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R/linreg_ADMM_LASSO_cv.R
In fdaSP: Sparse Functional Data Analysis Methods
Defines functions
linreg_ADMM_LASSO_cv
# 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
#' Cross-validation for Least Absolute Shrinkage and Selection Operator
#'
#' LASSO, or L1-regularized regression, solves the following optimization problem
#' \deqn{\textrm{min}_\beta ~ \frac{1}{2}\|y-X\beta-Z\gamma\|_2^2 + \lambda \|\beta\|_1}
#' to obtain a sparse coefficient vector \eqn{\beta\in\mathbb{R}^p} and a coefficient vector \eqn{\gamma\in\mathbb{R}^q}. The regularization path
#' is computed for the 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.
#'
#' @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 var_weights a vector of length \eqn{p} containing variable-specific weights. 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 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 cv.fold the number of folds - default is 5.
#' @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 model fitted on the full sample with \eqn{\lambda} corresponding to minimum cross-validated MSE.}
#' \item{coefficients}{a length-\eqn{q} solution vector for the parameters \eqn{\gamma}, which corresponds to the model fitted on the full sample with \eqn{\lambda} corresponding to minimum cross-validated MSE.
#' It is provided only when either the matrix \eqn{Z} in input is not NULL or the intercept is set to TRUE.}
#' \item{convergence}{logical. 1 denotes achieved convergence for the model fitted on the full sample.}
#' \item{iternum}{number of iterations for the model fitted on the full sample.}
#' \item{lambda}{sequence of lambda.}
#' \item{lambda.min}{value of lambda that attains the minimum cross-validated MSE.}
#' \item{indi.min.mse}{index of the lambda sequence corresponding to lambda.min.}
#' \item{mse}{cross-validated mean squared error.}
#' \item{min.mse}{minimum value of the cross-validated MSE for the sequence of lambdas.}
#' \item{mse.sd}{standard deviation of the cross-validated mean squared error.}
#' \item{elapsedTime}{elapsed time in seconds for the whole procedure.}
#' \item{foldid}{a vector of values between 1 and cv.fold identifying what fold each observation is in.}
#' }
#'
#' 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
#'
#' ### generate sample data
#' set.seed(2023)
#' n <- 50
#' p <- 30
#' X <- matrix(rnorm(n * p), n, p)
#' beta <- apply(matrix(rnorm(p, sd = 1), p, 1), 1, fdaSP:::softhresh, 1.5)
#' y <- X %*% beta + rnorm(n, sd = sqrt(crossprod(X %*% beta)) / 20)
#'
#' ### set the hyper-parameters of the ADMM algorithm
#' maxit <- 1000
#' adaptation <- TRUE
#' rho <- 1
#' reltol <- 1e-5
#' abstol <- 1e-5
#'
#' ### run cross-validation
#' mod_cv <- fdaSP:::linreg_ADMM_LASSO_cv(X = X, y = y, standardize.data = FALSE, intercept = FALSE,
#' cv.fold = 5, nlambda = 30,
#' control = list("adaptation" = adaptation, "rho" = rho,
#' "maxit" = maxit, "reltol" = reltol, "abstol" = abstol,
#' "print.out" = FALSE))
#'
#' ### graphical presentation
#' plot(log(mod_cv$lambda), mod_cv$mse, type = "l", col = "blue", lwd = 2, bty = "n",
#' xlab = TeX("$\\log(\\lambda)$"), ylab = "Prediction Error",
#' ylim = range(mod_cv$mse - mod_cv$mse.sd, mod_cv$mse + mod_cv$mse.sd),
#' main = "Cross-validated Prediction Error")
#' fdaSP:::confband(xV = log(mod_cv$lambda), yVmin = mod_cv$mse - mod_cv$mse.sd, yVmax = mod_cv$mse + mod_cv$mse.sd)
#' abline(v = log(mod_cv$lambda[which(mod_cv$lambda == mod_cv$lambda.min)]), col = "red", lwd = 1.0)
#'
#' ### comparison with oracle error
#' mod <- fdaSP:::linreg_ADMM_LASSO(X = X, y = y, standardize.data = FALSE, intercept = FALSE,
