test1 <- function(){
get.initial.alpha <- function(eps, J){
if(length(eps) != J)
stop("wrong eps input")
k <- (1 - sum(eps))/(J + sqrt(J))
alpha <- eps + k
return(alpha)
}
J <- 2
y <- 1
z <- matrix(c(2,3),1,2)
x <- matrix(0, 1,1)
eps <- rep(0.01/J, J)
reltol <- 10^-3
relerr <- 10^-3
a0 <- get.initial.alpha(rep(0.1/J, J), J)
e <- penoptpersp( x,y,z, a0 = a0, eps = eps, reltol = reltol, relerr = relerr, rho0 = NULL)
## e <- penoptpersp(x, y, z, eps, reltol ,relerr, a0 = get.initial.alpha(rep(0.1/J, J), J))
## optimal alpha is c(0, 1)
}
# This version modifies default starting alpha
test0 = function( n=30, J=10, K=11, eps = .01, seed = 20140422,relerr ){
# zero'th test of optimization
set.seed(seed)
z = matrix(runif(n*J),ncol=J) # must be positive
x = matrix(rnorm(n*K),ncol=K)
y = rnorm(n)
penoptpersp(x,y,z,eps,reltol=1e-8,relerr=relerr)
}
#' penalized optimization of the constrained linearized perspective function
#' @export
#' @param x \eqn{n \times K} matrix
#' @param y length \eqn{n} vector
#' @param z \eqn{n \times J} matrix
#' @param a0 length \eqn{J} vector
#' @param b0 length \eqn{K} vector
#' @param eps length \eqn{J} vector, default to be rep(0.1/J, J)
#' @param reltol relative tolerence for Newton step, between 0 to 1, default to be \eqn{10^{-3}}. For each inner loop, we optimize \eqn{f_0 + \rho \times \mathrm{pen} } for a fixed \eqn{\rho}, we stop when the Newton decrement \eqn{f(x) - inf_y \hat{f}(y) \leq f(x)* \mathrm{reltol}}, where \eqn{\hat{f}} is the second-order approximation of \eqn{f} at \eqn{x}
#' @param relerr stop when within (1+\emph{relerr}) of minimum variance, default to be \eqn{10^{-3}}, between 0 to 1.
#' @param rho0 initial value for \eqn{\rho}, default to be 1
#' @param maxin maximum number of inner iterations
#' @param maxout maximum number of outer iterations
#' @return a list of \describe{
#' \item{x}{input x}
#' \item{y}{input y}
#' \item{z}{input z}
#' \item{alpha}{optimized alpha}
#' \item{beta}{optimized beta}
#' \item{rho}{value of rho}
#' \item{f}{value of the objective function}
#' \item{rhopen}{value of rho*pen when returned}
#' \item{outer}{number of outer loops}
#' \item{relerr}{relative error}
#' \item{alphasum}{sum of optimized alpha}
#' }
#' @details To minimize \eqn{\sum_i \frac{(y_i - x_i^T \beta)^2}{z_i^T\alpha}} over \eqn{\alpha} and \eqn{\beta},
#' subject to \eqn{\alpha_j > \epsilon_j} for \eqn{j = 1, \cdots, J} and \eqn{\sum_{j=1}^J \alpha_j < 1},
#'
#' Instead we minimize \eqn{ \sum_i \frac{(y_i - x_i^T \beta)^2}{z_i^T\alpha} + \rho \times \mathrm{pen}} for a decreasing sequence of \eqn{\rho}
#'
#' where \eqn{ \mathrm{pen} = -( \sum_{j = 1}^J( \log(\alpha_j-\epsilon_j) ) + \log(1-\sum_{j = 1}^J \alpha_j) )}
#'
#' starting values are \eqn{\alpha = a0} and \eqn{\beta = b0}. They can be missing.
