Variables that consistently end up on the (same) boundary can be removed. In any case, a reasonable (linear) scaling of variables in x should still be taken care of when the sizes of boundary intervals and the (supposed) sensitivities do not agree. In the very same spirit, the boundary handling implemented in the C and Python codes uses a piecewise either linear or quadratic function to avoid numerical subtleties (however the implemented boundary handling does not shift the domain and therefore maps the domain center a + ( b-a) / 2 to itself). y remains between a and b also for any other value of x. Lower and upper bounds a, b can be achieved with y = a + ( b-a) × (1 – cos(π × x / 10)) / 2, where for 0 ≤ x ≤ 10 this is a strictly increasing bijective mapping into. In summary, to map the values into we have the alternative transformationsĪ simple way to achieve, for example, a lower bound a for a variable x is to use a + x 2 instead of x in the objective function call (as already mentioned above). In the case where only a lower bound at zero is necessary, a simple and natural transformation is x 2 × default_x, such that x=1 represents the default (or initial) value and x remains unbounded during optimization. In order to avoid the problem that changes of very small values have too less an impact, an alternative is to choose 10 –1 × ( x/10) 2 ≥ 0. Again, the idea is to have similar sensitivity: this makes sense if we expect the change from 10 –4 to 10 –3 to have an impact similar to the change from 10 –2 to 10 –1. More specifically, to achieve the parameter range, we use 10 –4×10 3 x/10 with x in. That is, for a parameter that must always be positive, with a ratio between typical upper and lower value being larger than 100, we might use 10 x instead of x to call the objective function. The natural encoding of (some of) the parameters can also be "logarithmic". With this transformation, a typical initial sigma will be ≈ 2, see also below. To achieve this on-the-fly, a linear scaling of variables is provided in the Scilab and Python codes below. The wrapper scales, for example, in each parameter/coordinate the value into the typical actual domain of the parameter/coordinate. Usually, the best approach is to write a wrapper around the objective function that transforms the parameters before the actual function call. Presumably similar sensitivity (this makes the identity as initial covariance matrix the right choice). All parameters should be rescaled such that they have In particular, a reasonable parameter encoding is essential. The specific formulation of a (real) optimization problem has a tremendous impact on the optimization performance. Minimalistic code for reading and education is provided in Matlab/Octave and Python. Python since version 0.9.92 implements selective mirrored sampling ( Auger et al, 2011), where a few of the worst solutions are mirrored.The implementations differ and also deviate considerably from the original paper. Matlab since version 3.40.beta and Python since version 0.9.92 implement the idea of active CMA ( Jastrebski and Arnold, 2006, see option CMA.active or CMA_active), where a "negative" covariance matrix update explicitly reduces the variance in directions where bad samples were found.Matlab since version 3.0 and Python since version 0.9.92 support an additional uncertainty handling ( Hansen et al, 2009, set option Noise.on='yes', however, don't rely on this option without a rough.Which might be useful for dimensionalities, say, larger than a hundred. Latter permits a faster learning of the diagonal elements and reduces the internal time complexity from quadratic to linear Initial phase, where the covariance matrix remains diagonal ( Ros and Hansen, 2008). All implementations, except for the minimalistic codeįor reading, optionally support a "separable/diagonal".Reading, provide an independent restart procedure with increasingĪnd Hansen, 2005) and standardized data output for All implementations, except for the minimalistic codes for.The default population size is small ( Hansen and Ostermeier,Ģ001). Non-uniform recombination weights ( Hansen and Kern,Ģ004) and an improved parameter setting for the step-size The rank-µ-update ( Hansen et al, 2003) with Support small and large population sizes, the latter by implementing All implementations provided in this page follow very closely Hansen (2006). This page provides implementations of the CMA-ES and links to libraries that contain such implementations. CMA Evolution Strategy Source Code UP (The CMA Evolution Strategy)
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