SNewton: safeguarded Newton methods for function minimization

In optimx: Expanded Replacement and Extension of the 'optim' Function

Safeguarded Newton algorithms

So-called Newton methods are among the most commonly mentioned in the solution of nonlinear equations or function minimization. However, as discussed in

https://en.wikipedia.org/wiki/Newton%27s_method#History,

the Newton or Newton-Raphson method as we know it today was not what either of its supposed originators knew.

This vignette discusses the development of simple safeguarded variants of the Newton method for function minimization in R. Note that there are some resources in R for solving nonlinear equations by Newton-like methods in the packages nleqslv and pracma.

The basic approach

If we have a function $f(x)$, with gradient $g(x)$ and second derivative (Hessian) $H(x)$ the first order condition for an extremum (min or max) is

$$g(x) = 0$$

To ensure a minimum, we want

$$ H(x) > 0 $$

The first order condition leads to a root-finding problem.

It turns out that $x$ need not be a scalar. We can consider it to be a vector of parameters to be determined. This renders $g(x)$ a vector also, and $H(x)$ a matrix. The conditions of optimality then require a zero gradient and positive-definite Hessian.

The Newton approach to such equations is to provide a guess to the root $x_try$ and to then solve the equation

$$ H(x_t) * s = - g(x_t)$$

for the search vector $s$. We update $x_t$ to $x_t + s$ and repeat until we have a very small gradient $g(x_t)$. If $H(x)$ is positive definite, we have a reasonable approximation to a (local) minimum.

Motivations

A particular interest in Newton-like methods its theoretical quadratic convergence. See https://en.wikipedia.org/wiki/Newton%27s_method. That is, the method will converge in one step for a quadratic function $f(x)$, and for "reasonable" functions will converge very rapidly. There are, however, a number of conditions, and practical programs need to include safequards against mis-steps in the iterations.

The principal issues concern the possiblity that $H(x)$ may not be positive definite, at least in some parts of the domain, and that the curvature may be such that a unit step $x_t + s$ does not reduce the function $f$. We therefore get a number of possible variants of the method when different possible safeguards are applied.

Algorithm possibilities

There are many choices we can make in building a practical code to implement the ideas above. In tandem with the two main issues expressed above, we will consider

• the modification of the solution of the main equation $$ H(x_t) * s = - g(x_t)$$ so that a reasonable search vector $s$ is always generated by avoiding Hessian matrices that are not positive definite.

• the selection of a new set of parameters $x_new = x_t + step * s$ so that the function value $f(x_new)$ is less than $f(x_t)$.

The second choice above could be made slightly more stringent so that the Armijo condition of sufficient-decrease is met. Adding a curvature requirement gives the Wolfe condisions. See https://en.wikipedia.org/wiki/Wolfe_conditions. The Armijo requirement is generally written

$$f(x_t + steps) < f(x_t) + c * step * g(x_t)^Ts$$

where c is some number less than 1. Typically $c = 1e-4 = 0.0001$. Note that the product of gradient times search vector is negative for any reasonable situation, since we are trying to go "downhill".

As a result of the ideas in this section, the code snewton() uses a solution of the Newton equations with the Hessian provided (if this is possible, else we stop), along with a backtracking line search. The code snewtonm uses a Marquardt stabilization of the Hessian to create

$$ Haug = H + 1_n * lambda$$

That is, we add $lambda$ times the unit matrix to $H$. Then we try the set of parameters found by adding the solution of the Newton equations with $Haug$ in place of $H$ to the current "best" set of parameters. If this new set of parameters has a higher function value than the "best" so far, we increase $lambda$ and try again. Note that we do not need to re-evaluate the gradient or Hessian to do this. Moreover, for some value of $lambda$, the step is clearly down the gradient (i.e., steepest descents) or we have converged and no progress is possible. This leads to a very compact and elegant code, which we name snewtonm() for Safeguarded Newton-Marquardt method. It is reliable, but may be less efficient than using the un-modified Hessian.

A choice to compute the search vector

The primary concern in solving for $s$ is that the Hessian may not be positive definite. This means that we cannot apply fast and stable methods like the Cholesky decomposition to the matrix. At the time of writing, we use the following approach:

• We attempt to solve

$$ H(x_t) * s = - g(x_t)$$ with R directly, and rely on internal checks to catch any cases where the solution fails. We then use try() to stop the program in this case.

