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[Help-gsl] On conjugate gradient algorithms in multidimentional minimisa

From: Max Belushkin
Subject: [Help-gsl] On conjugate gradient algorithms in multidimentional minimisation problems.
Date: Tue, 29 Nov 2005 20:50:45 +0100
User-agent: Thunderbird 1.5 (X11/20051025)

    Hello list,

I apologize in advance for this rather long email, it is a collective description of the common problem I have failed to gain an understanding of, having tried for many months now.

Over the last couple of years, I have made some observations on conjugate gradient algorithms that I would like to seek some advice about. This experience is related to gsl-1.6 and gsl-1.7, as of late.

-- Part 1 --

The typical problem at hand is a parameter space of 20 dimensions or more. That is, a chi-squared function is built, which is to be minimized, that depends on 20 or more independent parameters passed to the minimization routine.

The most common problem is the minimizer eventually bailing out with an error of "iterations not progressing toward a solution" (the exact message may be a bit different, I do not remember off-hand). Thus, I took up to restarting the minimizer with gsl_multimin_fdfminimizer_restart. However, it still does not help. Here is what happens. If we take a toy sample from the examples, and modify it a little, we get:
do {
  status = gsl_multimin_fdfminimizer_iterate(s);
if (status) {printf ("Restarting minimizer point 1\n"); gsl_fdfminimizer_restart(s); goto tryagain;};
  status = gsl_multimin_test_gradient (s->gradient, 1e-3);
  if (status == GSL_SUCCESS) { printf ("converged to minimum\n"); break;};
if (status!=GSL_CONTINUE) { gsl_fdfminimizer_restart(s); printf ("Restarted minimizer point 2\n"); status = GSL_CONTINUE;};
} while (status == GSL_CONTINUE && iter < 10000);

what happens is that we will always sit on "Restarting minimizer point 1", and status will always be "iterations not progressing toward a solution". Removing the goto has the same effect, only I [without any proof, just a feeling] believe it should also ruin the calls to test_gradient in some way.

Now, if we remove the restart at point 1 and break there, then take the parameter vector we ended up with, plug it in, and start the program all over again, it will run for another 5000 iterations or so, then bail out with the same error at the same point. Repeat. And repeat...

At this point, I was somehow thinking it may be an issue with my parameter space. However, recently, my colleague, who has some code in Fortran, asked me to come up with a C library employing GSL routines that would minimize his problem, which is completely different from mine, but the number of parameters is also large, around 15. So I came up with a module that could be linked with a Fortran program, and the exact same phenomenon appears there.

I must admit I am no huge expert in the mathematics underlying the methods (Fletcher-Reeves, Polak-Ribiere and Broyden-Fletcher-Goldfarb-Shanno give the exact same results, up to an arbitrary constant in terms of the number of iterations), but I do know they have to be restarted from time to time. I did things like "if (iter % 50) gsl_fdfminimizer_restart(s)". No effect. So why does restarting the iteration all over (restarting the program) from that point works, and sometimes reduces the chi squared quite significantly over several such runs? And is there any way to force the minimizer to keep going, or have any more control over the convergence process?

-- Part 2 --

This is somewhat related to Part 1. If I look at the convergence by the value chi squared takes per iteration, a typical [example] picture looks like this:
50->30->20.1->20.0->19.99->19.981->19.9809->....<keep basically idle
for some 100s of iterations>->19.97->19.5->19.0->18.3->
17.8->17.79->....<hang here for another century>...->...

The iteration process shows huge plateaus. What can this be an indication of? This is also very common and is seen on a wide range of problems, thus I am assuming this is a well known fact. Perhaps someone has a solution of speeding up the iteration and not sitting on those plateaus where nothing seems to be happening much?

I apologize once again for the long email, and would be most grateful for any help on this subject.


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