of ones.
#' @param standardize.data logical. Should data be standardized?
#' @param lambda either a regularization parameter or a vector of regularization parameters. In this latter case the routine computes the whole path. If it is NULL the path values for lambda are provided by the routine.
#' @param alpha the sparse group-LASSO mixing parameter, with \eqn{0\leq\alpha\leq1}. The penalty is defined as
#' \eqn{\alpha = 1} is the LASSO penalty, and \eqn{\alpha = 0} the group-LASSO penalty.
#' @param lambda.min.ratio smallest value for lambda, as a fraction of the maximum lambda value, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero).
#' The default depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.0001,
#' close to zero. If nobs < nvars, the default is 0.01. A very small value of lambda.min.ratio will lead to a saturated fit
#' in the nobs < nvars case.
#' @param nlambda the number of lambda values - default is 30.
#' @param intercept logical. If it is TRUE, a column of ones is added to the design matrix.
#' @param control a list of control parameters. See ‘Details’.
#'
#' @return A named list containing \describe{
#' \item{sp.coefficients}{a length-\eqn{p} solution vector for the parameters \eqn{\beta}, which corresponds to the minimum in-sample MSE.}
#' \item{coefficients}{a length-\eqn{q} solution vector for the parameters \eqn{\gamma}, which corresponds to the minimum in-sample MSE.
#' It is provided only when either the matrix \eqn{Z} in input is not NULL or the intercept is set to TRUE.}
#' \item{sp.coef.path}{an \eqn{(nlambda\times p)} matrix of estimated \eqn{\beta} coefficients for each lambda of the provided sequence.}
#' \item{coef.path}{an \eqn{(nlambda\times q)} matrix of estimated \eqn{\gamma} coefficients for each lambda of the provided sequence.
#' It is provided only when either the matrix \eqn{Z} in input is not NULL or the intercept is set to TRUE.}
#' \item{lambda}{sequence of lambda.}
#' \item{lambda.min}{value of lambda that attains the minimum in-sample MSE.}
#' \item{mse}{in-sample mean squared error.}
#' \item{min.mse}{minimum value of the in-sample MSE for the sequence of lambdas.}
#' \item{convergence}{logical. 1 denotes achieved convergence.}
#' \item{elapsedTime}{elapsed time in seconds.}
#' \item{iternum}{number of iterations.}
#' }
#'
#' When you run the algorithm, output returns not only the solution, but also the iteration history recording
#' following fields over iterates:
#' \describe{
#' \item{objval}{objective function value}
#' \item{r_norm}{norm of primal residual}
#' \item{s_norm}{norm of dual residual}
#' \item{eps_pri}{feasibility tolerance for primal feasibility condition}
#' \item{eps_dual}{feasibility tolerance for dual feasibility condition.}
#' }
#' Iteration stops when both \code{r_norm} and \code{s_norm} values
#' become smaller than \code{eps_pri} and \code{eps_dual}, respectively.
#' @section Details: The control argument is a list that can supply any of the following components:\describe{
#' (To understand exactly what these do see the source code: higher levels give more detail.).
