[R] Noisy objective functions
Hans W
hwborcher@ @end|ng |rom gm@||@com
Sun Aug 13 21:28:19 CEST 2023
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)
(1+rnorm(1, sd=0.01)) * adagio::fnRosenbrock(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]
neldermead = nloptr::neldermead,
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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