svm.R

##' Solving Regularized Support Vector Machine
##'
##' @description
##' This function solves the regularized support vector machine
##' of the following form:
##' \deqn{
##' \min_{\beta}\ \frac{C}{n}\sum_{i=1}^n \max(1-y_i x_i^T \beta,0) +
##' \frac{1}{2}\Vert \beta \Vert_2^2
##' }
##' where \eqn{\beta\in\mathbb{R}^d}{\beta} is a length-\eqn{d} vector,
##' \eqn{x_i\in\mathbb{R}^d}{x_i} is the feature vector for the \eqn{i}-th observation,
##' \eqn{y_i\in\{-1,1\}}{y_i \in {-1, 1}} is a binary label,
##' and \eqn{C} is the cost parameter.
##'
##' @param x          The data matrix \eqn{X=(x_1,\ldots,x_n)^T} of size
##'                   \eqn{n\times d}{n * d}, representing \eqn{n} observations
##'                   and \eqn{d} features.
##' @param y          The length-\eqn{n} response vector.
##' @param C          The cost parameter.
##' @param max_iter   Maximum number of iterations.
##' @param tol        Tolerance parameter for convergence test.
##' @param shrink     Whether to use the shrinkage algorithm.
##' @param verbose    Level of verbosity.
##' @param trace_freq Trace objective function values every \code{trace_freq}
##'                   iterations. Only works if \code{verbose > 0}.
##'
##' @return A list of the following components:
##' \item{beta}{Optimized value of the \eqn{\beta} vector.}
##' \item{xi,Lambda,Gamma}{Values of dual variables.}
##' \item{niter}{Number of iterations used.}
##' \item{dual_objfns}{Dual objective function values during the optimization process.}
##' \item{primal_objfns}{Primal objective function values during the optimization process.}
##'
##' @author Yixuan Qiu \url{https://statr.me}
##'
##'         Ben Dai \url{https://bendai.org}
##'
##' @examples
##' # Simulate a data set
##' set.seed(123)
##' n = 5000
##' p = 100
##' x1 = matrix(rnorm(n / 2 * p, -0.25, 0.1), n / 2)
##' x2 = matrix(rnorm(n / 2 * p, 0.25, 0.1), n / 2)
##' x = rbind(x1, x2)
##' beta = 0.1 * rnorm(p)
##' prob = plogis(c(x %*% beta))
##' y = 2 * rbinom(n, 1, prob) - 1
##' C = 0.5
##'
##' # Compute using ReHLine
##' time1 = system.time(
##'     res1 <- svm(x, y, C = C * n, max_iter = 1000, tol = 1e-3, verbose = 0)
##' )
##' beta1 = res1$beta
##' print(time1)
##'
##' # Compare the result with LiblineaR
##' if (require(LiblineaR))
##' {
##'     time2 = system.time(
##'         res2 <- LiblineaR(x, y, type = 3, cost = C, epsilon = 1e-3, bias = 0, verbose = FALSE)
##'     )
##'     beta2 = as.numeric(res2$W)
##'     print(time2)
##'
##'     plot(beta1, beta2)
##'     abline(0, 1, col = "red")
##' }
svm = function(x, y, C = 1, max_iter = 1000, tol = 1e-5, shrink = TRUE, verbose = 0, trace_freq = 100)
{
    n = nrow(x)

    Umat = -C / n * matrix(y, nrow = 1)
    Vmat = matrix(C / n, 1, n)

    res = rehline(x, Umat, Vmat, max_iter = max_iter, tol = tol,
                  shrink = shrink, verbose = verbose, trace_freq = trace_freq)
    res
}