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, according to the well-recognized Nature review of Evolutionary Computation.
- 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 tuned 12>>> cmaes = CMAES(problem, options) # to initialize the optimizer class 13>>> results = cmaes.optimize() # to run the optimization/evolution process 14>>> # to return the number of function evaluations and the best-so-far fitness 15>>> print(f"CMAES: {results['n_function_evaluations']}, {results['best_so_far_y']}") 16CMAES: 5000, 0.0017836312093795592
For its correctness checking of coding, refer to this code-based repeatability report for more details.
- 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
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., 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.
See one lightweight Python implementation of CMA-ES from cyberagent.ai: https://github.com/CyberAgentAILab/cmaes
Refer to the official Python implementation of CMA-ES from Hansen, N.: https://github.com/CMA-ES/pycma