#'
#' The optimization stops when within (1+\emph{relerr}) of minimum variance.
penoptpersp = function( x,y,z, a0 = NULL, b0 = NULL, eps = NULL, reltol = NULL, relerr = NULL, rho0 = NULL, maxin= NULL, maxout = NULL){
J = ncol(z)
K = ncol(x)
# Verify that inputs are ok and feasible
if(is.null(eps))
eps <- rep(0.1/J, J)
else if( length(eps)==1 )
eps = rep(eps,J)
else if(length(eps) != J)
stop("Wrong dimension of eps")
if( min(eps)<0 )stop("Negative epsilon")
if(sum(eps) >= 1) stop("sum of epsilon >= 1")
if(is.null(reltol) | reltol > 1 | reltol < 0) reltol <- 10^-3
if(is.null(relerr) | relerr > 1 | relerr < 0) relerr <- 10^-3
if(is.null(rho0)) rho0 <- 1
if(is.null(maxin)) maxin <- 20
if(is.null(maxout)) maxout <- 30
if( !is.null(a0) ){
if( any(a0<eps) )stop("Infeasible start point, any(a0<eps)")
if( sum(a0)>1 )stop("Infeasible start point, sum(a0)>1")
}
if(any(apply(z, 1, function(x) sum(x^2)) == 0)) stop("Some rows of z is zero")
## library(Matrix)
if(rankMatrix(x, tol = .Machine$double.eps^2) < min(dim(x))) warning("x may be rank deficient")
delta = (1-sum(eps))/(J+1)
if( delta < 0 )stop("No feasible alpha")
if( K==0 )warning("No regression portion")
# main workhorse function
fgH = function(alpha,beta,rho,do=3){
# do = 3 function, gradient, Hessian
# do = 2 function, gradient
# do = 1 function
# do = 0 feasibility
# The penalty
if( any(alpha<eps) )
pen = Inf
else if( sum(alpha)>1 )
pen = Inf
else
pen = -( sum( log(alpha-eps) ) + log(1-sum(alpha) ) )
if( do <= 0 )return( pen < Inf )
# The objective function
res = y - x%*%beta # n-vector of residuals
zal = z%*%alpha # n-vector of mixture probabilities
ebz = res/zal
f0 = sum( res^2/zal )
f = f0 + rho * pen
if( is.nan(f) ){
print("Hit a NaN")
print(pen)
print(alpha)
print(sum(alpha))
}
if( do <=1 )return(list(f=f, f0 = f0))
# The gradient
gpen = -(1/(alpha-eps) - 1/(1-sum(alpha)))
gbeta = -2 * t(x) %*% ebz
galph = -t(z) %*% ebz^2
g0 <- c(gbeta, galph)
gphi <- c(rep(0, K), rho*gpen)
g = c( gbeta, galph + rho*gpen )
if( do <=2 )return(list(f=f,g=g,f0 = f0, g0 = g0, gphi = gphi))
# The Hessian
xbyrootq = x
for( j in 1:ncol(x) )
xbyrootq[,j] = xbyrootq[,j]/sqrt(zal)
Hbb = 2*t(xbyrootq) %*% xbyrootq
zebyqrootq = z
for( j in 1:ncol(z) )
zebyqrootq[,j] = zebyqrootq[,j] * res /zal^(3/2)
Haa = 2 * t(zebyqrootq) %*% zebyqrootq
Hba = 2 * t(xbyrootq) %*% zebyqrootq
Hpen = diag( (alpha-eps)^-2 ) + 1/(1-sum(alpha))^2
H = rbind( cbind( Hbb, Hba ),
cbind( t(Hba), Haa + rho*Hpen ) )
return(list(f=f,g=g,H=H, f0 = f0, g0 = g0, gphi = gphi))
}
testgradient = function(){
# this function tests whether the gradient
# is really the derivative of the function
# the code passed, but save this function for
# rechecking if necessary.
alpha = .5*rep(1,J)/J # start at subdistribution
beta = rep(0,K)
ans = fgH(alpha,beta,rho,do=3)
del = 0.0001
print("testing gradient wrt beta")
for( k in 1:K ){
bup = beta
bup[k] = beta[k]+del
ansup = fgH(alpha,bup,rho,do=1)
bdn = beta
bdn[k] = beta[k]-del
ansdn = fgH(alpha,bdn,rho,do=1)
print(k)
print(c(ans$g[k],(ansup$f-ansdn$f)/(2*del) ))
}
print("testing gradient wrt alpha")
for( j in 1:J ){
aup = alpha
aup[j] = alpha[j]+del
ansup = fgH(aup,beta,rho,do=1)
adn = alpha
adn[j] = alpha[j]-del
ansdn = fgH(adn,beta,rho,do=1)
print(j)
print(c(ans$g[K+j],(ansup$f-ansdn$f)/(2*del) ))
