Covariance Matrix Adaptation Evolution Strategy (CMAES)

class pypop7.optimizers.es.cmaes.CMAES(problem, options)[source]

Covariance Matrix Adaptation Evolution Strategy (CMAES).

Note

CMAES is widely recognized as one of State-Of-The-Art (SOTA) evolutionary algorithms for continuous black-box optimization (BBO), according to the well-recognized Nature review of Evolutionary Computation.

For some (rather all) interesting applications of CMA-ES, please refer to e.g., [ICLR-2024 Spotlight], [TMRB-2024], [LWC-2024], [RSIF-2024], [MNRAS-2024], [Medical Physics-2024], [Wolff, 2024], [Jankowski et al., 2024], [Martin, 2024, Ph.D. Dissertation (Harvard University)], [Milekovic et al., 2023, Nature Medicine], [Chen et al., 2023, Science Robotics], [Falk et al., 2023, PNAS], [Thamm&Rosenow, 2023, PRL], [Brea et al., 2023, Nature Communications], [Ghafouri&Biros, 2023], [Barral, 2023, Ph.D. Dissertation (University of Oxford)], [Slade et al., 2022, Nature], [Rudolph et al., 2022, Nature Communications], [Cazenille et al., 2022, Bioinspiration & Biomimetics], [Franks et al., 2021], [Yuan et al., 2021, MNRAS], [Löffler et al., 2021, Nature Communications], [Papadopoulou et al., 2021, JPCB], [Schmucker et al., 2021, PLoS Comput Biol], [Barkley, 2021, Ph.D. Dissertation (Harvard University)], [Fernandes, 2021, Ph.D. Dissertation (Harvard University)], [Quinlivan, 2021, Ph.D. Dissertation (Harvard University)], [Vasios et al., 2020, Soft Robotics], [Pal et al., 2020], [Lei, 2020, Ph.D. Dissertation (University of Oxford)], [Pisaroni et al., 2019, Journal of Aircraft], [Yang et al., 2019, Journal of Aircraft], [Ong et al., 2019, PLOS Computational Biology], [Zhang et al., 2017, Science], [Wei&Mahadevan, 2016, Soft Matter], [Loshchilov&Hutter, 2016], [Molinari et al., 2014, AIAAJ], [Melton, 2014, Acta Astronautica], [Khaira et al., 2014, ACS Macro Lett.], to name a few.

Parameters:

  • problem (dict) –

    problem arguments with the following common settings (keys):

    • ’fitness_function’ - objective function to be minimized (func),

    • ’ndim_problem’ - number of dimensionality (int),

    • ’upper_boundary’ - upper boundary of search range (array_like),

    • ’lower_boundary’ - lower boundary of search range (array_like).

  • options (dict) –

    optimizer options with the following common settings (keys):

    • ’max_function_evaluations’ - maximum of function evaluations (int, default: np.inf),

    • ’max_runtime’ - maximal runtime to be allowed (float, default: np.inf),

    • ’seed_rng’ - seed for random number generation needed to be explicitly set (int);

    and with the following particular settings (keys):

    • ’sigma’ - initial global step-size, aka mutation strength (float),

    • ’mean’ - initial (starting) point, aka mean of Gaussian search distribution (array_like),

      • if not given, it will draw a random sample from the uniform distribution whose search range is bounded by problem[‘lower_boundary’] and problem[‘upper_boundary’].

    • ’n_individuals’ - number of offspring, aka offspring population size (int, default: 4 + int(3*np.log(problem[‘ndim_problem’]))),

    • ’n_parents’ - number of parents, aka parental population size (int, default: int(options[‘n_individuals’]/2)).

Examples

Use the black-box optimizer CMAES to minimize the well-known test function Rosenbrock:

 1>>> import numpy  # engine for numerical computing
 2>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
 3>>> from pypop7.optimizers.es.cmaes import CMAES
 4>>> problem = {'fitness_function': rosenbrock,  # to define problem arguments
 5...            'ndim_problem': 2,
 6...            'lower_boundary': -5.0*numpy.ones((2,)),
 7...            'upper_boundary': 5.0*numpy.ones((2,))}
 8>>> options = {'max_function_evaluations': 5000,  # to set optimizer options
 9...            'seed_rng': 2022,
10...            'mean': 3.0*numpy.ones((2,)),
11...            'sigma': 3.0}  # global step-size may need to be fine-tuned for better performance
12>>> cmaes = CMAES(problem, options)  # to initialize the optimizer class
13>>> results = cmaes.optimize()  # to run the optimization/evolution process
14>>> print(f"CMAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
15CMAES: 5000, 0.0017

For its correctness checking of Python coding, please refer to this code-based repeatability report for all details. For pytest-based automatic testing, please see test_cmaes.py.

best_so_far_x

final best-so-far solution found during entire optimization.

Type:

array_like

best_so_far_y

final best-so-far fitness found during entire optimization.

Type:

array_like

mean

initial (starting) point, aka mean of Gaussian search distribution.

Type:

array_like

n_individuals

number of offspring, aka offspring population size / sample size.

Type:

int

n_parents

number of parents, aka parental population size / number of positively selected search points.

Type:

int

sigma

final global step-size, aka mutation strength (updated during optimization).

Type:

float

References

https://cma-es.github.io/

Hansen, N., 2023. The CMA evolution strategy: A tutorial. arXiv preprint arXiv:1604.00772.

Ollivier, Y., Arnold, L., Auger, A. and Hansen, N., 2017. Information-geometric optimization algorithms: A unifying picture via invariance principles. Journal of Machine Learning Research, 18(18), pp.1-65.

Hansen, N., Atamna, A. and Auger, A., 2014, September. How to assess step-size adaptation mechanisms in randomised search. In International Conference on Parallel Problem Solving From Nature (pp. 60-69). Springer, Cham.

Kern, S., Müller, S.D., Hansen, N., Büche, D., Ocenasek, J. and Koumoutsakos, P., 2004. Learning probability distributions in continuous evolutionary algorithms–a comparative review. Natural Computing, 3, pp.77-112.

Hansen, N., Müller, S.D. and Koumoutsakos, P., 2003. Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evolutionary Computation, 11(1), pp.1-18.

Hansen, N. and Ostermeier, A., 2001. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 9(2), pp.159-195.

Hansen, N. and Ostermeier, A., 1996, May. Adapting arbitrary normal mutation distributions in evolution strategies: The covariance matrix adaptation. In Proceedings of IEEE International Conference on Evolutionary Computation (pp. 312-317). IEEE.

Please refer to its lightweight Python implementation from cyberagent.ai: https://github.com/CyberAgentAILab/cmaes

Please refer to its official Python implementation from Hansen, N.: https://github.com/CMA-ES/pycma

Basic Information of CMA-ES

RoboCup 3D Simulation League Competition Champions.

Some of High-Quality Tutorials for CMA-ES

Some Applications of CMA-ES

Some Interesting Applications in Robotics

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