Matrix Adaptation Evolution Strategy (MAES)

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Matrix Adaptation Evolution Strategy (MAES).


MAES is a powerful simplified version of the well-established CMA-ES algorithm nearly without significant performance loss, designed in 2017 by Beyer and Sendhoff. One obvious advantage of such a simplification is to help better understand the underlying working principles (e.g., invariance and unbias) of CMA-ES, which are often thought to be rather complex for newcomers. It is highly recommended to first attempt more advanced ES variants (e.g., LMCMA, LMMAES) for large-scale black-box optimization, since MAES has a cubic time complexity (w.r.t. each sampling). Note that another improved version called FMAES provides a relatively more efficient implementation for MAES with quadratic time complexity (w.r.t. each sampling).

  • 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)).


Use the optimizer MAES 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 import MAES
 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>>> maes = MAES(problem, options)  # initialize the optimizer class
13>>> results = maes.optimize()  # run the optimization process
14>>> # return the number of function evaluations and best-so-far fitness
15>>> print(f"MAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16MAES: 5000, 2.129367016460251e-19

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


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




number of offspring, aka offspring population size.




number of parents, aka parental population size.




final global step-size, aka mutation strength.




Beyer, H.G., 2020, July. Design principles for matrix adaptation evolution strategies. In Proceedings of ACM Conference on Genetic and Evolutionary Computation Companion (pp. 682-700).

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.

Beyer, H.G. and Sendhoff, B., 2017. Simplify your covariance matrix adaptation evolution strategy. IEEE Transactions on Evolutionary Computation, 21(5), pp.746-759.

See the official Matlab version from Prof. Beyer: