nlm function fails with analytic Hessian - r

Question

Some background: the nlm function in R is a general purpose optimization routine that uses Newton's method. To optimize a function, Newton's method requires the function, as well as the first and second derivatives of the function (the gradient vector and the Hessian matrix, respectively). In R the nlm function allows you to specify R functions that correspond to calculations of the gradient and Hessian, or one can leave these unspecified and numerical solutions are provided based on numerical derivatives (via the deriv function). More accurate solutions can be found by supplying functions to calculate the gradient and Hessian, so it's a useful feature.
My problem: the nlm function is slower and often fails to converge in a reasonable amount of time when the analytic Hessian is supplied. I'm guessing this is some sort of bug in the underlying code, but I'd be happy to be wrong. Is there a way to make nlm work better with an analytic Hessian matrix?
Example: my R code below demonstrates this problem using a logistic regression example, where
log(Pr(Y=1)/Pr(Y=0)) = b0 + Xb
where X is a multivariate normal of dimension N by p and b is a vector of coefficients of length p.
library(mvtnorm)
# example demonstrating a problem with NLM
expit <- function(mu) {1/(1+exp(-mu))}
mk.logit.data <- function(N,p){
set.seed(1232)
U = matrix(runif(p*p), nrow=p, ncol=p)
S = 0.5*(U+t(U)) + p*diag(rep(1,p))
X = rmvnorm(N, mean = runif(p, -1, 1), sigma = S)
Design = cbind(rep(1, N), X)
beta = sort(sample(c(rep(0,p), runif(1))))
y = rbinom(N, 1, expit(Design%*%beta))
list(X=X,y=as.numeric(y),N=N,p=p)
}
# function to calculate gradient vector at given coefficient values
logistic_gr <- function(beta, y, x, min=TRUE){
mu = beta[1] + x %*% beta[-1]
p = length(beta)
n = length(y)
D = cbind(rep(1,n), x)
gri = matrix(nrow=n, ncol=p)
for(j in 1:p){
gri[,j] = D[,j]*(exp(-mu)*y-1+y)/(1+exp(-mu))
}
gr = apply(gri, 2, sum)
if(min) gr = -gr
gr
}
# function to calculate Hessian matrix at given coefficient values
logistic_hess <- function(beta, y, x, min=TRUE){
# allow to fail with NA, NaN, Inf values
mu = beta[1] + x %*% beta[-1]
p = length(beta)
n = length(y)
D = cbind(rep(1,n), x)
h = matrix(nrow=p, ncol=p)
for(j in 1:p){
for(k in 1:p){
h[j,k] = -sum(D[,j]*D[,k]*(exp(-mu))/(1+exp(-mu))^2)
}
}
if(min) h = -h
h
}
# function to calculate likelihood (up to a constant) at given coefficient values
logistic_ll <- function(beta, y,x, gr=FALSE, he=FALSE, min=TRUE){
mu = beta[1] + x %*% beta[-1]
lli = log(expit(mu))*y + log(1-expit(mu))*(1-y)
ll = sum(lli)
if(is.na(ll) | is.infinite(ll)) ll = -1e16
if(min) ll=-ll
# the below specification is required for using analytic gradient/Hessian in nlm function
if(gr) attr(ll, "gradient") <- logistic_gr(beta, y=y, x=x, min=min)
if(he) attr(ll, "hessian") <- logistic_hess(beta, y=y, x=x, min=min)
ll
}
First example, with p=3:
dat = mk.logit.data(N=100, p=3)
The glm function estimates are for reference. nlm should give the same answer, allowing for small errors due to approximation.
(glm.sol <- glm(dat$y~dat$X, family=binomial()))$coefficients
> (Intercept) dat$X1 dat$X2 dat$X3
> 0.00981465 0.01068939 0.04417671 0.01625381
# works when correct analytic gradient is specified
(nlm.sol1 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, y=dat$y, x=dat$X))$estimate
> [1] 0.009814547 0.010689396 0.044176627 0.016253966
# works, but less accurate when correct analytic hessian is specified (even though the routine notes convergence is probable)
(nlm.sol2 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, he=TRUE, y=dat$y, x=dat$X, hessian = TRUE, check.analyticals=TRUE))$estimate
> [1] 0.009827701 0.010687278 0.044178416 0.016255630
But the problem becomes apparent when p is larger, here it is 10
dat = mk.logit.data(N=100, p=10)
Again, glm solution for reference. nlm should give the same answer, allowing for small errors due to approximation.
(glm.sol <- glm(dat$y~dat$X, family=binomial()))$coefficients
> (Intercept) dat$X1 dat$X2 dat$X3 dat$X4 dat$X5 dat$X6 dat$X7
> -0.07071882 -0.08670003 0.16436630 0.01130549 0.17302058 0.03821008 0.08836471 -0.16578959
> dat$X8 dat$X9 dat$X10
> -0.07515477 -0.08555075 0.29119963
# works when correct analytic gradient is specified
(nlm.sol1 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, y=dat$y, x=dat$X))$estimate
> [1] -0.07071879 -0.08670005 0.16436632 0.01130550 0.17302057 0.03821009 0.08836472
> [8] -0.16578958 -0.07515478 -0.08555076 0.29119967
# fails to converge in 5000 iterations when correct analytic hessian is specified
(nlm.sol2 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, he=TRUE, y=dat$y, x=dat$X, hessian = TRUE, iterlim=5000, check.analyticals=TRUE))$estimate
> [1] 0.31602065 -0.06185190 0.10775381 -0.16748897 0.05032156 0.34176104 0.02118631
> [8] -0.01833671 -0.20364929 0.63713991 0.18390489
Edit: I should also add that I have confirmed I have the correct Hessian matrix through multiple different approaches

