Fast Matrix Adaptation Evolution Strategy (FMAES)

class pypop7.optimizers.es.fmaes.FMAES(problem, options)[source]

Fast Matrix Adaptation Evolution Strategy (FMAES).

Note

FMAES is a more efficient implementation of MAES with quadractic time complexity w.r.t. each sampling, which replaces the computationally expensive matrix-matrix multiplication (cubic time complexity) with the combination of matrix-matrix addition and matrix-vector multiplication (quadractic time complexity) for transformation matrix adaptation. It is highly recommended to first attempt more advanced ES variants (e.g., LMCMA, LMMAES) for large-scale black-box optimization, since FMAES still has a computationally intensive quadratic time complexity (w.r.t. each sampling).

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 optimizer FMAES to minimize the well-known test function Rosenbrock:

 1>>> import numpy
 2>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
 3>>> from pypop7.optimizers.es.fmaes import FMAES
 4>>> problem = {'fitness_function': rosenbrock,  # 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,  # set optimizer options
 9...            'seed_rng': 2022,
10...            'mean': 3.0*numpy.ones((2,)),
11...            'sigma': 0.1}  # the global step-size may need to be tuned for better performance
12>>> fmaes = FMAES(problem, options)  # initialize the optimizer class
13>>> results = fmaes.optimize()  # run the optimization process
14>>> # return the number of function evaluations and best-so-far fitness
15>>> print(f"FMAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16FMAES: 5000, 2.1296244414852865e-19

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

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

final global step-size, aka mutation strength.

Type:

float

References

Beyer, H.G., 2020, July. Design principles for matrix adaptation evolution strategies. In Proceedings of Annual Conference on Genetic and Evolutionary Computation Companion (pp. 682-700). https://dl.acm.org/doi/abs/10.1145/3377929.3389870

Loshchilov, I., Glasmachers, T. and Beyer, H.G., 2019. Large scale black-box optimization by limited-memory matrix adaptation. IEEE Transactions on Evolutionary Computation, 23(2), pp.353-358. https://ieeexplore.ieee.org/abstract/document/8410043

Beyer, H.G. and Sendhoff, B., 2017. Simplify your covariance matrix adaptation evolution strategy. IEEE Transactions on Evolutionary Computation, 21(5), pp.746-759. https://ieeexplore.ieee.org/document/7875115

See the official Matlab version from Prof. Beyer: https://homepages.fhv.at/hgb/downloads/ForDistributionFastMAES.tar