[R] Not nice behaviour of nlminb (windows 32 bit, version 2.11.1)
Ravi Varadhan
rvaradhan at jhmi.edu
Sat Jul 10 05:29:20 CEST 2010
Hi,
The absolute function stopping criterion is not meant for any positive objective function. It is meant for functions whose minimum is 0. Here is what David Gay's documentation from PORT says:
"6 - absolute function convergence: |f (x)| < V(AFCTOL) = V(31). This test is only of interest in
problems where f (x) = 0 means a ‘‘perfect fit’’, such as nonlinear least-squares problems."
For example, let us try a positive objective function:
> nlminb( obj = function(x) x^2 + 1, start=1, lower=-Inf, upper=Inf, control=list(trace=TRUE))
0: 2.0000000: 1.00000
1: 1.0000000: 0.00000
2: 1.0000000: 0.00000
$par
[1] 0
$objective
[1] 1
$convergence
[1] 0
$message
[1] "relative convergence (4)"
$iterations
[1] 2
$evaluations
function gradient
3 2
Here the absolute function criterion does not kicks in.
Now let us try a function whose minimum value is 0.
> nlminb( obj = function(x) x^2, start=6, grad=function(x) 2*x, lower=-Inf, upper=Inf, control=list(trace=TRUE) )
0: 36.000000: 6.00000
1: 4.0000000: 2.00000
2: 4.9303807e-32: 2.22045e-16
$par
[1] 2.220446e-16
$objective
[1] 4.930381e-32
$convergence
[1] 0
$message
[1] "absolute function convergence (6)"
$iterations
[1] 2
$evaluations
function gradient
4 3
We see that convergence is attained and that the stoppage is due to absolute function criterion.
Suppose, we now set abs.tol=0:
> nlminb( obj = function(x) x^2, start=6, grad=function(x) 2*x, lower=-Inf, upper=Inf, control=list(trace=TRUE, abs.tol=0) )
0: 36.000000: 6.00000
1: 4.0000000: 2.00000
2: 4.9303807e-32: 2.22045e-16
3: 2.4308653e-63: -4.93038e-32
4: 2.9962729e-95: -5.47382e-48
5:1.4772766e-126: 1.21543e-63
6:1.8208840e-158: 1.34940e-79
7:8.9776511e-190: -2.99627e-95
8:1.1065809e-221: -3.32653e-111
9:5.4558652e-253: 7.38638e-127
10:6.7248731e-285: 8.20053e-143
11:3.3156184e-316: -1.82088e-158
12: 0.0000000: -2.02159e-174
13: 0.0000000: -2.02159e-174
$par
[1] -2.021587e-174
$objective
[1] 0
$convergence
[1] 0
$message
[1] "X-convergence (3)"
$iterations
[1] 13
$evaluations
function gradient
15 13
>
Now, we see that it takes a while to stop, eventhough it is clear that convergence has been attained after 2 iterations. This demonstrates the need for the absolute function criterion for obj functions whose minimum is exactly 0. Although, there is nothing wrong with setting abs.tol=0, except for some loss of computational efficiency.
Now, let us get back to Matthew' example:
> nlminb( obj = function(x) x, start=1, lower=-2, upper=2, control=list(trace=TRUE))
0: 1.0000000: 1.00000
1: 0.0000000: 0.00000
$par
[1] 0
$objective
[1] 0
$convergence
[1] 0
$message
[1] "absolute function convergence (6)"
$iterations
[1] 1
$evaluations
function gradient
2 2
> nlminb( obj = function(x) x, start=1, lower=-2, upper=2, control=list(trace=TRUE, abs.tol=0))
0: 1.0000000: 1.00000
1: 0.0000000: 0.00000
2: -2.0000000: -2.00000
3: -2.0000000: -2.00000
$par
[1] -2
$objective
[1] -2
$convergence
[1] 0
$message
[1] "both X-convergence and relative convergence (5)"
$iterations
[1] 3
$evaluations
function gradient
3 3
Thus it is evident that setting abs.tol=0 is a reasonable, general solution for functions whose minimum value is non-zero, because it protects against premature termination at iteration `n' whenever |f(x_n)| < abs.tol. The only limitation being that of loss of efficiency in problems where f(x*) = 0. where x* is the local minimum.
Ravi.
____________________________________________________________________
Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology
School of Medicine
Johns Hopkins University
Ph. (410) 502-2619
email: rvaradhan at jhmi.edu
----- Original Message -----
From: Duncan Murdoch <murdoch.duncan at gmail.com>
Date: Friday, July 9, 2010 6:54 pm
Subject: Re: [R] Not nice behaviour of nlminb (windows 32 bit, version 2.11.1)
To: Matthew Killeya <matthewkilleya at googlemail.com>
Cc: Peter Ehlers <ehlers at ucalgary.ca>, Ravi Varadhan <rvaradhan at jhmi.edu>, r-help at r-project.org, bates at stat.wisc.edu
> On 09/07/2010 6:09 PM, Matthew Killeya wrote:
> >Yes clearly a bug... there are numerous variations ... problem seems
> to be
> >for a linear function whenever the first function valuation is 1.
> >
>
> Not at all. You can get the same problem on a quadratic that happens
> to have a zero at an inconvenient place, e.g.
>
> nlminb( obj = function(x) x^2-25, start=6, lower=-Inf, upper=Inf )
>
> Ravi's workaround of setting the abs.tol to zero fixes this example,
> but I think it's pretty clear from the documentation that the whole
> thing was designed for positive objective functions, so I wouldn't
> count on his workaround solving all the problems.