Answer 1

I tried the code, but at first it seemed to be using a different rmvnorm than I can find on CRAN. I found one rmvnorm in dae package, then one in the mvtnorm package. The latter is the one to use.
nlm() was patched about the time of the above posting. I'm currently trying to verify the patches and it now seems to work OK. Note that I am author of a number of R's optimization codes, including 3/5 in optim().
nashjc at uottawa.ca
Code is below.

Answer 2

Revised code:
# example demonstrating a problem with NLM
expit <- function(mu) {1/(1+exp(-mu))}
mk.logit.data <- function(N,p){
set.seed(1232)
U = matrix(runif(p*p), nrow=p, ncol=p)
S = 0.5*(U+t(U)) + p*diag(rep(1,p))
X = rmvnorm(N, mean = runif(p, -1, 1), sigma = S)
Design = cbind(rep(1, N), X)
beta = sort(sample(c(rep(0,p), runif(1))))
y = rbinom(N, 1, expit(Design%*%beta))
list(X=X,y=as.numeric(y),N=N,p=p)
}
# function to calculate gradient vector at given coefficient values
logistic_gr <- function(beta, y, x, min=TRUE){
mu = beta[1] + x %*% beta[-1]
p = length(beta)
n = length(y)
D = cbind(rep(1,n), x)
gri = matrix(nrow=n, ncol=p)
for(j in 1:p){
gri[,j] = D[,j]*(exp(-mu)*y-1+y)/(1+exp(-mu))
}
gr = apply(gri, 2, sum)
if(min) gr = -gr
gr
}
# function to calculate Hessian matrix at given coefficient values
logistic_hess <- function(beta, y, x, min=TRUE){
# allow to fail with NA, NaN, Inf values
mu = beta[1] + x %*% beta[-1]
p = length(beta)
n = length(y)
D = cbind(rep(1,n), x)
h = matrix(nrow=p, ncol=p)
for(j in 1:p){
for(k in 1:p){
h[j,k] = -sum(D[,j]*D[,k]*(exp(-mu))/(1+exp(-mu))^2)
}
}
if(min) h = -h
h
}
# function to calculate likelihood (up to a constant) at given coefficient values
logistic_ll <- function(beta, y,x, gr=FALSE, he=FALSE, min=TRUE){
mu = beta[1] + x %*% beta[-1]
lli = log(expit(mu))*y + log(1-expit(mu))*(1-y)
ll = sum(lli)
if(is.na(ll) | is.infinite(ll)) ll = -1e16
if(min) ll=-ll
# the below specification is required for using analytic gradient/Hessian in nlm function
if(gr) attr(ll, "gradient") <- logistic_gr(beta, y=y, x=x, min=min)
if(he) attr(ll, "hessian") <- logistic_hess(beta, y=y, x=x, min=min)
ll
}
##!!!! NOTE: Must have this library loaded
library(mvtnorm)
dat = mk.logit.data(N=100, p=3)
(glm.sol <- glm(dat$y~dat$X, family=binomial()))$coefficients
# works when correct analytic gradient is specified
(nlm.sol1 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, y=dat$y, x=dat$X))$estimate
# works, but less accurate when correct analytic hessian is specified (even though the routine notes convergence is probable)
(nlm.sol2 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, he=TRUE, y=dat$y, x=dat$X, hessian = TRUE, check.analyticals=TRUE))$estimate
dat = mk.logit.data(N=100, p=10)
# Again, glm solution for reference. nlm should give the same answer, allowing for small errors due to approximation.
(glm.sol <- glm(dat$y~dat$X, family=binomial()))$coefficients
# works when correct analytic gradient is specified
(nlm.sol1 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, y=dat$y, x=dat$X))$estimate
# fails to converge in 5000 iterations when correct analytic hessian is specified
(nlm.sol2 <- nlm(p=runif(dat$p+1), f=logistic_ll, gr=TRUE, he=TRUE, y=dat$y, x=dat$X, hessian = TRUE, iterlim=5000, check.analyticals=TRUE))$estimate