#' nlambda = 30, control = list("adaptation" = adaptation, "rho" = rho,
#' "maxit" = maxit, "reltol" = reltol, "abstol" = abstol,
#' "print.out" = FALSE))
#' err_mod <- apply(mod$sp.coef.path, 1, function(x) sum((x - beta)^2))
#' plot(log(mod$lambda), err_mod, type = "l", col = "blue", lwd = 2, xlab = TeX("$\\log(\\lambda)$"),
#' ylab = "Estimation Error", main = "True Estimation Error", bty = "n")
#' abline(v = log(mod$lambda[which(err_mod == min(err_mod))]), col = "red", lwd = 1.0)
#' abline(v = log(mod_cv$lambda[which(mod_cv$lambda == mod_cv$lambda.min)]), col = "red", lwd = 1.0, lty = 2)
#'
#' @references
#' \insertRef{tibshirani_regression_1996a}{fdaSP}
#'
#' \insertRef{boyd_etal.2011}{fdaSP}
#'
#' \insertRef{hastie_etal.2015}{fdaSP}
#'
#' \insertRef{lin_etal.2022}{fdaSP}
#'
#' \insertRef{zou.2006}{fdaSP}
#'
#' @noRd
linreg_ADMM_LASSO_cv <- function(X, Z = NULL, y, var_weights = NULL, standardize.data = TRUE, lambda = NULL, nlambda = NULL, lambda.min.ratio = NULL,
cv.fold = 5, intercept = TRUE, control = list()) {
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get dimensions
n <- dim(X)[1]
p <- dim(X)[2]
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing
# data validity
if (!check_data_matrix(X)) {
stop("* linreg_ADMM_LASSO_cv : input 'X' is invalid data matrix.")
}
if (!check_data_vector(y)) {
stop("* linreg_ADMM_LASSO_cv : input 'y' is invalid data vector.")
}
y = as.vector(y)
# data size
if (n != length(y)) {
stop("* linreg_ADMM_LASSO_cv : two inputs 'X' and 'y' have non-matching dimension.")
}
if (!is.null(var_weights)) {
if (!check_data_positive_vector(var_weights)) {
stop("* linreg_ADMM_LASSO : input 'var_weights' is invalid vector of weights.")
} else {
adalasso <- TRUE
}
} else {
adalasso <- FALSE
var_weights <- rep(1L, p)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get dimensions
if (standardize.data == TRUE) {
if (intercept == TRUE){
intercept = FALSE
warning("* linreg_ADMM_LASSO_cv : intercept should be set to FALSE is data are standardized!\n")
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# add constant
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)
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# 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
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# data validity
if (!check_param_constant_multiple(c(abstol, reltol))) {
stop("* linreg_ADMM_LASSO_cv : tolerance level is invalid.")
}
if (!check_param_integer(maxit, 0.0)) {
stop("* linreg_ADMM_LASSO_cv : 'maxiter' should be a positive integer.")
}
maxiter = as.integer(maxit)
if (!check_param_constant(rho, 0.0)) {
stop("* linreg_ADMM_LASSO_cv : 'rho' should be a positive real number.")
}
if (!check_param_constant(mu.ada, 0.0)) {
stop("* linreg_ADMM_LASSO_cv : 'alpha' should be a positive real number.")
}
if ((tau.ada < 1.0) || (tau.ada > 2.0)) {
warning("* linreg_ADMM_LASSO_cv : 'tau' value is suggested to be in [1,2].")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# 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_ADALASSO(y = y.std, X = X.std, Z = Z, var_weights = var_weights, lambda.min.ratio = lambda.min.ratio, maxl = nlambda)$lambda.seq
}
if (is.null(nlambda) && is.null(lambda)) {
# get the smallest value of such that the regression
# coefficients estimated by the lasso are all equal to zero
lambda <- lm_lambdamax_ADALASSO(y = y.std, X = X.std, Z = Z, var_weights = var_weights, lambda.min.ratio = lambda.min.ratio, maxl = 30)$lambda.seq
}
if (!is.null(lambda)) {
nlambda <- length(lambda)
}
# other parameters
meps = (.Machine$double.eps)
negsmall = -meps
if (!check_param_constant(min(lambda), negsmall)) {
stop("* linreg_ADMM_LASSO_cv : reg. parameter 'lambda' is invalid.")
