# R/opt.R In optismixture: Optimal Mixture Weights in Multiple Importance Sampling

#### Documented in penoptpersp

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) ))
}

for( j in 1:J ){
aup    = alpha
aup[j] = alpha[j]+del
ansup  = fgH(aup,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)
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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optismixture documentation built on May 29, 2017, 1:02 p.m.