205 lines
7.8 KiB
R
205 lines
7.8 KiB
R
#' Regularized Matrix Regression
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#'
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#' Solved the regularized problem
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#' \deqn{min h(B) = l(B) + J(B)}
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#' for a matrix \eqn{B}.
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#' where \eqn{l} is a loss function; for the GLM, we use the negative
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#' log-likelihood as the loss. \eqn{J(B) = f(\sigma(B))}, where \eqn{f} is a
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#' function of the singular values of \eqn{B}.
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#'
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#' Currently, only the least squares problem with nuclear norm penalty is
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#' implemented.
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#'
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#' In case of \code{lambda = Inf} the maximum penalty \eqn{\lambda} is computed.
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#' In this case the return value is only estimate as a single value.
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#'
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#' @param X the singnal data ether as a 3D tensor or a 2D matrix. In case of a
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#' 3D tensor the axis are assumed to be \eqn{n\times p\times q} meaning the
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#' first dimension are the observations while the second and third are the
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#' `image' dimensions. When the data is provided as a matix it's assumed to be
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#' of shape \eqn{n\times p q} where each observation is the vectorid `image'.
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#' @param Z additional covariate vector (can be \code{NULL} if not required.
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#' For regression with intercept set \code{Z = rep(1, n)})
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#' @param y univariate response vector
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#' @param lambda penalty term, if set to \code{Inf} max lambda is computed.
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#' @param max.iter maximum number of gadient updates
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#' @param max.line.iter maximum number of line search iterations
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#' @param shape Shape of the matrix valued predictors. Required iff the
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#' predictors \code{X} are provided in vectorized form, e.g. as a 2D matrix.
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#' @param step.size max. stepsize for gradient updates
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#' @param B0 initial value for optimization. Matrix of dimensions \eqn{p\times q}
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#' @param beta0 initial value of additional covatiates coefficient for \eqn{Z}
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#' @param alpha iterative Nesterov momentum scaling values
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#' @param eps precition for main loop break conditions
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#' @param logger logging callback invoced after every line search before break
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#' condition checks. The expected function signature is of the form
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#' \code{function(iter, loss, penalty, B, beta, step.size)}.
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#'
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#' @export
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RMReg <- function(X, Z, y, lambda = 0, max.iter = 500L, max.line.iter = 50L,
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shape = dim(X)[-1], step.size = 1e-3,
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B0 = array(0, dim = shape),
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beta0 = rep(0, NCOL(Z)),
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alpha = function(a, t) { (1 + sqrt(1 + (2 * a)^2)) / 2 },
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eps = .Machine$double.eps,
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logger = NULL
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) {
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# Define loss (without penalty)
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loss <- function(B, beta, X, Z, y) 0.5 * sum((y - Z %*% beta - X %*% c(B))^2)
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# gradient of loss (without penalty)
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grad <- function(B, beta, X, Z, y) {
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inner <- X %*% c(B) + Z %*% beta - y
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list(beta = c(crossprod(inner, Z)), B = c(crossprod(inner, X)))
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}
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# # and the penalty function (as function of singular values)
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# penalty <- function(sigma) sum(sigma)
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# Check (prepair) params
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stopifnot(nrow(X) == length(y))
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if (!missing(shape)) {
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stopifnot(ncol(X) == prod(shape))
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} else {
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stopifnot(length(dim(X)) == 3)
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dim(X) <- c(nrow(X), prod(shape))
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}
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if (missing(Z) || is.null(Z)) {
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Z <- matrix(0, nrow(X), 1)
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} else if (!is.matrix(Z)) {
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Z <- as.matrix(Z)
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}
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# Set singular values of start matrix predictor coefficients
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if (missing(B0)) {
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B1.sv <- rep(0, min(shape))
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} else {
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B1.sv <- La.svd(B0, 0, 0)$d
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}
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# initialize current and previous coefficients (start position)
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B1 <- B0
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beta1 <- beta0
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alpha0 <- 0
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alpha1 <- 1
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loss0 <- loss1 <- loss(B1, beta1, X, Z, y)
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# main descent loop
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no.nesterov <- FALSE
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for (iter in seq_len(max.iter)) {
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if (no.nesterov) {
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# classic gradient step as fallback
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S <- B1
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s <- beta1
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} else {
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# momentum step (extrapolation using previous direction)
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S <- B1 + ((alpha0 - 1) / alpha1) * (B1 - B0)