}
if (min(lambda) < meps) {
message("* linreg_ADMM_LASSO_cv : since 'lambda' is effectively zero, a least-squares solution is returned.")
xsol = as.vector(aux_pinv(X) %*% matrix(y))
output = list()
output$x = xsol
return(output)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Output
vMse <- rep(0, nlambda)
vSd <- rep(0, nlambda)
vRunTime <- rep(0, nlambda)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Store
u_store <- matrix(0L, nrow = p, ncol = nlambda)
z_store <- matrix(0L, nrow = p, ncol = nlambda)
rho_store <- rep(0L, nlambda)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get groups
ret.gr <- groups.cv(n = n, k = cv.fold)
# store Lasso regression for each fold
vMse_cv <- matrix(0.0, cv.fold, nlambda)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Initial time
startTime <- Sys.time()
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Run lasso
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Run overlap Group Lasso
for (kt in 1:cv.fold) {
# run ADMM
mX_ <- X.std[ret.gr$groups.cv[[kt]],]
vY_ <- y.std[ret.gr$groups.cv[[kt]]]
if (is.null(Z)) {
if (adalasso == FALSE) {
ret <- .admm_lasso_fast(A = mX_, b = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso_fast",
# A = mX_, b = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso_fast(A = mX_, b = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_adalasso_fast",
# A = mX_, b = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# fit
mSpRegP_ <- ret$coef.path
fit <- X.std[ret.gr$groups.pred[[kt]],] %*% t(mSpRegP_)
} else {
mZ_ <- Z[ret.gr$groups.cv[[kt]],]
if (adalasso == FALSE) {
ret <- .admm_lasso_cov_fast(W = mX_, Z = mZ_, y = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso_cov_fast",
# W = mX_, Z = mZ_, y = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso_cov_fast(W = mX_, Z = mZ_, y = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_adalasso_cov_fast",
# W = mX_, Z = mZ_, y = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# fit
mSpRegP_ <- ret$sp.coef.path
mRegP_ <- ret$coef.path
fit <- X.std[ret.gr$groups.pred[[kt]],] %*% t(mSpRegP_) + Z[ret.gr$groups.pred[[kt]],] %*% t(mRegP_)
}
# Evaluate MSE
n_ <- length(y.std[ret.gr$groups.pred[[kt]]])
mse <- apply(fit, 2, function(x) {sum((y.std[ret.gr$groups.pred[[kt]]] - x)^2)/n_})
vMse_cv[kt,] <- mse
}
# get mean squared error
vMse <- apply(vMse_cv, 2, mean)
vSd <- apply(vMse_cv, 2, sd)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# system time
endTime <- Sys.time()
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# compute run time
eltime <- difftime(time1 = endTime, time2 = startTime, units = "secs")
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get lasso output
min.mse <- min(vMse)
indi.min.mse <- which(vMse == min.mse)
lambda.min <- lambda[indi.min.mse]
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# get estimate over the full sample
#rho_ <- ret$rho[indi.min.mse]
rho_ <- ret$rho[[indi.min.mse]]
if (is.null(Z)) {
if (adalasso == FALSE) {
ret <- .admm_lasso(A = X.std, b = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso",
# A = X.std, b = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso(A = X.std, b = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("_admm_adalasso",
# A = X.std, b = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# get estimated parameters
vSpRegP <- ret$x
} else {
if (adalasso == FALSE) {
ret <- .admm_lasso_cov(W = X.std, Z = Z, y = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso_cov",
# W = X.std, Z = Z, y = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso_cov(W = X.std, Z = Z, y = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_adalasso_cov",
# W = X.std, Z = Z, y = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# get estimated parameters
vSpRegP <- ret$x
