Schwefel’s Self-Adaptation Evolution Strategy (SSAES)

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

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

Note

SSAES adapts all the individual step-sizes (aka coordinate-wise standard deviations) on-the-fly, proposed by Schwefel (one recipient of IEEE Evolutionary Computation Pioneer Award 2002 and IEEE Frank Rosenblatt Award 2011). Since it often needs a relatively large population (e.g., larger than number of dimensionality) for reliable self-adaptation, SSAES suffers easily from slow convergence for large-scale black-box optimization. Therefore, it is recommended to first attempt more advanced ES variants (e.g., LMCMA, LMMAES) for large-scale black-box optimization. Here we include SSAES mainly for benchmarking and theoretical purpose. Currently the restart process is not implemented owing to its typically slow convergence.

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 black-box optimizer SSAES 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.ssaes import SSAES
 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>>> ssaes = SSAES(problem, options)  # to initialize the black-box optimizer class
13>>> results = ssaes.optimize()  # to run the optimization/evolution process
14>>> # to return the number of function evaluations and the best-so-far fitness
15>>> print(f"SSAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16SSAES: 5000, 0.00023558230456829403

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

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 (updated during optimization).

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.

Beyer, H.G. and Schwefel, H.P., 2002. Evolution strategies–A comprehensive introduction. Natural Computing, 1(1), pp.3-52.

Schwefel, H.P., 1988. Collective intelligence in evolving systems. In Ecodynamics (pp. 95-100). Springer, Berlin, Heidelberg.

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.