}
}
testhessian = function(){
# this function tests whether the Hessian
# is really the derivative of the gradient
# the code passed, but save this function for
# rechecking if necessary.
# call was test0(J=4,K=3)
#
alpha = .5*rep(1,J)/J # start at subdistribution
beta = rep(0,K)
ans = fgH(alpha,beta,rho,do=3)
del = 0.0001
print("testing Hessian wrt beta")
for( k in 1:K ){
bup = beta
bup[k] = beta[k]+del
ansup = fgH(alpha,bup,rho,do=2)
bdn = beta
bdn[k] = beta[k]-del
ansdn = fgH(alpha,bdn,rho,do=2)
print(k)
print(ans$H[,k])
print((ansup$g-ansdn$g)/(2*del))
}
print("testing Hessian wrt alpha")
for( j in 1:J ){
aup = alpha
aup[j] = alpha[j]+del
ansup = fgH(aup,beta,rho,do=2)
adn = alpha
adn[j] = alpha[j]-del
ansdn = fgH(adn,beta,rho,do=2)
print(j)
print(ans$H[,K+j])
print((ansup$g-ansdn$g)/(2*del))
}
}
#testgradient() # worked
#testhessian() # worked
linesearch = function(alpha,beta,rho,direction,oldf,oldg,LSalpha=.15,LSbeta=0.45){
# backtracking line search, Boyd and Vandenberghe p 464, Alg 9.2
# line search parameters:
# LSbeta typically between .1 (crude search) and .8
# LSalpha typically between .01 and 0.30
#print("in linesearch")
#print(oldf)
#print(oldg)
#print(direction)
tval = 1
while(1){
newbeta = beta + tval * direction[1:K]
newalpha = alpha + tval * direction[K+(1:J)]
newfg = fgH(newalpha,newbeta,rho,do=1)
# print("newfg");print(newfg)
if( fgH(newalpha,newbeta,rho,do=0) &&
(newfg$f <= oldf + LSalpha * tval * direction%*%oldg) )
break
tval = tval * LSbeta
# print(tval)
# print("newalpha");print(newalpha)
# print("newbeta");print(newbeta)
}
tval
}
svdsolve = function(A,b,epsrel=1e-9,epsabs=1e-100){
# Solve Av = b via SVD
A.svd <- svd(A)
d <- A.svd$d
index1 <- which(d>=(epsrel*d[1] +epsabs))
index2 <- which(d<(epsrel*d[1] +epsabs))
d.trun.inv <- d
d.trun.inv[index1] <- 1/d[index1]
d.trun.inv[index2] <- 0
A.inv <- A.svd$v%*%diag(d.trun.inv)%*%t(A.svd$u)
return(A.inv%*%b)
}
preconditionsolve = function(A, b, reltol=.Machine$double.eps){
J <- (dim(A)[1] + 1)/2
p <- sqrt(median(abs(A[1:(J-1), 1:(J-1)]))/median(abs(A[J:(2*J-1), J:(2*J - 1)])))
P.vec <- c(rep(1, J - 1), rep(p, J))
P <- diag(P.vec)
A.pc <- P%*%A%*%P
if(kappa(A.pc) < kappa(A)){
x <- try(as.vector(P %*% svdsolve(A.pc, P%*%b)), silent = TRUE)
}else{
x <- try(as.vector(svdsolve(A, b)), silent = TRUE)
}
return(x)
}
dampednewton = function(alpha,beta,rho,reltol){
# From Boyd and Vandenberghe p 487, Alg 9.5
done = FALSE
inct <- 0
oldf <- -Inf
while( 1 ){
vals = fgH(alpha,beta,rho,do=3)
if(identical(oldf,vals$f)){
solstatus <- "exact"
break
}
oldf <- vals$f
# print(vals)
## print(svd(vals$H)$d)
# newtonstep = -solve(vals$H,vals$g,reltol=1e-50) # aggressively small reltol here
## print("singular values of vals$H")
## print(svd(vals$H)$d)
## file <- "../Routput/singularvalH"
## sink(file)
## print("singular values of vals$H")
## print(svd(vals$H)$d)
## sink()
newtonstep = try(-preconditionsolve(vals$H,vals$g), silent = TRUE)
# print("newtonstep"); print(newtonstep)
if(class(newtonstep) == "try-error"){
if(isTRUE(all.equal(vals$g, rep(0, length(vals$g))))){
solstatus <- "exact"
break
}else{
## save(alpha, beta, vals, file = "exdata/errorvals.RData")
print("returning inexact solution")
solstatus <- "inexact"
break
