Schwefel’s Self-Adaptation Evolution Strategy (SSAES)

class pypop7.optimizers.es.ssaes.SSAES(problem, options)

Schwefel’s Self-Adaptation Evolution Strategy (SSAES).

Note

SSAES adapts all the individual step-sizes on-the-fly, proposed by Schwefel, one recipient of IEEE Evolutionary Computation Pioneer Award 2002. Since it needs a relatively large population (e.g. larger than number of dimensionality) for reliable self-adaptation, SSAES easily suffers from very slow convergence for large-scale black-box optimization (LSBBO). Therefore, it is highly recommended to first attempt more advanced ES variants (e.g. LMCMA, LMMAES) for LSBBO. Here we include it mainly for benchmarking and theoretical purpose.

Note that the restart process is not implemented owing to typically slow convergence of SSAES.

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: 5*problem[‘ndim_problem’]),

  • ’n_parents’ - number of parents, aka parental population size (int, default: int(options[‘n_individuals’]/4)),

  • ’lr_sigma’ - learning rate of global step-size self-adaptation (float, default: 1.0/np.sqrt(problem[‘ndim_problem’])),

  • ’lr_axis_sigmas’ - learning rate of individual step-sizes self-adaptation (float, default: 1.0/np.power(problem[‘ndim_problem’], 1.0/4.0)).

Examples

Use the optimizer SSAES to minimize the well-known test function Rosenbrock:

>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
>>> from pypop7.optimizers.es.ssaes import SSAES
>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
...            'ndim_problem': 2,
...            'lower_boundary': -5*numpy.ones((2,)),
...            'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000,  # set optimizer options
...            'seed_rng': 2022,
...            'mean': 3*numpy.ones((2,)),
...            'sigma': 0.1}  # the global step-size may need to be tuned for better performance
>>> ssaes = SSAES(problem, options)  # initialize the optimizer class
>>> results = ssaes.optimize()  # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"SSAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
SSAES: 5000, 0.13131869620062903

For its correctness checking of coding, refer to this code-based repeatability report for more details.

lr_axis_sigmas

learning rate of individual step-sizes self-adaptation.

Type:

float

lr_sigma

learning rate of global step-size self-adaptation.

Type:

float

mean

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

Type:

array_like

n_individuals

number of offspring, aka offspring population size.

Type:

int

n_parents

number of parents, aka parental population size.

Type:

int

sigma

initial global step-size, aka mutation strength.

Type:

float

_axis_sigmas

final individuals step-sizes.

Type:

array_like

References

Hansen, N., Arnold, D.V. and Auger, A., 2015. Evolution strategies. In Springer Handbook of Computational Intelligence (pp. 871-898). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007%2F978-3-662-43505-2_44

Beyer, H.G. and Schwefel, H.P., 2002. Evolution strategies–A comprehensive introduction. Natural Computing, 1(1), pp.3-52. https://link.springer.com/article/10.1023/A:1015059928466

Schwefel, H.P., 1988. Collective intelligence in evolving systems. In Ecodynamics (pp. 95-100). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-642-73953-8_8

Schwefel, H.P., 1984. Evolution strategies: A family of non-linear optimization techniques based on imitating some principles of organic evolution. Annals of Operations Research, 1(2), pp.165-167. https://link.springer.com/article/10.1007/BF01876146