vRegP <- ret$v
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Store output: full sample estimate
niter <- ret$niter
if (niter < maxiter) {
converged <- TRUE
} else {
converged <- FALSE
}
dConv <- ifelse(converged == TRUE, 1, 0)
nIterN <- ret$niter
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Retrieve estimated parameters
if (standardize.data == TRUE) {
vSpRegP_ <- matrix(solve(mU) %*% vSpRegP %*% mV, nrow = p, ncol = 1)
if (!is.null(Z)) {
vRegP <- vRegP %*% mV
}
} else {
vSpRegP_ <- matrix(vSpRegP, nrow = p, ncol = 1)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Print to screen
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
if (print.out == TRUE) {
cat("\n\n\n")
cat("Linear regression model with LASSO penalty, Cross Validation\n")
cat("Alternating direction method of multipliers\n")
cat("running time (for ", reltol, " relative error):",
endTime - startTime, "seconds \n\n\n")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get output
res.names <- c("sp.coefficients",
"coefficients",
"mse",
"mse.sd",
"min.mse",
"lambda",
"lambda.min",
"convergence",
"elapsedTime",
"iternum",
"indi.min.mse",
"foldid")
res <- vector(mode = "list", length = length(res.names))
names(res) <- res.names
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get output
ret <- NULL
ret$sp.coefficients <- vSpRegP_
if (!is.null(Z)) {
ret$coefficients <- vRegP
}
ret$mse <- vMse
ret$mse.sd <- vSd
ret$min.mse <- min.mse
ret$lambda <- lambda
ret$lambda.min <- lambda.min
ret$convergence <- dConv
ret$elapsedTime <- eltime
ret$iternum <- nIterN
ret$indi.min.mse <- indi.min.mse
ret$foldid <- ret.gr$foldid
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Return output
return(ret)
}
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Defines functions linreg_ADMM_LASSO_cv
# 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
#' Cross-validation for Least Absolute Shrinkage and Selection Operator
#'
#' LASSO, or L1-regularized regression, solves the following optimization problem
#' \deqn{\textrm{min}_\beta ~ \frac{1}{2}\|y-X\beta-Z\gamma\|_2^2 + \lambda \|\beta\|_1}
#' to obtain a sparse coefficient vector \eqn{\beta\in\mathbb{R}^p} and a coefficient vector \eqn{\gamma\in\mathbb{R}^q}. The regularization path
#' is computed for the 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.
#'
#' @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 var_weights a vector of length \eqn{p} containing variable-specific weights. 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 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 cv.fold the number of folds - default is 5.
#' @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 model fitted on the full sample with \eqn{\lambda} corresponding to minimum cross-validated MSE.}
#' \item{coefficients}{a length-\eqn{q} solution vector for the parameters \eqn{\gamma}, which corresponds to the model fitted on the full sample with \eqn{\lambda} corresponding to minimum cross-validated MSE.
#' It is provided only when either the matrix \eqn{Z} in input is not NULL or the intercept is set to TRUE.}
#' \item{convergence}{logical. 1 denotes achieved convergence for the model fitted on the full sample.}
#' \item{iternum}{number of iterations for the model fitted on the full sample.}
#' \item{lambda}{sequence of lambda.}
#' \item{lambda.min}{value of lambda that attains the minimum cross-validated MSE.}
#' \item{indi.min.mse}{index of the lambda sequence corresponding to lambda.min.}
#' \item{mse}{cross-validated mean squared error.}