Choosing the step size

The traditional Newton approach is that the stepsize is taken to be 1. In practice, this can sometimes mean that the function value is not reduced. As an alternative, we can use a simple backtrack search. We start with $step = 1$ (actually the program allows for the element defstep of the control list to be set to a value other than 1). If the Armijo condition is not met, we replace $step$ with $ r * step $ where $r$ is less than 1. Here we suggest control$stepdec = 0.2. We repeat until $x_t$ satisfies the Armijo condition or $x_t$ is essentially unchanged by the step.

Here "essentially unchanged" is determined by a test using an offset value, that is, the test

$$ (x_t + offset) == (x_t + step * d + offset) $$

where $d$ is the search direction. control$offset = 100 is used. We could also, and almost equivalently, use the R identical function.

This approach has been coded into the snewton() function.

Examples

These examples are coded as a test to the interim package snewton, but as at 2018-7-10 are part of the optimx package. We call these below mostly via the optimr() function to allow compact output to be used, but please note that some count information on the number of hessian evaluations and "iterations" (which generally is an algorithm-specific measure) is not then returned.

A simple example

The following example is trivial, in that the Hessian is a constant matrix, and we achieve convergence immediately.

x0<-c(1,2,3,4)
fnt <- function(x, fscale=10){
  yy <- length(x):1
  val <- sum((yy*x)^2)*fscale
}
grt <- function(x, fscale=10){
  nn <- length(x)
  yy <- nn:1
  #    gg <- rep(NA,nn)
  gg <- 2*(yy^2)*x*fscale
  gg
}

hesst <- function(x, fscale=10){
  nn <- length(x)
  yy <- nn:1
  hh <- diag(2*yy^2*fscale)
  hh
}

require(optimx)
t1 <- snewton(x0, fnt, grt, hesst, control=list(trace=0), fscale=3.0)
print(t1)
# we can also use nlm and nlminb
fght <- function(x, fscale=10){
  ## combine f, g and h into single function for nlm
     ff <- fnt(x, fscale)
     gg <- grt(x, fscale)
     hh <- hesst(x, fscale)
     attr(ff, "gradient") <- gg
     attr(ff, "hessian") <- hh
     ff
}

t1nlm <- nlm(fght, x0, fscale=3.0, hessian=TRUE, print.level=0)
print(t1nlm)

## BUT ... it looks like nlminb is NOT using a true Newton-type method
t1nlminb <- nlminb(x0, fnt, gradient=grt, hessian=hesst, fscale=3.0, 
                   control=list(trace=0))
print(t1nlminb)

# and call them from optimx (i.e., test this gives same results)
t1so <-  optimr(x0, fnt, grt, hess=hesst, method="snewton", fscale=3.0, 
                 control=list(trace=0))
proptimr(t1so)

t1nlmo <- optimr(x0, fnt, grt, hess=hesst, method="nlm", fscale=3.0, 
                 control=list(trace=0))
proptimr(t1nlmo)

tst <- try(t1nlminbo <- optimr(x0, fnt, grt, hess=hesst, method="nlminb", 
                               fscale=3.0, control=list(trace=0)))
if (class(tst) == "try-error"){
    cat("try-error on attempt to run nlminb in optimr()\n")
} else { proptimr(t1nlminbo) }

From the number of function and gradient evaluations, it appears nlminb() is not using the Hessian information. Note that the snewton() and snewtonm() functions return count information for iterations and hessian evaluations, but these are not part of the standard optim() (and thus optimr()) result objects.