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s <- beta1 + ((alpha0 - 1) / alpha1) * (beta1 - beta0)
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}
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# compute (nesterov) gradient
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G <- grad(S, s, X, Z, y)
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# backtracking line search (executed at least once)
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for (delta in step.size * 0.5^seq(0, max.line.iter - 1L)) {
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# Gradient step with step size delta
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A <- S - delta * G$B
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beta.temp <- s - delta * G$beta
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if (lambda == Inf) {
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# Application of Corollary 1 for estimation of max lambda
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# Return max lambda estimate
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return(max(La.svd(A, 0, 0)$d) / delta)
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} else if (lambda > 0) {
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# SVD of (potential) next step
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svdA <- La.svd(A)
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# Next (possible) penalized iterate
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B.temp.sv <- pmax(0, svdA$d - delta * lambda)
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B.temp <- svdA$u %*% (B.temp.sv * svdA$vt)
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} else {
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# in case of no penalization (pure least squares)
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B.temp.sv <- La.svd(A, 0, 0)$d
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B.temp <- A
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}
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# Check line search condition
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# h(B.temp) <= g(B.temp | S, delta)
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# \_ left _/ \_____ right _____/
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# where g(B.temp | S, delta) is the first order approx. of the loss
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# l(S) + <grad l(S), B - S> + | B - S |_F^2 / 2 delta + J(B)
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left <- loss(B.temp, beta.temp, X, Z, y)
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right <- loss(S, s, X, Z, y) +
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sum(G$B * (B.temp - S)) + sum(G$beta * (beta.temp - s)) +
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(norm(B.temp - S, 'F')^2 + sum((beta.temp - s)^2)) / (2 * delta)
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if (left <= right) {
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break
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}
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}
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# Evaluate loss to ensure descent after line search
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loss.temp <- loss(B.temp, beta.temp, X, Z, y)
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# logging callback
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if (is.function(logger)) {
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logger(iter, loss.temp, lambda * sum(B.temp.sv),
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B.temp, beta.temp, delta)
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}
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# after line search enforce descent
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if (loss.temp + lambda * sum(B.temp.sv) <= loss1 + lambda * sum(B1.sv)) {
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B0 <- B1
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B1 <- array(B.temp, shape)
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B1.sv <- B.temp.sv
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beta0 <- beta1
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beta1 <- beta.temp
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loss0 <- loss1
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loss1 <- loss.temp
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no.nesterov <- FALSE # always reset
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} else if (!no.nesterov) {
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no.nesterov <- TRUE # retry without momentum
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next
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} else {
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break # failed even without momentum -> stop
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}
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# check break conditions
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if (sum(B1.sv) < eps) {
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break # estimate is (numerically) zero -> stop
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}
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if ((sum(G$B^2) + sum(G$beta^2)) < eps * sum(unlist(Map(length, G)))) {
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break # mean squared gradient is smaller than epsilon -> stop
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}
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if (abs(loss0 - loss1) < eps) {
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break # decrease is smaller than epsilon -> stop
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}
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# update momentum scaling
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alpha0 <- alpha1
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alpha1 <- alpha(alpha1, iter)
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# set step size to two times current delta
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step.size <- 2 * delta
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}
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# Degrees of Freedom estimate (TODO: this is like in `matrix_sparsereg.m`)
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sigma <- c(La.svd(A, 0, 0)$d, rep(0, max(shape) - min(shape)))
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df <- length(beta1)
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for (i in seq_len(sum(B1.sv > 0))) {
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df <- df + 1 + sigma[i] * (sigma[i] - delta * lambda) * (
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sum(ifelse((1:shape[1]) != i, 1 / (sigma[i]^2 - sigma[1:shape[1]]^2), 0)) +
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sum(ifelse((1:shape[2]) != i, 1 / (sigma[i]^2 - sigma[1:shape[2]]^2), 0)))
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}
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# return estimates and some additional stats
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list(
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B = B1,
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beta = beta1,
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singular.values = B1.sv,
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iter = iter,
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df = df,
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loss = loss1,
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lambda = lambda,
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AIC = loss1 + 2 * df, # TODO: check this!
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BIC = loss1 + log(nrow(X)) * df, # TODO: check this!
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call = match.call() # invocing function call, collects params like lambda
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)
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}
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