[R] Cautioning optim() users about "Nelder-Mead" default - (originally) Optim instability

[ProfJCNash]

Agreed on the default algorithm issue. That is important for users to
know, and I'm happy to underline it. Also that CG (which is based on one
of my codes) should be deprecated. BFGS (also based on one of my codes
from long ago) does much better than I would ever have expected.

Over the years I've tried different Nelder-Mead implementations. Cannot
say I've found any that is always better than that in optim() (also
based on an old code of mine), though nmkb() from dfoptim package seems
to do better a lot of the time, and it has a transformation method for
bounds, which may be useful, but does have the awkwardness that one
cannot start on a bound. For testing a function, I don't think it makes
a lot of difference which variant of NM one uses if the trace is on to
catch never-ending runs. For production use, it is a really good idea to
try different methods on a sample of likely cases and choose a method
that does well. That is the motivation for the optimx package or the
opm() function of the newer optimz (on R-forge) that I'm still
polishing. optimz has a function optimr() that has the same call as
optim() but incorporates over a dozen optimizers via method = "(selected
method)".

As a gradient-free choice, the Powell codes from minqa or other packages
(there are several implementations) can sometimes have spectacular
performance, but they also flub rather more regularly than Nelder-Mead
in my experience. That is, when they are good, they are very very good,
and when they are not they are horrid. (Plagiarism!)

JN

On 15-11-15 12:46 PM, Ravi Varadhan wrote:
> Hi John,
> My main point is not about Nelder-Mead per se.  It is *primarily* about the Nelder-Mead implementation in optim().  
> 
> The users of optim() should be cautioned regarding the default algorithm and that they should consider alternatives such as "BFGS" in optim(), or other implementations of Nelder-Mead.
> 
> Best regards,
> Ravi
> ________________________________________
> From: ProfJCNash <profjcnash at gmail.com>
> Sent: Sunday, November 15, 2015 12:21 PM
> To: Ravi Varadhan; 'r-help at r-project.org'; lorenzo.isella at gmail.com
> Cc: bhh at xs4all.nl; Gabor Grothendieck
> Subject: Re: Cautioning optim() users about "Nelder-Mead" default - (originally) Optim instability
> 
> Not contradicting Ravi's message, but I wouldn't say Nelder-Mead is
> "bad" per se. It's issues are that it assumes the parameters are all on
> the same scale, and the termination (not convergence) test can't use
> gradients, so it tends to get "near" the optimum very quickly -- say
> only 10% of the computational effort -- then spends an awful amount of
> effort deciding it's got there. It often will do poorly when the
> function has nearly "flat" zones e.g., long valley with very low slope.
> 
> So my message is still that Nelder-Mead is an unfortunate default -- it
> has been chosen I believe because it is generally robust and doesn't
> need gradients. BFGS really should use accurate gradients, preferably
> computed analytically, so it would only be a good default in that case
> or with very good approximate gradients (which are costly
> computationally).
> 
> However, if you understand what NM is doing, and use it accordingly, it
> is a valuable tool. I generally use it as a first try BUT turn on the
> trace to watch what it is doing as a way to learn a bit about the
> function I am minimizing. Rarely would I use it as a production minimizer.
> 
> Best, JN
> 
> On 15-11-15 12:02 PM, Ravi Varadhan wrote:
>> Hi,
>>
>>
>>
>> While I agree with the comments about paying attention to parameter
>> scaling, a major issue here is that the default optimization algorithm,
>> Nelder-Mead, is not very good.  It is unfortunate that the optim
>> implementation chose this as the "default" algorithm.  I have several
>> instances where people have come to me with poor results from using
>> optim(), because they did not realize that the default algorithm is
>> bad.  We (John Nash and I) have pointed this out before, but the R core
>> has not addressed this issue due to backward compatibility reasons.
>>
>>
>>
>> There is a better implementation of Nelder-Mead in the "dfoptim" package.
>>
>>
>>
>> ​require(dfoptim)
>>
>> mm_def1 <- nmk(par = par_ini1, min.perc_error, data = data)
>>
>> mm_def2 <- nmk(par = par_ini2, min.perc_error, data = data)
>>
>> mm_def3 <- nmk(par = par_ini3, min.perc_error, data = data)
>>
>> print(mm_def1$par)
>>
>> print(mm_def2$par)
>>
>> print(mm_def3$par)
>>
>>
>>
>> In general, better implementations of optimization algorithms are
>> available in packages such as "optimx", "nloptr".  It is unfortunate
>> that most naïve users of optimization in R do not recognize this.
>> Perhaps, there should be a "message" in the optim help file that points
>> this out to the users.
>>
>>
>>
>> Hope this is helpful,
>>
>> Ravi
>>
>>
>