#' \item{adaptation}{logical. If it is TRUE, ADMM with adaptation is performed. The default value is TRUE.}
#' \item{rho}{an augmented Lagrangian parameter. The default value is 1.}
#' \item{tau.ada}{an adaptation parameter in [1,2]. Only needed if adaptation = TRUE. The default value is 2. See Boyd et al. (2011) for details.}
#' \item{mu.ada}{an adaptation parameter. Only needed if adaptation = TRUE. The default value is 10. See Boyd et al. (2011) for details.}
#' \item{abstol}{absolute tolerance stopping criterion. The default value is sqrt(sqrt(.Machine$double.eps)).}
#' \item{reltol}{relative tolerance stopping criterion. The default value is sqrt(.Machine$double.eps).}
#' \item{maxit}{maximum number of iterations. The default value is 100.}
#' \item{print.out}{logical. If it is TRUE, a message about the procedure is printed. The default value is TRUE.}
#' }
#'
#' @examples
#' \donttest{
#'
#' ### generate sample data
#' set.seed(2023)
#' n <- 50
#' p <- 30
#' X <- matrix(rnorm(n*p), n, p)
#' beta <- c(rep(4, 12), rep(0, p - 13), -2)
#' y <- X %*% beta + rnorm(n, sd = sqrt(crossprod(X %*% beta)) / 20)
#'
#' ### define groups of dimension 3 each
#' group1 <- rep(1:10, each = 3)
#'
#' ### set regularization parameter grid
#' lam <- 10^seq(1, -2, length.out = 30)
#'
#' ### set the alpha parameter
#' alpha <- 0.5
#'
#' ### set the hyper-parameters of the ADMM algorithm
#' maxit <- 1000
#' adaptation <- TRUE
#' rho <- 1
#' reltol <- 1e-5
#' abstol <- 1e-5
#'
#' ### run example
#' mod <- fdaSP:::linreg_ADMM_spGLASSO(X = X, y = y, groups = group1, standardize.data = FALSE, intercept = FALSE,
#' lambda = lam, alpha = alpha, control = list("adaptation" = adaptation,
#' "rho" = rho, "maxit" = maxit,
#' "abstol" = abstol, "reltol" = reltol,
#' "print.out" = FALSE))
#'
#' ### graphical presentation
#' matplot(log(lam), mod$sp.coef.path, type = "l", main = "Sparse Group Lasso solution path",
#' bty = "n", xlab = TeX("$\\log(\\lambda)$"), ylab = "")
#' }
#'
#' @references
#' \insertRef{simon_etal.2013}{fdaSP}
#'
#' \insertRef{yuan_lin.2006}{fdaSP}
#'
#' \insertRef{zou_hastie.2005}{fdaSP}
#'
#' \insertRef{zou.2006}{fdaSP}
#'
#' \insertRef{hastie_etal.2015}{fdaSP}
#'
#' \insertRef{boyd_etal.2011}{fdaSP}
#'
#' \insertRef{lin_etal.2022}{fdaSP}
#'
#' @noRd
linreg_ADMM_spGLASSO <- function(X, Z = NULL, y, groups, group_weights = NULL, var_weights = NULL, var_weights_L1 = NULL,
standardize.data = TRUE, lambda = NULL, alpha = NULL, lambda.min.ratio = NULL, nlambda = 30,
intercept = FALSE, control = list()) {
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# add constant
if (standardize.data == TRUE) {
if (intercept == TRUE){
intercept = FALSE
warning("* linreg_ADMM_spGLASSO : intercept should be set to FALSE is data are standardized!\n")
}
}
if (intercept == TRUE) {
if (!is.null(Z)) {
Z <- cbind(matrix(data = 1, nrow = length(y), ncol = 1), Z)
} else {
Z <- matrix(data = 1, nrow = length(y), ncol = 1)
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: data validity
if (!check_data_matrix(X)) {
stop("* linreg_ADMM_spGLASSO : input 'X' is invalid data matrix.")
}
if (!check_data_vector(y)) {
stop("* linreg_ADMM_spGLASSO : input 'y' is invalid data vector.")
}
y <- as.vector(y)
# data size
if (dim(X)[1] != length(y)) {
stop("* linreg_ADMM_spGLASSO : two inputs 'X' and 'Y' have non-matching dimension.")
}
if (!is.null(Z)) {
if (dim(Z)[1] != length(y)) {
stop("* linreg_ADMM_spGLASSO : two inputs 'Z' and 'Y' have non-matching dimension.")
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: lambda parameter (in case of a single lambda)
if (!is.null(lambda)) {
if (length(lambda) == 1) {
meps <- (.Machine$double.eps)
negsmall <- -meps
if (!check_param_constant(lambda, negsmall)) {
stop("* linreg_ADMM_spGLASSO : reg. parameter 'lambda' is invalid.")