## H <- vals$H
## save(x, y, z, H, alpha, beta, rho, eps,file = "../errorworkspace/largeconditionerrorH.RData")
## print("largeconditionerrorH.RData saved")
## stop("condition number still too large after preconditioning")
}
}
decrement = - (vals$g %*% newtonstep)[1,1]/2
## print(paste("vals$f", vals$f))
## print(paste("vals$f0", vals$f0))
## print(paste("decrement", decrement))
reldecrement = decrement/vals$f
if( reldecrement <= reltol ){
solstatus <- "exact"
break
}
tval = linesearch(alpha,beta,rho,newtonstep,vals$f,vals$g)
# print("tval from linesearch");print(tval)
beta = beta + tval * newtonstep[1:K]
alpha = alpha + tval * newtonstep[K+(1:J)]
# print("alpha");print(alpha)
# print("beta");print(beta)
inct = inct + 1
if(inct >= maxin){
print("Reaching maximum inner iterations")
solstatus <- "semiexact"
print(paste("decrement", decrement))
print(paste("vals$f", vals$f))
break
}
}
## print(paste("inct", inct))
## print(paste("sum(alpha)", sum(alpha)))
list(alpha=alpha,beta=beta, solstatus = solstatus, inct = inct)
}
if(is.null(a0)|missing(b0)){
alpha = eps + delta
}else
alpha = a0
if(is.null(b0)|missing(b0)){
beta = rep(0,K)
}else
beta = b0
# Use Boyd and Vandenberghe p 569, Alg 11.1, Barrier method
# with comments on choice of parameters
## print("starting alpha")
## print(alpha)
initfgH <- fgH(alpha, beta, rho=1, do=2)
## rho0 <- lm(initfgH$g0 ~ initfgH$gphi - 1)$coef[[1]] # initial value of penalty factor (their 1/t)
rho <- rho0
mu = 10 # penalty increase factor
outer.ct <- 0
inct.vec <- c()
while(1){
thedn = dampednewton(alpha,beta,rho,reltol=reltol)
inct.vec <- c(inct.vec, thedn$inct)
if(thedn$solstatus == "inexact"){
warning("cannot achieve required relerr")
thevar = fgH(alpha,beta,rho=0,do=1)$f
dual.opt <- thevar - (J+1)*rho*mu
if(dual.opt > 0){
relerr <- (J+1)*rho*mu/dual.opt
print("relative error of the last available exact solution")
print(relerr)
}else{
abserr <- (J+1)*rho*mu
print("absolute error of the last available exact solution")
print(abserr)
print("thevar at last available exact solution, i.e. the value of the unpenalized objective function at the solution")
print(thevar)
}
break
}
alpha = thedn$alpha
beta = thedn$beta
thevar = fgH(alpha,beta,rho=0,do=1)$f
# print(thevar)
dual.opt <- thevar - (J+1)*rho
if( (J+1)*rho < dual.opt*relerr ){ #Boyd p 242 relerr
## print("relative error reached relerr")
relerr <- (J+1)*rho/thevar
break
}
f = fgH(alpha,beta,rho=0,do=1)$f
rhopen = fgH(alpha,beta,rho=rho,do=1)$f-f
rho = rho/mu
outer.ct <- outer.ct + 1
if(outer.ct >= maxout){
print("Reaching maximum outer iterations")
break
}
}
## print("alpha")
## print(alpha)
## print("sum(alpha)")
## print(sum(alpha))
## print("number of outer iterations")
## print(outer.ct)
## print("number of inner iterations")
## print(inct.vec)
## print("total inner iterations")
## print(inner)
f = fgH(alpha,beta,rho=0,do=1)$f
rhopen = fgH(alpha,beta,rho=rho,do=1)$f-f
## print(paste("outer loops", log(rho0/rho,mu)))
list(x=x, y=y, z=z, alpha=alpha,beta=beta,rho=rho,f=f,rhopen=rhopen, outer.count =outer.ct, inner.counts = inct.vec, relerr = relerr, alphasum = sum(alpha))
#list(allalpha=allalpha,allbeta=allbeta,fgH=fgH(alpha,beta,rho,do=3))
}
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