# [R] Noisy objective functions

Ben Bolker bbo|ker @end|ng |rom gm@||@com
Sun Aug 13 21:39:43 CEST 2023

```   This is a huge topic.

Differential evolution (DEoptim package) would be one good starting
point; there is a simulated annealing method built into optim() (method
= "SANN") but it usually requires significant tuning.

Also genetic algorithms.

You could look at the NLopt list of algorithms
the options for derivative-free global optimization , and then use them
via the nloptr package.

Good luck ...

On 2023-08-13 3:28 p.m., Hans W wrote:
> While working on 'random walk' applications, I got interested in
> optimizing noisy objective functions. As an (artificial) example, the
> following is the Rosenbrock function, where Gaussian noise of standard
> deviation `sd = 0.01` is added to the function value.
>
>      fn <- function(x)
>
> To smooth out the noise, define another function `fnk(x, k = 1)` that
> calls `fn` k times and returns the mean value of those k function
> applications.
>
>      fnk <- function(x, k = 1) {     # fnk(x) same as fn(x)
>          rv = 0.0
>          for (i in 1:k) rv <- rv + fn(x)
>          return(rv/n)
>      }
>
> When we apply several optimization solvers to this noisy and smoothed
> noise functions we get for instance the following results:
> (Starting point is always `rep(0.1, 5)`, maximal number of iterations 5000,
>   relative tolerance 1e-12, and the optimization is successful if the
> function value at the minimum is below 1e-06.)
>
>        k   nmk       anms neldermead     ucminf optim_BFGS
>       ---------------------------------------------------
>        1  0.21       0.32       0.13       0.00       0.00
>        3  0.52       0.63       0.50       0.00       0.00
>       10  0.81       0.91       0.87       0.00       0.00
>
> Solvers: nmk = dfoptim::nmk, anms = pracma::anms [both Nelder-Mead codes]
>           ucminf = ucminf::ucminf, optim_BFGS = optim with method "BFGS"
>
> Read the table as follows: `nmk` will be successful in 21% of the
> trials, while for example `optim` will never come close to the true
> minimum.
>
> I think it is reasonable to assume that gradient-based methods do
> poorly with noisy objectives, though I did not expect to see them fail
> so clearly. On the other hand, Nelder-Mead implementations do quite
> well if there is not too much noise.
>
> In real-world applications, it will often not be possible to do the
> same measurement several times. That is, we will then have to live
> with `k = 1`. In my applications with long 'random walks', doing the
> calculations several times in a row will really need some time.
>
> QUESTION: What could be other approaches to minimize noisy functions?
>
> I looked through some "Stochastic Programming" tutorials and did not
> find them very helpful in this situation. Of course, I might have
> looked into these works too superficially.
>
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