Ravi Varadhan wrote:
>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."
> Okay, I've taken a more careful look at the docs, and they do say
that the return code 6 does not necessarily indicate convergence: "The
desirable return codes are 3, 4, 5, and sometimes 6". So we shouldn't by
default terminate on it, we should allow users to choose that if they want
faster convergence to perfect fits.
Would changing the default for abs.tol to zero be a reasonable solution?
Duncan Murdoch
>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: rvarad...@jhmi.edu
>
>
>----- Original Message -----
>From: Duncan Murdoch <murdoch.dun...@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 <matthewkill...@googlemail.com>
>Cc: Peter Ehlers <ehl...@ucalgary.ca>, Ravi Varadhan <
rvarad...@jhmi.edu>, r-help@r-project.org, ba...@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 <ehl...@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-boun...@r-project.org [
>> >>>On
>> >>>Behalf Of Ravi Varadhan
>> >>>Sent: Friday, July 09, 2010 4:42 PM
>> >>>To: 'Duncan Murdoch'; 'Matthew Killeya'
>> >>>Cc: r-help@r-project.org; ba...@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-boun...@r-project.org [
>> >>>On
>> >>>Behalf Of Duncan Murdoch
>> >>>Sent: Friday, July 09, 2010 3:45 PM
>> >>>To: Matthew Killeya
>> >>>Cc: r-help@r-project.org; ba...@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]]
>> >>>>
>> >>>>
>> >>>> >
>> >
