BIPOP should restart from re-sampled x0

[beniz]

I missed this in the original paper. This poses the question of how to specify a distribution on the initial x0 while not asking too much from the user.

The following points will be assessed before proceedings with a solution:

• benchmark BIPOP with and without re-sampling x0 upon restart
• benchmark re-sampling a new x0 from within the search trace of past runs
  Benchmarking should especially check against the multi-modal functions of BBOB.

[beniz]

FTR, comparison of three resampling schemes for x0 on
each aBIPOP restart on BBOB's multi-modal functions is available in PDF (1):

• blue: resampling within [-4,4]
• red: resampling from collected history (*)
• greenish: taking the best known search state from collected history
  (i.e. the one with current min f-value)

These results seem to suggest that we may want to provide a resampling
scheme around initial x0 for BIPOP. It may be best to have it
done without user intervention.

(1) http://juban.free.fr/stuff/libcmaes/bipop_sampl_compar.pdf

[beniz]

My understanding is that CMA's first step upon every restart already samples from a Gaussian bump around x0 of standard deviation 1 and scaled by the initial value of sigma.

So it could make sense to either re-sample x0 itself from a distribution to be defined (or user-defined, though this seems error-prone), or to modify the initial standard deviation so that the initial population possibly spreads to a larger area.

[nikohansen]

I agree, in contrast to the input to CMA-ES, which is an initial solution and a standard deviation, the standard input to BIPOP-CMA-ES should rather be a domain. (A distribution for X0 is not quite sufficient, because we need to be able to infer the initial standard deviation). Another common approach is to define a standard domain, where the user must transform the presented problem to be defined in. In BBOB our standard domain is [-5, 5]^dim, I have also regularly seen the use of [0, 1]^dim as standard domain.

[beniz]

Yes, so the good news is that this is actually already in the lib as it does support setting [x0min,x0max] as the initial domain of x0. An initial candidate is then sampled uniformly from this domain upon (re-)creation of the initial (empty) CMASolution object. I kind of forgot about that, and from looking at the code, both IPOP and BIPOP are already taking advantage of this resampling if x0min and x0max are supplied.

All I believe I have to do then is to update the documentation so the user is informed that BIPOP would yield potentially better results by giving an initial domain.

[beniz]

Added a paragraph to https://github.com/beniz/libcmaes/wiki/Practical-hints
Will add just another set_x0 function with a signature for STL vectors, and this should then close this issue.