[R] Dealing with -Inf in a maximisation problem.

[William Dunlap]

Perhaps the C function Rf_logspace_sum(double *x, int n) would help in
computing log(b).  It computes log(sum(exp(x_i))) for i in 1..n, avoiding
unnecessary under- and overflow.

Bill Dunlap
TIBCO Software
wdunlap tibco.com

On Sun, Nov 6, 2016 at 5:25 PM, Rolf Turner <r.turner at auckland.ac.nz> wrote:

> On 07/11/16 13:07, William Dunlap wrote:
>
>> Have you tried reparameterizing, using logb (=log(b)) instead of b?
>>
>
> Uh, no.  I don't think that that makes any sense in my context.
>
> The "b" values are probabilities and must satisfy a "sum-to-1"
> constraint.  To accommodate this constraint I re-parametrise via a
> "logistic" style parametrisation --- basically
>
>    b_i = exp(z_i)/[sum_j exp(z_j)], j = 1, ... n
>
> with the parameters that the optimiser works with being z_1, ..., z_{n-1}
> (and with z_n == 0 for identifiability).  The objective function is of the
> form sum_i(a_i * log(b_i)), so I transform back
> from the z_i to the b_i in order calculate the value of the objective
> function.  But when the z_i get moderately large-negative, the b_i become
> numerically 0 and then log(b_i) becomes -Inf.  And the optimiser falls over.
>
> cheers,
>
> Rolf
>
>
>> Bill Dunlap
>> TIBCO Software
>> wdunlap tibco.com <http://tibco.com>
>>
>> On Sun, Nov 6, 2016 at 1:17 PM, Rolf Turner <r.turner at auckland.ac.nz
>> <mailto:r.turner at auckland.ac.nz>> wrote:
>>
>>
>>     I am trying to deal with a maximisation problem in which it is
>>     possible for the objective function to (quite legitimately) return
>>     the value -Inf, which causes the numerical optimisers that I have
>>     tried to fall over.
>>
>>     The -Inf values arise from expressions of the form "a * log(b)",
>>     with b = 0.  Under the *starting* values of the parameters, a must
>>     equal equal 0 whenever b = 0, so we can legitimately say that a *
>>     log(b) = 0 in these circumstances.  However as the maximisation
>>     algorithm searches over parameters it is possible for b to take the
>>     value 0 for values of
>>     a that are strictly positive.  (The values of "a" do not change during
>>     this search, although they *do* change between "successive searches".)
>>
>>     Clearly if one is *maximising* the objective then -Inf is not a value
>> of
>>     particular interest, and we should be able to "move away".  But the
>>     optimising function just stops.
>>
>>     It is also clear that "moving away" is not a simple task; you can't
>>     estimate a gradient or Hessian at a point where the function value
>>     is -Inf.
>>
>>     Can anyone suggest a way out of this dilemma, perhaps an optimiser
>>     that is equipped to cope with -Inf values in some sneaky way?
>>
>>     Various ad hoc kludges spring to mind, but they all seem to be
>>     fraught with peril.
>>
>>     I have tried changing the value returned by the objective function
>> from
>>     "v" to exp(v) --- which maps -Inf to 0, which is nice and finite.
>>     However this seemed to flatten out the objective surface too much,
>>     and the search stalled at the 0 value, which is the antithesis of
>>     optimal.
>>
>>     The problem arises in a context of applying the EM algorithm where
>>     the M-step cannot be carried out explicitly, whence numerical
>>     optimisation.
>>     I can give more detail if anyone thinks that it could be relevant.
>>
>>     I would appreciate advice from younger and wiser heads! :-)
>>
>>     cheers,
>>
>>     Rolf Turner
>>
>>     --
>>     Technical Editor ANZJS
>>     Department of Statistics
>>     University of Auckland
>>     Phone: +64-9-373-7599 ext. 88276 <tel:%2B64-9-373-7599%20ext.%2
>> 088276>
>>
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>>
>>
>
> --
> Technical Editor ANZJS
> Department of Statistics
> University of Auckland
> Phone: +64-9-373-7599 ext. 88276
>

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