The Rosenbrock function

require(optimx)
#Rosenbrock banana valley function
f <- function(x){
return(100*(x[2] - x[1]*x[1])^2 + (1-x[1])^2)
}
#gradient
gr <- function(x){
return(c(-400*x[1]*(x[2] - x[1]*x[1]) - 2*(1-x[1]), 200*(x[2] - x[1]*x[1])))
}
#Hessian
h <- function(x) {
a11 <- 2 - 400*x[2] + 1200*x[1]*x[1]; a21 <- -400*x[1]
return(matrix(c(a11, a21, a21, 200), 2, 2))
}
x0 <- c(-1.2, 1)
# sink("mbrn1-170408.txt", split=TRUE)
t1 <- snewton(x0, fn=f, gr=gr, hess=h, control=list(trace=0))
print(t1)

# we can also use nlm and nlminb
fght <- function(x){
  ## combine f, g and h into single function for nlm
     ff <- f(x)
     gg <- gr(x)
     hh <- h(x)
     attr(ff, "gradient") <- gg
     attr(ff, "hessian") <- hh
     ff
}

# COULD TRY: t1nlm <- nlm(fght, x0, hessian=TRUE, print.level=2, iterlim=10000)
t1nlmo <- optimr(x0, f, gr, hess=h, method="nlm", control=list(trace=0))
proptimr(t1nlmo)

t1so <- optimr(x0, f, gr, hess=h, method="snewton", control=list(trace=0))
proptimr(t1so)

t1smo <-  optimr(x0, f, gr, hess=h, method="snewtonm", control=list(trace=0))
proptimr(t1smo)


## Again, nlminb probably not using hessian
tst <- try(t1nlminbo <- optimr(x0, f, gr, hess=h, method="nlminb", 
                               control=list(trace=0)))
if (class(tst) == "try-error"){
    cat("try-error on attempt to run nlminb in optimr()\n")
} else { proptimr(t1nlminbo) }

The Wood function

For nlm() the "standard" start takes more than 100 iterations and returns a non-optimal solution.

#Example: Wood function
#
wood.f <- function(x){
  res <- 100*(x[1]^2-x[2])^2+(1-x[1])^2+90*(x[3]^2-x[4])^2+(1-x[3])^2+
    10.1*((1-x[2])^2+(1-x[4])^2)+19.8*(1-x[2])*(1-x[4])
  return(res)
}
#gradient:
wood.g <- function(x){
  g1 <- 400*x[1]^3-400*x[1]*x[2]+2*x[1]-2
  g2 <- -200*x[1]^2+220.2*x[2]+19.8*x[4]-40
  g3 <- 360*x[3]^3-360*x[3]*x[4]+2*x[3]-2
  g4 <- -180*x[3]^2+200.2*x[4]+19.8*x[2]-40
  return(c(g1,g2,g3,g4))
}
#hessian:
wood.h <- function(x){
  h11 <- 1200*x[1]^2-400*x[2]+2;    h12 <- -400*x[1]; h13 <- h14 <- 0
  h22 <- 220.2; h23 <- 0;    h24 <- 19.8
  h33 <- 1080*x[3]^2-360*x[4]+2;    h34 <- -360*x[3]
  h44 <- 200.2
  H <- matrix(c(h11,h12,h13,h14,h12,h22,h23,h24,
                h13,h23,h33,h34,h14,h24,h34,h44),ncol=4)
  return(H)
}

wood.fgh <- function(x){
      fval <- wood.f(x)
      gval <- wood.g(x)
      hval <- wood.h(x)
      attr(fval,"gradient") <- gval
      attr(fval,"hessian")<- hval
      fval
}

#################################################
x0 <- c(-3,-1,-3,-1) # Wood standard start

require(optimx)
# In 100 iterations, not converged
t1nlm <- nlm(wood.fgh, x0, print.level=0)
print(t1nlm)
# But both newton approaches do work
wd <- snewton(x0, fn=wood.f, gr=wood.g, hess=wood.h, control=list(trace=0))
print(wd)
wdm <- snewtonm(x0, fn=wood.f, gr=wood.g, hess=wood.h, control=list(trace=0))
print(wdm)

## AND again nlminb not likely using hessian information
## t1nlminb <- nlminb(x0, wood.f, gradient=wood.g, hess=wood.h, control=list(trace=0))
## print(t1nlminb)
# and call them from optimx (i.e., test this gives same results)

# But optimr uses a larger iteration limit, and gets to solution
t1nlmo <- optimr(x0, wood.f, wood.g, hess=wood.h, method="nlm", control=list(trace=0))
proptimr(t1nlmo)

tst<-try(t1nlminbo <- optimr(x0, wood.f, wood.g, hess=wood.h, method="nlminb", control=list(trace=0)))
if (class(tst) == "try-error"){
    cat("try-error on attempt to run nlminb in optimr()\n")
} else { proptimr(t1nlminbo) }

A generalized Rosenbrock function

There are several generalizations of the Rosenbrock function (??ref)

# genrosa function code -- attempts to match the rosenbrock at gs=100 and x=c(-1.2,1)
genrosa.f<- function(x, gs=NULL){ # objective function
## One generalization of the Rosenbrock banana valley function (n parameters)
    n <- length(x)
        if(is.null(gs)) { gs=100.0 }
        # Note do not at 1.0 so min at 0
    fval<-sum (gs*(x[1:(n-1)]^2 - x[2:n])^2 + (x[1:(n-1)] - 1)^2)
}

genrosa.g <- function(x, gs=NULL){
# vectorized gradient for genrose.f
# Ravi Varadhan 2009-04-03
    n <- length(x)
        if(is.null(gs)) { gs=100.0 }
    gg <- as.vector(rep(0, n))
    tn <- 2:n
    tn1 <- tn - 1
    z1 <- x[tn] - x[tn1]^2
    z2 <- 1 - x[tn1]
        # f = gs*z1*z1 + z2*z2
    gg[tn] <- 2 * (gs * z1)
    gg[tn1] <- gg[tn1] - 4 * gs * x[tn1] * z1 - 2 *z2 
    return(gg)
}

genrosa.h <- function(x, gs=NULL) { ## compute Hessian
   if(is.null(gs)) { gs=100.0 }
    n <- length(x)
    hh<-matrix(rep(0, n*n),n,n)
    for (i in 2:n) {
        z1<-x[i]-x[i-1]*x[i-1]
#       z2<-1.0 - x[i-1]
                hh[i,i]<-hh[i,i]+2.0*(gs+1.0)
                hh[i-1,i-1]<-hh[i-1,i-1]-4.0*gs*z1-4.0*gs*x[i-1]*(-2.0*x[i-1])
                hh[i,i-1]<-hh[i,i-1]-4.0*gs*x[i-1]
                hh[i-1,i]<-hh[i-1,i]-4.0*gs*x[i-1]
    }
        return(hh)
}

require(optimx)
cat("Generalized Rosenbrock tests\n")

cat("original n and x0")

x0 <- c(-1.2, 1)
solorig <- snewton(x0, genrosa.f, genrosa.g, genrosa.h)
print(solorig)
print(eigen(solorig$Hess)$values)
solorigm <- snewtonm(x0, genrosa.f, genrosa.g, genrosa.h)
print(solorigm)
print(eigen(solorigm$Hess)$values)

cat("Start with 50 values of pi and scale factor 10\n")
x0 <- rep(pi, 50)
sol50pi <- optimr(x0, genrosa.f, genrosa.g, genrosa.h, method="snewton", gs=10)
proptimr(sol50pi)
hhi <- genrosa.h(sol50pi$par, gs=10)
print(eigen(hhi)$values)
sol50pim <- optimr(x0, genrosa.f, genrosa.g, genrosa.h, method="snewtonm", gs=10)
proptimr(sol50pim)
hhm <- genrosa.h(sol50pim$par, gs=10)
print(eigen(hhm)$values)

The Hobbs weed infestation problem

This problem is described in @cnm79. It has various nasty properties. Note that one starting point causes failure of the snewton() optimizer.

## Optimization test function HOBBS
## ?? refs (put in .doc??)
## Nash and Walker-Smith (1987, 1989) ...
require(optimx)

hobbs.f<- function(x){ # # Hobbs weeds problem -- function
    if (abs(12*x[3]) > 500) { # check computability
       fbad<-.Machine$double.xmax
       return(fbad)
    }
    res<-hobbs.res(x)
    f<-sum(res*res)
}


hobbs.res<-function(x){ # Hobbs weeds problem -- residual
# This variant uses looping
    if(length(x) != 3) stop("hobbs.res -- parameter vector n!=3")
    y<-c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443, 38.558, 50.156, 62.948,
         75.995, 91.972)
    t<-1:12
    if(abs(12*x[3])>50) {
       res<-rep(Inf,12)
    } else {
       res<-x[1]/(1+x[2]*exp(-x[3]*t)) - y
    }
}