>
> Duncan Murdoch
>
>
> >e.g. two more examples:
> > nlminb( obj = function(x) x+1, start=0, lower=-Inf, upper=Inf )
> > nlminb( obj = function(x) x+2, start=-1, lower=-Inf, upper=Inf )
> >
> >(I wasn't sure where best to report a bug, so emailed the help list)
> >
> >On 9 July 2010 22:10, Peter Ehlers <ehlers at ucalgary.ca> wrote:
> >
> >
> >>Actually, it looks like any value other than 1.0
> >>(and in (lower, upper)) for start will work.
> >>
> >> -Peter Ehlers
> >>
> >>
> >>On 2010-07-09 14:45, Ravi Varadhan wrote:
> >>
> >>
> >>>Setting abs.tol = 0 works! This turns-off the absolute function
> >>>convergence
> >>>criterion.
> >>>
> >>>
> >>> nlminb( objective=function(x) x, start=1, lower=-2, upper=2,
> >>> control=list(abs.tol=0))
> >>>$par
> >>>[1] -2
> >>>
> >>>$objective
> >>>[1] -2
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "both X-convergence and relative convergence (5)"
> >>>
> >>>$iterations
> >>>[1] 3
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 3 3
> >>>
> >>>
> >>>This is clearly a bug.
> >>>
> >>>
> >>>Ravi.
> >>>
> >>>-----Original Message-----
> >>>From: r-help-bounces at r-project.org [
> >>>On
> >>>Behalf Of Ravi Varadhan
> >>>Sent: Friday, July 09, 2010 4:42 PM
> >>>To: 'Duncan Murdoch'; 'Matthew Killeya'
> >>>Cc: r-help at r-project.org; bates at stat.wisc.edu
> >>>Subject: Re: [R] Not nice behaviour of nlminb (windows 32 bit, version
> >>>2.11.1)
> >>>
> >>>Duncan, `nlminb' is not intended for non-negative functions only.
> There
> >>>is
> >>>indeed something strange happening in the algorithm!
> >>>
> >>>start<- 1.0 # converges to wrong minimum
> >>>
> >>>startp<- 1.0 + .Machine$double.eps # correct
> >>>
> >>>startm<- 1.0 - .Machine$double.eps # correct
> >>>
> >>> nlminb( objective=obj, start=start, lower=-2, upper=2)
> >>> $par
> >>>[1] 0
> >>>
> >>>$objective
> >>>[1] 0
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "absolute function convergence (6)"
> >>>
> >>>$iterations
> >>>[1] 1
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 2 2
> >>>
> >>>
> >>>
> >>>>nlminb( objective=obj, start=startp, lower=-2, upper=2)
> >>>>
> >>>>
> >>>$par
> >>>[1] -2
> >>>
> >>>$objective
> >>>[1] -2
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "both X-convergence and relative convergence (5)"
> >>>
> >>>$iterations
> >>>[1] 3
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 3 3
> >>>
> >>>
> >>>
> >>>>nlminb( objective=obj, start=startm, lower=-2, upper=2)
> >>>>
> >>>>
> >>>$par
> >>>[1] -2
> >>>
> >>>$objective
> >>>[1] -2
> >>>
> >>>$convergence
> >>>[1] 0
> >>>
> >>>$message
> >>>[1] "both X-convergence and relative convergence (5)"
> >>>
> >>>$iterations
> >>>[1] 3
> >>>
> >>>$evaluations
> >>>function gradient
> >>> 3 3
> >>>
> >>>
> >>> From the convergence message the `absolute function convergence'
> seems to
> >>> be
> >>>the culprit, although I do not understand why that stopping
> criterion is
> >>>becoming effective, when the algorithm is started at x=1, but not
> at any
> >>>other values. The documentation in IPORT makes it clear that this
> >>>criterion
> >>>is effective only for functions where f(x*) = 0, where x* is a local
> >>>minimum. In this example, x=0 is not a local minimum for f(x), so
> that
> >>>criterion should not apply.
> >>>
> >>>
> >>>Ravi.
> >>>
> >>>
> >>>-----Original Message-----
> >>>From: r-help-bounces at r-project.org [
> >>>On
> >>>Behalf Of Duncan Murdoch
> >>>Sent: Friday, July 09, 2010 3:45 PM
> >>>To: Matthew Killeya
> >>>Cc: r-help at r-project.org; bates at stat.wisc.edu
> >>>Subject: Re: [R] Not nice behaviour of nlminb (windows 32 bit, version
> >>>2.11.1)
> >>>
> >>>On 09/07/2010 10:37 AM, Matthew Killeya wrote:
> >>>
> >>>
> >>>> nlminb( obj = function(x) x, start=1, lower=-Inf, upper=Inf )
> >>>>
> >>>>
> >>>>
> >>>If you read the PORT documentation carefully, you'll see that their
> >>>convergence criteria are aimed at minimizing positive functions.
> (They
> >>>never state this explicitly, as far as I can see.) So one stopping
> >>>criterion is that |f(x)|< abs.tol, and that's what it found for
> you. I
> >>>don't know if there's a way to turn this off.
> >>>
> >>>Doug or Deepayan, do you know if nlminb can be made to work on functions
> >>>that go negative?
> >>>
> >>>Duncan Murdoch
> >>>
> >>> $par
> >>>
> >>>>[1] 0
> >>>>
> >>>>$objective
> >>>>[1] 0
> >>>>
> >>>>$convergence
> >>>>[1] 0
> >>>>
> >>>>$message
> >>>>[1] "absolute function convergence (6)"
> >>>>
> >>>>$iterations
> >>>>[1] 1
> >>>>
> >>>>$evaluations
> >>>>function gradient
> >>>> 2 2
> >>>>
> >>>> [[alternative HTML version deleted]]
> >>>>
> >>>>
> >>>>
> >
> >
>
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