#' \item{min.mse}{minimum value of the cross-validated MSE for the sequence of lambdas.}
#' \item{mse.sd}{standard deviation of the cross-validated mean squared error.}
#' \item{elapsedTime}{elapsed time in seconds for the whole procedure.}
#' \item{foldid}{a vector of values between 1 and cv.fold identifying what fold each observation is in.}
#' }
#'
#' 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$double.eps).}
#' \item{maxit}{maximum number of iterations. The default value is 100.}
#' \item{print.out}{logical. If it is TRUE, a message about the procedure is printed. The default value is TRUE.}
#' }
#' @examples
#'
#' ### generate sample data
#' set.seed(2023)
#' n <- 50
#' p <- 30
#' X <- matrix(rnorm(n * p), n, p)
#' beta <- apply(matrix(rnorm(p, sd = 1), p, 1), 1, fdaSP:::softhresh, 1.5)
#' y <- X %*% beta + rnorm(n, sd = sqrt(crossprod(X %*% beta)) / 20)
#'
#' ### set the hyper-parameters of the ADMM algorithm
#' maxit <- 1000
#' adaptation <- TRUE
#' rho <- 1
#' reltol <- 1e-5
#' abstol <- 1e-5
#'
#' ### run cross-validation
#' mod_cv <- fdaSP:::linreg_ADMM_LASSO_cv(X = X, y = y, standardize.data = FALSE, intercept = FALSE,
#' cv.fold = 5, nlambda = 30,
#' control = list("adaptation" = adaptation, "rho" = rho,
#' "maxit" = maxit, "reltol" = reltol, "abstol" = abstol,
#' "print.out" = FALSE))
#'
#' ### graphical presentation
#' plot(log(mod_cv$lambda), mod_cv$mse, type = "l", col = "blue", lwd = 2, bty = "n",
#' xlab = TeX("$\\log(\\lambda)$"), ylab = "Prediction Error",
#' ylim = range(mod_cv$mse - mod_cv$mse.sd, mod_cv$mse + mod_cv$mse.sd),
#' main = "Cross-validated Prediction Error")
#' fdaSP:::confband(xV = log(mod_cv$lambda), yVmin = mod_cv$mse - mod_cv$mse.sd, yVmax = mod_cv$mse + mod_cv$mse.sd)
#' abline(v = log(mod_cv$lambda[which(mod_cv$lambda == mod_cv$lambda.min)]), col = "red", lwd = 1.0)
#'
#' ### comparison with oracle error
#' mod <- fdaSP:::linreg_ADMM_LASSO(X = X, y = y, standardize.data = FALSE, intercept = FALSE,
#' nlambda = 30, control = list("adaptation" = adaptation, "rho" = rho,
#' "maxit" = maxit, "reltol" = reltol, "abstol" = abstol,
#' "print.out" = FALSE))
#' err_mod <- apply(mod$sp.coef.path, 1, function(x) sum((x - beta)^2))
#' plot(log(mod$lambda), err_mod, type = "l", col = "blue", lwd = 2, xlab = TeX("$\\log(\\lambda)$"),
#' ylab = "Estimation Error", main = "True Estimation Error", bty = "n")
#' abline(v = log(mod$lambda[which(err_mod == min(err_mod))]), col = "red", lwd = 1.0)
#' abline(v = log(mod_cv$lambda[which(mod_cv$lambda == mod_cv$lambda.min)]), col = "red", lwd = 1.0, lty = 2)
#'
#' @references
#' \insertRef{tibshirani_regression_1996a}{fdaSP}
#'
#' \insertRef{boyd_etal.2011}{fdaSP}
#'
#' \insertRef{hastie_etal.2015}{fdaSP}
#'
#' \insertRef{lin_etal.2022}{fdaSP}
#'
#' \insertRef{zou.2006}{fdaSP}
#'
#' @noRd
linreg_ADMM_LASSO_cv <- function(X, Z = NULL, y, var_weights = NULL, standardize.data = TRUE, lambda = NULL, nlambda = NULL, lambda.min.ratio = NULL,
cv.fold = 5, intercept = TRUE, control = list()) {
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get dimensions
n <- dim(X)[1]
p <- dim(X)[2]
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Pre-processing
# data validity
if (!check_data_matrix(X)) {
stop("* linreg_ADMM_LASSO_cv : input 'X' is invalid data matrix.")
}
if (!check_data_vector(y)) {
stop("* linreg_ADMM_LASSO_cv : input 'y' is invalid data vector.")
}
y = as.vector(y)
# data size
if (n != length(y)) {
stop("* linreg_ADMM_LASSO_cv : two inputs 'X' and 'y' have non-matching dimension.")
}
if (!is.null(var_weights)) {
if (!check_data_positive_vector(var_weights)) {
stop("* linreg_ADMM_LASSO : input 'var_weights' is invalid vector of weights.")
