Cumulative Step-size Adaptation Evolution Strategy (CSAES)

class, options)

Cumulative Step-size self-Adaptation Evolution Strategy (CSAES).


CSAES adapts all the individual step-sizes on-the-fly with a relatively small population, according to the well-known CSA rule from the Evolutionary Computation community. The default setting (i.e., using a small population) may result in relatively fast (local) convergence, but with the risk of getting trapped in suboptima on multi-modal fitness landscape (which can be alleviated via restart).

AKA cumulative (evolution) path-length control.

  • 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(np.floor(3*np.log(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 adaptation (float, default: np.sqrt(options[‘n_parents’]/(problem[‘ndim_problem’] + options[‘n_parents’]))).


Use the optimizer CSAES 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 import CSAES
 4>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
 5...            'ndim_problem': 2,
 6...            'lower_boundary': -5*numpy.ones((2,)),
 7...            'upper_boundary': 5*numpy.ones((2,))}
 8>>> options = {'max_function_evaluations': 5000,  # set optimizer options
 9...            'seed_rng': 2022,
10...            'mean': 3*numpy.ones((2,)),
11...            'sigma': 0.1}  # the global step-size may need to be tuned for better performance
12>>> csaes = CSAES(problem, options)  # initialize the optimizer class
13>>> results = csaes.optimize()  # run the optimization process
14>>> # return the number of function evaluations and best-so-far fitness
15>>> print(f"CSAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16CSAES: 5000, 0.010143683086819875

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


learning rate of global step-size adaptation.




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




number of offspring, aka offspring population size.




number of parents, aka parental population size.




initial global step-size, aka mutation strength.




Hansen, N., Arnold, D.V. and Auger, A., 2015. Evolution strategies. In Springer Handbook of Computational Intelligence (pp. 871-898). Springer, Berlin, Heidelberg.

Kern, S., Müller, S.D., Hansen, N., Büche, D., Ocenasek, J. and Koumoutsakos, P., 2004. Learning probability distributions in continuous evolutionary algorithms–a comparative review. Natural Computing, 3, pp.77-112.

Ostermeier, A., Gawelczyk, A. and Hansen, N., 1994, October. Step-size adaptation based on non-local use of selection information In International Conference on Parallel Problem Solving from Nature (pp. 189-198). Springer, Berlin, Heidelberg.