}
if (lambda < meps){
message("* linreg_ADMM_spGLASSO : since 'lambda' is effectively zero, a least-squares solution is returned.")
xsol <- as.vector(aux_pinv(X) %*% matrix(y))
output <- list()
output$x <- xsol
return(output)
}
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# get data dimensions
n <- dim(X)[1]
p <- dim(X)[2]
if (!is.null(Z)) {
q <- dim(Z)[2]
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: alpha parameter
if (is.null(alpha)) {
alpha <- 0.5
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Standardise response and design matrix
if (standardize.data == TRUE) {
res <- standardizemat(X, y)
X.std <- res$X.std
y.std <- res$y.std
mU <- res$mU
mV <- res$mV
} else {
X.std <- X
y.std <- y
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# check for inputs
npar <- dim(X.std)[2]
con <- list(maxit = 100L,
abstol = sqrt(.Machine$double.eps),
reltol = sqrt(sqrt(.Machine$double.eps)),
adaptation = TRUE,
rho = 1,
tau.ada = 2,
mu.ada = 10,
print.out = TRUE)
nmsC <- names(con)
con[(namc <- names(control))] <- control
if (length(noNms <- namc[!namc %in% nmsC])) {
warning("unknown names in control: ", paste(noNms, collapse = ", "))
}
# set default values (for inputs)
maxit <- con$maxit
abstol <- con$abstol
reltol <- con$reltol
adaptation <- con$adaptation
rho <- con$rho
tau.ada <- con$tau.ada
mu.ada <- con$mu.ada
print.out <- con$print.out
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: other parameters
if (!check_param_constant_multiple(c(abstol, reltol))) {
stop("* linreg_ADMM_spGLASSO : tolerance level is invalid.")
}
if (!check_param_integer(maxit, 0.0)) {
stop("* linreg_ADMM_spGLASSO : 'maxiter' should be a positive integer.")
}
maxit = as.integer(maxit)
if (!check_param_constant(rho, 0.0)) {
stop("* linreg_ADMM_spGLASSO : 'rho' should be a positive real number.")
}
if (!check_param_constant(mu.ada, 0.0)) {
stop("* linreg_ADMM_spGLASSO : 'mu.ada' should be a positive real number.")
}
if ((tau.ada < 1.0) || (tau.ada > 2.0)) {
warning("* linreg_spADMM_GLASSO : 'tau.ada' value is suggested to be in [1,2].")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: variable weights
check_weights(x = var_weights, n = dim(X)[2], algname = "linreg_ADMM_spGLASSO", funname = "var_weights")
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing: variable weights (L1)
check_weights(x = var_weights_L1, n = dim(X)[2], algname = "linreg_ADMM_spGLASSO", funname = "var_weights_L1")
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage groups
if (is.list(groups) && !is.null(groups$Glen)) {
nG <- groups$Glen
groups_ <- groups$groups
GRmat <- groups2mat_GLASSO(groups = groups_, Glen = nG)
G <- dim(GRmat)[1]
} else {
groups_ <- groups
GRmat <- groups2mat_GLASSO(groups = groups_, Glen = NULL)
G <- dim(GRmat)[1]
nG <- rowSums(GRmat)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Check overlapping groups
res.method <- check_group_overlaps(GRmat)
if (res.method == "ovglasso") {
stop("* linreg_ADMM_spGLASSO : this routine is for fitting the linear regression model with sparse group-LASSO penalty.
Use routine 'linreg_ADMM_OVGLASSO' for fitting the linear regression model with overlap group-LASSO penalty!")
}
if (res.method == "lasso") {
warning("* linreg_spADMM_GLASSO : this routine is for fitting the linear regression model with sparse group-LASSO penalty.
Use (faster) routine 'linreg_ADMM_LASSO' instead or check the provided groups!")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage group weights
if (is.null(group_weights)) {
# Define the weights as in Yuan and Lin 2006, JRSSB
group_weights <- sqrt(diag(tcrossprod(GRmat)))
} else {
check_weights(x = group_weights, n = G, algname = "linreg_ADMM_spGLASSO", funname = "group_weights")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage variable weights
if (is.null(var_weights)) {
var_weights <- 1.0 / colSums(GRmat)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Manage variable weights: adaptive Lasso
if (is.null(var_weights_L1)) {
var_weights_L1 <- rep(1, p)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get the sequence of lambdas
if (!is.null(nlambda) && is.null(lambda)) {
# get the smallest value of such that the regression
# coefficients estimated by the lasso are all equal to zero
lambda <- lm_lambdamax_spGLASSO(y = y.std, X = X.std, Z = Z,
groups = groups, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda.min.ratio = lambda.min.ratio, maxl = nlambda, alpha = alpha)$lambda.seq
}
if (is.null(nlambda) && is.null(lambda)) {
# get the smallest value of such that the regression
# coefficients estimated by the lasso are all equal to zero
lambda <- lm_lambdamax_spGLASSO(y = y.std, X = X.std, Z = Z,
groups = groups, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda.min.ratio = lambda.min.ratio, maxl = 30, alpha = alpha)$lambda.seq
}
if (!is.null(lambda)) {
nlambda <- length(lambda)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Run lasso
if (is.null(Z)) {
ret <- .admm_spglasso_fast(A = X.std, b = y.std, groups = GRmat,
group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_spglasso_fast",
# A = X.std, b = y.std, groups = GRmat,
# group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
# lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# get estimated coefficients and path
mSpRegP <- t(ret$coef.path)
vSpRegP <- ret$coefficients
} else {
ret <- .admm_spglasso_cov_fast(W = X.std, Z = Z, y = y.std,
groups = GRmat, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_spglasso_cov_fast",
# W = X.std, Z = Z, y = y.std,
# groups = GRmat, group_weights = group_weights, var_weights = var_weights, var_weights_L1 = var_weights_L1,
# lambda = lambda, alpha = alpha, rho_adaptation = adaptation, rho = rho, tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# get estimated coefficients and path
mSpRegP <- t(ret$sp.coef.path)
mRegP <- t(ret$coef.path)
vSpRegP <- ret$sp.coefficients
vRegP <- ret$coefficients
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get the path and retrieve the scaled estimates
if (standardize.data == TRUE) {
# if (alpha == 0) {
# dim_ <- sum(nG)
# } else if (alpha == 1) {
# dim_ <- p
# } else {
# # dim_ <- sum(nG) + p
# dim_ <- p
# }
dim_ <- p
sp.path <- matrix(nlambda, dim_, data = t(apply(mSpRegP, 2, function(x) solve(mU) %*% x %*% mV)))
vSpRegP <- solve(mU) %*% vSpRegP %*% mV
if (!is.null(Z)) {
path <- matrix(nlambda, q, data = t(apply(mRegP, 2, function(x) x %*% mV)))
vRegP <- vRegP %*% mV
}
} else {
vSpRegP <- matrix(vSpRegP, length(vSpRegP), 1)
sp.path <- t(mSpRegP)
if (!is.null(Z)) {
path <- t(mRegP)
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Print to screen
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
if (print.out == TRUE) {
cat("\n\n\n")
cat("Linear regression model with sparse group-LASSO penalty\n")
cat("Alternating direction method of multipliers\n")
cat("running time (for ", reltol, " relative error):",
sum(ret$elapsedTime), "seconds \n\n\n")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get output
res.names <- c("sp.coefficients",
"sp.coef.path",
"coefficients",
"coef.path",
"lambda.min",
"lambda",
"mse",
"min.mse",
"convergence",
"elapsedTime",
"iternum",
"objfun",
"r_norm",
"s_norm",
"err_pri",
"err_dual",
"rho")
res <- vector(mode = "list", length = length(res.names))
names(res) <- res.names
if (!is.null(Z)) {
res$coefficients <- vRegP
res$coef.path <- path
}
res$sp.coefficients <- vSpRegP
res$sp.coef.path <- sp.path
res$lambda.min <- ret$lambda.min
res$lambda <- ret$lambda
res$mse <- ret$mse
res$min.mse <- ret$min.mse
res$convergence <- ret$convergence
res$elapsedTime <- ret$elapsedTime
res$iternum <- ret$iternum
res$objfun <- ret$objfun
res$r_norm <- ret$r_norm
res$s_norm <- ret$s_norm
res$err_pri <- ret$err_pri
res$err_dual <- ret$err_dual
res$rho <- ret$rho
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Return output
return(res)
}
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