} else {
adalasso <- TRUE
}
} else {
adalasso <- FALSE
var_weights <- rep(1L, p)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get dimensions
if (standardize.data == TRUE) {
if (intercept == TRUE){
intercept = FALSE
warning("* linreg_ADMM_LASSO_cv : intercept should be set to FALSE is data are standardized!\n")
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# add constant
if (intercept == TRUE) {
if (!is.null(Z)) {
Z <- cbind(matrix(data = 1, nrow = length(y), ncol = 1), Z)
} else {
Z <- matrix(data = 1, nrow = length(y), ncol = 1)
}
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Standardise response and design matrix
if (standardize.data == TRUE) {
res <- standardizemat(X, y)
X.std <- res$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$print.out
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# data validity
if (!check_param_constant_multiple(c(abstol, reltol))) {
stop("* linreg_ADMM_LASSO_cv : tolerance level is invalid.")
}
if (!check_param_integer(maxit, 0.0)) {
stop("* linreg_ADMM_LASSO_cv : 'maxiter' should be a positive integer.")
}
maxiter = as.integer(maxit)
if (!check_param_constant(rho, 0.0)) {
stop("* linreg_ADMM_LASSO_cv : 'rho' should be a positive real number.")
}
if (!check_param_constant(mu.ada, 0.0)) {
stop("* linreg_ADMM_LASSO_cv : 'alpha' should be a positive real number.")
}
if ((tau.ada < 1.0) || (tau.ada > 2.0)) {
warning("* linreg_ADMM_LASSO_cv : 'tau' value is suggested to be in [1,2].")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get the sequence of lambdas
if (!is.null(nlambda) && is.null(lambda)) {
# get the smallest value of such that the regression
# coefficients estimated by the lasso are all equal to zero
lambda <- lm_lambdamax_ADALASSO(y = y.std, X = X.std, Z = Z, var_weights = var_weights, lambda.min.ratio = lambda.min.ratio, maxl = nlambda)$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_ADALASSO(y = y.std, X = X.std, Z = Z, var_weights = var_weights, lambda.min.ratio = lambda.min.ratio, maxl = 30)$lambda.seq
}
if (!is.null(lambda)) {
nlambda <- length(lambda)
}
# other parameters
meps = (.Machine$double.eps)
negsmall = -meps
if (!check_param_constant(min(lambda), negsmall)) {
stop("* linreg_ADMM_LASSO_cv : reg. parameter 'lambda' is invalid.")
}
if (min(lambda) < meps) {
message("* linreg_ADMM_LASSO_cv : since 'lambda' is effectively zero, a least-squares solution is returned.")
xsol = as.vector(aux_pinv(X) %*% matrix(y))
output = list()
output$x = xsol
return(output)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Output
vMse <- rep(0, nlambda)
vSd <- rep(0, nlambda)
vRunTime <- rep(0, nlambda)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Store
u_store <- matrix(0L, nrow = p, ncol = nlambda)
z_store <- matrix(0L, nrow = p, ncol = nlambda)
rho_store <- rep(0L, nlambda)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get groups
ret.gr <- groups.cv(n = n, k = cv.fold)
# store Lasso regression for each fold
vMse_cv <- matrix(0.0, cv.fold, nlambda)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Initial time
startTime <- Sys.time()
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Run lasso
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Run overlap Group Lasso
for (kt in 1:cv.fold) {
# run ADMM
mX_ <- X.std[ret.gr$groups.cv[[kt]],]
vY_ <- y.std[ret.gr$groups.cv[[kt]]]
if (is.null(Z)) {
if (adalasso == FALSE) {
ret <- .admm_lasso_fast(A = mX_, b = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso_fast",
# A = mX_, b = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso_fast(A = mX_, b = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_adalasso_fast",
# A = mX_, b = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# fit
mSpRegP_ <- ret$coef.path
fit <- X.std[ret.gr$groups.pred[[kt]],] %*% t(mSpRegP_)
} else {
mZ_ <- Z[ret.gr$groups.cv[[kt]],]
if (adalasso == FALSE) {
ret <- .admm_lasso_cov_fast(W = mX_, Z = mZ_, y = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso_cov_fast",
