An update: I have programmed a version of SPSA and indeed it is working fine (after some tuning). Thanks, @dlakelan! Of course I will make it public soon, after including some more tests.

I am wondering about the following: suppose that for a given parameter \theta, I am simulating N individuals to obtain some average statistic

\bar{h}(\theta) = \frac1N \sum_{i = 1}^N g(\theta)

and my objective is some \bar{h}(\theta) - h_0.

Intuitively, it would make sense to use a low N for he initial phase for finding the right region, and a large N for the later phase, because in the beginning I care more about convergence than accuracy, then I ran refine later. Eg 10k vs 100k seems to work well. Is there any systematic, theoretically supported way of doing this, eg an N(k) function scheme?

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glad you got it working, itās a very clever algorithm. I donāt know about the idea of N(k) it seems this would be very problem specific. For example suppose your simulation is just this side of having infinite varianceā¦ N will need to be very large. On the other hand if your simulation is well behaved it could be much smaller.

Central Limit Theorem should give some idea for what `N`

should be in order to bound the error in \bar{h}(\theta) at a certain level, especially if the variance of g(\theta) does not change a lot with \theta.

Are you getting better results with your SPSA than nloptās CRS2?

Iāve been working on some higher dimensional problems where CRS2 currently has the best convergence, so Iām curious to try your SPSA when itās ready.

Are there any new developments on this type of problems?

I am currently using @Tamas_Papp MultistartOptimization.jl which works like a charm, but I am always interested intrying new things.