CMA Evolution Strategy Source Code

This page provides implementations of the CMA-ES and links to libraries that contain such implementations. All implementations provided in this page follow very closely Hansen (2006). They support small and large population sizes, the latter by implementing the rank-µ-update (Hansen et al, 2003) with non-uniform weights (Hansen and Kern, 2004) and an improved parameter setting for the step-size adaptation (Hansen and Kern, 2004). The default population size is small (Hansen and Ostermeier, 2001). Further (optionally) supported features:

1. All implementations, except for the minimalistic code for reading, provide an independent restart procedure with increasing population size (Auger and Hansen, 2005) and standardized data output for visualization.
2. All implementations, except for the minimalistic code for reading and the C version, optionally support a "separable" initial phase, where the covariance matrix remains diagonal (Ros and Hansen, 2008). The latter permits a faster learning of the diagonal elements and reduces the internal time complexity from quadratic to linear which might be useful for dimensionalities, say, larger than a hundred.
3. Matlab version 3.x supports an additional uncertainty handling (Hansen et al, 2009, set option Noise.on='yes').
4. Matlab version since 3.40.beta implements the idea of active CMA (Jastrebski and Arnold, 2006), where a rank-µ matrix is substracted from the original covariance matrix, therefore explicitely reducing the variance in directions where bad samples were found. The implementation deviates from the original paper.

Code for Reading

C (ANSI)

cmaes_c.tar.gz, cmaes_c.tar provides a pure ANSI C implementation in a slightly object-oriented way, where the iteration step can be explicitly controlled. Therefore, the integration into any existing program in C or C++, should be fairly easy.

C++

• Shark library
• EO Evolving Objects evolutionary computation framework
• Open BEAGLE
• PACLib Perception-Action Components Library

Fortran

Java

• Documentation
• Code example and an example properties (parameter) file (both part of the source below)
• Source as jar file, as gzipped tar file.

Matlab and Octave

cmaes.m (version 3, revised June 2008, former versions) provides an interface similar to Matlab's optimization routines, e.g. fminsearch and fminunc. The following (optional) variants/extensions are implemented.

• option LBounds and UBounds: coordinate wise boundary handling (Hansen et al, 2009)
• option Noise.on: uncertainty (noise) handling (Hansen et al, 2009)
• option CMA.active: a substractive update of the covariance matrix, similar to Jastrebski and Arnold (2006) (set option CMA.active = 1 with check for positive definiteness or CMA.active = 2 without checking). This version might become the default in near future.
• option DiagonalOnly: only the diagonal of the covariance matrix is adapted for a number of initial iterations (Ros and Hansen, 2008) which leads to a faster learning of a variable scaling and also reduces the internal time complexity of the algorithm from quadratic to linear (meanwhile no correlations are learned)

cmaesV255.m (version 2.55, July 2007) is the former version which saves the data record history internally. plotcmaes.m and dispcmaes.m are stand alone functions to plot/display data output from cmaes.m before version 3. They can be used to investigate the output data while the optimization is still running.

For Octave the package octave-forge is needed (thanks to Luís Correia for the advise).

Python

The preliminary documentation is online. Contact me, hansen (AT) lri.fr, for the source code.

Scilab

Two interfaces are provided. An object oriented "ask-and-tell" interface (cma_new etc.), where the loop control remains with the user, and a functional interface (cma_optim).

• Online documentation
• Source v0.997 as tar file

Sample Output

A sample output from a run on the 12-dimensional Rosenbrock function, using plotcmaesdat.m.

The lower figures show the square root of eigenvalues (left) and of diagonal elements (right) of the covariance matrix C. The actual sample distribution has the covariance matrix σ² C.

Testing your own source code (under construction)

Testing a stochastic optimization code is intricate because the algorithm might be "robust" with respect to bugs. A CMA-ES code might run well on quite a few test functions however an unobserved coding error might later compromise its performance.

A simple and most effective way of testing is, to compare two codes, furnished with the same "random" numbers. One can use a simple one-line random number generator. The numbers need not to be normal, but should be continuous and symmetric about zero. Tracking the internal state variables for two iterations must lead to identical results (apart from numerical effects that are usually in the order of 10^-15). Internal state variables are the distribution mean, the step-size and the covariance matrix, and furthermore the two evolution paths. For comparison, one of the above codes can be used.

Further, the code should also be tested on functions that can be found, for example, in purecmaes.m, and the results should be compared to one of the above codes.

1. On a random objective function frand (e.g. f(x) is uniform in [0,1] and independent of its input x). Step-size σ and the eigenvalues of C should be tracked and plotted. log(σ) must conduct a symmetric (unbiased) random walk. The iteration-wise sorted log(eigenvalues) will slowly drift apart, quite symmetrically but with the overall dendency to decrease. Finally the condition number of C will exceed numerically feasible limits, which should be covered in the code as termination criterion.
2. The linear function fplane (e.g. f(x) = x(1)) should be tested (X0=ones,sigma0=1,stopfitness=-1e5).
3. The sphere function fsphere should be tested (X0=ones,sigma0=1,stopfitness=1e-10).
4. The Ellipsoid felli should be tested (X0=ones, sigma0=1, stopfitness=1e-10). The eigenvalues get a nice regular configuration on this function.
5. The Cigar fcigar should be tested (X0=ones, sigma0=1, stopfitness=1e-10).
6. The Rosenbrock function frosen is a good test case (X0=zeros, sigma0=0.1, stopfitness=1e-10). Occasionally the local minimum will be discovered with a function value close under four.
7. Finally, frastrigin10 (X0=5*ones, sigma0=5, stopfitness=1e-10), where only large populations (about larger 10n) will find the global optimum with a reasonable probability.

I appreciate any feedback regarding the provided source code and regarding successful and unsuccessful applications or modifications of the CMA-ES.

Mail to hansen AT lri.fr .