# W = mX_, Z = mZ_, y = vY_, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso_cov_fast(W = mX_, Z = mZ_, y = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_adalasso_cov_fast",
# W = mX_, Z = mZ_, y = vY_, var_weights = var_weights, lambda = lambda, rho_adaptation = adaptation, rho = rho,
# tau = tau.ada, mu = mu.ada, reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# fit
mSpRegP_ <- ret$sp.coef.path
mRegP_ <- ret$coef.path
fit <- X.std[ret.gr$groups.pred[[kt]],] %*% t(mSpRegP_) + Z[ret.gr$groups.pred[[kt]],] %*% t(mRegP_)
}
# Evaluate MSE
n_ <- length(y.std[ret.gr$groups.pred[[kt]]])
mse <- apply(fit, 2, function(x) {sum((y.std[ret.gr$groups.pred[[kt]]] - x)^2)/n_})
vMse_cv[kt,] <- mse
}
# get mean squared error
vMse <- apply(vMse_cv, 2, mean)
vSd <- apply(vMse_cv, 2, sd)
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# system time
endTime <- Sys.time()
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# compute run time
eltime <- difftime(time1 = endTime, time2 = startTime, units = "secs")
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get lasso output
min.mse <- min(vMse)
indi.min.mse <- which(vMse == min.mse)
lambda.min <- lambda[indi.min.mse]
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# get estimate over the full sample
#rho_ <- ret$rho[indi.min.mse]
rho_ <- ret$rho[[indi.min.mse]]
if (is.null(Z)) {
if (adalasso == FALSE) {
ret <- .admm_lasso(A = X.std, b = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso",
# A = X.std, b = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso(A = X.std, b = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("_admm_adalasso",
# A = X.std, b = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# get estimated parameters
vSpRegP <- ret$x
} else {
if (adalasso == FALSE) {
ret <- .admm_lasso_cov(W = X.std, Z = Z, y = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_lasso_cov",
# W = X.std, Z = Z, y = y.std, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
} else {
ret <- .admm_adalasso_cov(W = X.std, Z = Z, y = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
# ret <- .Call("admm_adalasso_cov",
# W = X.std, Z = Z, y = y.std, var_weights = var_weights, u = t(ret$U[indi.min.mse,]), z = t(ret$Z[indi.min.mse,]), lambda = lambda.min,
# rho_adaptation = adaptation, rho = rho_[length(rho_)], tau = tau.ada, mu = mu.ada,
# reltol = reltol, abstol = abstol, maxiter = maxit, ping = 0)
}
# get estimated parameters
vSpRegP <- ret$x
vRegP <- ret$v
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Store output: full sample estimate
niter <- ret$niter
if (niter < maxiter) {
converged <- TRUE
} else {
converged <- FALSE
}
dConv <- ifelse(converged == TRUE, 1, 0)
nIterN <- ret$niter
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Retrieve estimated parameters
if (standardize.data == TRUE) {
vSpRegP_ <- matrix(solve(mU) %*% vSpRegP %*% mV, nrow = p, ncol = 1)
if (!is.null(Z)) {
vRegP <- vRegP %*% mV
}
} else {
vSpRegP_ <- matrix(vSpRegP, nrow = p, ncol = 1)
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Print to screen
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
if (print.out == TRUE) {
cat("\n\n\n")
cat("Linear regression model with LASSO penalty, Cross Validation\n")
cat("Alternating direction method of multipliers\n")
cat("running time (for ", reltol, " relative error):",
endTime - startTime, "seconds \n\n\n")
}
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get output
res.names <- c("sp.coefficients",
"coefficients",
"mse",
"mse.sd",
"min.mse",
"lambda",
"lambda.min",
"convergence",
"elapsedTime",
"iternum",
"indi.min.mse",
"foldid")
res <- vector(mode = "list", length = length(res.names))
names(res) <- res.names
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Get output
ret <- NULL
ret$sp.coefficients <- vSpRegP_
if (!is.null(Z)) {
ret$coefficients <- vRegP
}
ret$mse <- vMse
ret$mse.sd <- vSd
ret$min.mse <- min.mse
ret$lambda <- lambda
ret$lambda.min <- lambda.min
ret$convergence <- dConv
ret$elapsedTime <- eltime
ret$iternum <- nIterN
ret$indi.min.mse <- indi.min.mse
ret$foldid <- ret.gr$foldid
# ::::::::::::::::::::::::::::::::::::::::::::::::::::::::
# Return output
return(ret)
}
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