hobbs.jac<-function(x){ # Jacobian of Hobbs weeds problem
   jj<-matrix(0.0, 12, 3)
   t<-1:12
    yy<-exp(-x[3]*t)
    zz<-1.0/(1+x[2]*yy)
     jj[t,1] <- zz
     jj[t,2] <- -x[1]*zz*zz*yy
     jj[t,3] <- x[1]*zz*zz*yy*x[2]*t
   return(jj)
}

hobbs.g<-function(x){ # gradient of Hobbs weeds problem
    # NOT EFFICIENT TO CALL AGAIN
    jj<-hobbs.jac(x)
    res<-hobbs.res(x)
    gg<-as.vector(2.*t(jj) %*% res)
    return(gg)
}


hobbs.rsd<-function(x) { # Jacobian second derivative
    rsd<-array(0.0, c(12,3,3))
    t<-1:12
    yy<-exp(-x[3]*t)
    zz<-1.0/(1+x[2]*yy)
    rsd[t,1,1]<- 0.0
    rsd[t,2,1]<- -yy*zz*zz
    rsd[t,1,2]<- -yy*zz*zz
    rsd[t,2,2]<- 2.0*x[1]*yy*yy*zz*zz*zz
    rsd[t,3,1]<- t*x[2]*yy*zz*zz
    rsd[t,1,3]<- t*x[2]*yy*zz*zz
    rsd[t,3,2]<- t*x[1]*yy*zz*zz*(1-2*x[2]*yy*zz)
    rsd[t,2,3]<- t*x[1]*yy*zz*zz*(1-2*x[2]*yy*zz)
##    rsd[t,3,3]<- 2*t*t*x[1]*x[2]*x[2]*yy*yy*zz*zz*zz
    rsd[t,3,3]<- -t*t*x[1]*x[2]*yy*zz*zz*(1-2*yy*zz*x[2])
    return(rsd)
}


hobbs.h <- function(x) { ## compute Hessian
#   cat("Hessian not yet available\n")
#   return(NULL)
    H<-matrix(0,3,3)
    res<-hobbs.res(x)
    jj<-hobbs.jac(x)
    rsd<-hobbs.rsd(x)
##    H<-2.0*(t(res) %*% rsd + t(jj) %*% jj)
    for (j in 1:3) {
       for (k in 1:3) {
          for (i in 1:12) {
             H[j,k]<-H[j,k]+res[i]*rsd[i,j,k]
          }
       }
    }
    H<-2*(H + t(jj) %*% jj)
    return(H)
}

require(optimx)
x0 <- c(200, 50, .3)
cat("Start for Hobbs:")
print(x0)
solx0 <- snewton(x0, hobbs.f, hobbs.g, hobbs.h)
## Note that we exceed count limit, but have answer
print(solx0)
print(eigen(solx0$Hess)$values)
## Note that we exceed count limit, but have answer

## Setting relative check offset larger gets quicker convergence
solx0a <- snewton(x0, hobbs.f, hobbs.g, hobbs.h, control=list(offset=1000.))
print(solx0a)


x1s <- c(100, 10, .1)
cat("Start for Hobbs:")
print(x1s)
solx1s <- snewton(x1s, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0))
print(solx1s)
print(eigen(solx1s$Hess)$values)
solx1m <- snewton(x1s, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0))
print(solx1m)
print(eigen(solx1m$Hess)$values)

cat("Following test fails in snewton with ERROR -- Why?\n")
x3 <- c(1, 1, 1)
cat("Start for Hobbs:")
print(x3)
ftest <- try(solx3 <- snewton(x3, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0)))
if (class(ftest) != "try-error") {
   print(solx3)
   print(eigen(solx3$Hess)$values)
}
cat("But Marquardt variant succeeds\n")
solx3m <- snewtonm(x3, hobbs.f, hobbs.g, hobbs.h, control=list(trace=0))
print(solx3m)
print(eigen(solx3m$Hess)$values)



# we can also use nlm and nlminb and call them from optimx

optimx documentation built on June 14, 2019, 3:01 p.m.