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CVE/CVE_R/R/cve_linesearch.R

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#' Implementation of the CVE method using curvilinear linesearch with Armijo-Wolfe
#' conditions.
#'
#' @keywords internal
#' @export
cve_linesearch <- function(X, Y, k,
nObs = sqrt(nrow(X)),
h = NULL,
tau = 1.0,
tol = 1e-3,
rho1 = 0.1,
rho2 = 0.9,
slack = 0,
epochs = 50L,
attempts = 10L,
max.linesearch.iter = 10L
) {
# Set `grad` functions environment to enable if to find this environments
# local variabels, needed to enable the manipulation of this local variables
# from within `grad`.
environment(grad) <- environment()
# Setup histories.
loss.history <- matrix(NA, epochs, attempts)
error.history <- matrix(NA, epochs, attempts)
tau.history <- matrix(NA, epochs, attempts)
# Get dimensions.
n <- nrow(X)
p <- ncol(X)
q <- p - k
# Save initial learning rate `tau`.
tau.init <- tau
# Addapt tolearance for break condition.
tol <- sqrt(2 * q) * tol
# Estaimate bandwidth if not given.
if (missing(h) | !is.numeric(h)) {
h <- estimate.bandwidth(X, k, nObs)
}
# Compute persistent data.
# Compute lookup indexes for symmetrie, lower/upper
# triangular parts and vectorization.
pair.index <- elem.pairs(seq(n))
i <- pair.index[, 1] # `i` indices of `(i, j)` pairs
j <- pair.index[, 2] # `j` indices of `(i, j)` pairs
# Matrix of vectorized indices. (vec(index) -> seq)
index <- matrix(seq(n * n), n, n)
lower <- index[lower.tri(index)]
upper <- t(index)[lower]
# Create all pairewise differences of rows of `X`.
X_diff <- X[i, , drop = F] - X[j, , drop = F]
# Identity matrix.
I_p <- diag(1, p)
# Init tracking of current best (according multiple attempts).
V.best <- NULL
loss.best <- Inf
# Start loop for multiple attempts.
for (attempt in 1:attempts) {
# Sample a `(p, q)` dimensional matrix from the stiefel manifold as
# optimization start value.
V <- rStiefl(p, q)
# Initial loss and gradient.
loss <- Inf
G <- grad(X, Y, V, h, loss.out = TRUE, persistent = TRUE)
# Set last loss (aka, loss after applying the step).
loss.last <- loss
## Start optimization loop.
for (epoch in 1:epochs) {
# Cayley transform matrix `A`
A <- (G %*% t(V)) - (V %*% t(G))
# Directional derivative of the loss at current position, given
# as `Tr(G^T \cdot A \cdot V)`.
loss.prime <- -0.5 * norm(A, type = 'F')^2
# Linesearch
tau.upper <- Inf
tau.lower <- 0
tau <- tau.init
for (iter in 1:max.linesearch.iter) {
# Apply learning rate `tau`.
A.tau <- (tau / 2) * A
# Parallet transport (on Stiefl manifold) into direction of `G`.
inv <- solve(I_p + A.tau)
V.tau <- inv %*% ((I_p - A.tau) %*% V)
# Loss at position after a step.
loss <- Inf # aka loss.tau
G.tau <- grad(X, Y, V.tau, h, loss.out = TRUE, persistent = TRUE)
# Armijo condition.
if (loss > loss.last + (rho1 * tau * loss.prime)) {
tau.upper <- tau
tau <- (tau.lower + tau.upper) / 2
next()
}
V.prime.tau <- -0.5 * inv %*% A %*% (V + V.tau)
loss.prime.tau <- sum(G * V.prime.tau) # Tr(grad(tau)^T \cdot Y^'(tau))
# Wolfe condition.
if (loss.prime.tau < rho2 * loss.prime) {
tau.lower <- tau
if (tau.upper == Inf) {
tau <- 2 * tau.lower
} else {
tau <- (tau.lower + tau.upper) / 2
}
} else {
break()
}
}
# Compute error.
error <- norm(V %*% t(V) - V.tau %*% t(V.tau), type = "F")
# Write history.
loss.history[epoch, attempt] <- loss
error.history[epoch, attempt] <- error
tau.history[epoch, attempt] <- tau
# Check break condition (epoch check to skip ignored gradient calc).
# Note: the devision by `sqrt(2 * k)` is included in `tol`.
if (error < tol | epoch >= epochs) {
# take last step and stop optimization.
V <- V.tau
break()
}
# Perform the step and remember previous loss.
V <- V.tau
loss.last <- loss
G <- G.tau
}
# Check if current attempt improved previous ones
if (loss < loss.best) {
loss.best <- loss
V.best <- V
}
}
return(list(
loss.history = loss.history,
error.history = error.history,
tau.history = tau.history,
loss = loss.best,
V = V.best,
B = null(V.best),
h = h
))
}