Rechenberg’s (1+1)-Evolution Strategy (RES)

class pypop7.optimizers.es.res.RES(problem, options)

Rechenberg’s (1+1)-Evolution Strategy with 1/5th success rule (RES).

Note

RES is the first evolution strategy with self-adaptation of the global step-size (designed by Rechenberg, one recipient of IEEE Evolutionary Computation Pioneer Award 2002). As theoretically investigated in his seminal Ph.D. dissertation at Technical University of Berlin, the existence of narrow evolution window explains the necessarity of global step-size adaptation to maximize the local convergence progress, if possible.

Since there is only one parent and only one offspring for each generation (iteration), RES generally shows limited exploration ability for large-scale black-box optimization (LBO). Therefore, it is recommended to first attempt more advanced ES variants (e.g., LMCMA, LMMAES) for LBO. Here we include RES (AKA two-membered ES) mainly for benchmarking and theoretical purposes.

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’].

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

Examples

Use the black-box optimizer RES to minimize the well-known test function Rosenbrock:

>>> import numpy  # engine for numerical computing
>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
>>> from pypop7.optimizers.es.res import RES
>>> problem = {'fitness_function': rosenbrock,  # to define problem arguments
...            'ndim_problem': 2,
...            'lower_boundary': -5.0*numpy.ones((2,)),
...            'upper_boundary': 5.0*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000,  # to set optimizer options
...            'seed_rng': 2022,
...            'mean': 3.0*numpy.ones((2,)),
...            'sigma': 3.0}  # global step-size may need to be tuned
>>> res = RES(problem, options)  # to initialize the black-box optimizer class
>>> results = res.optimize()  # to run its optimization/evolution process
>>> # to return the number of function evaluations and the best-so-far fitness
>>> print(f"RES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
RES: 5000, 0.00011689296624022443

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_sigma

learning rate of global step-size self-adaptation.

Type:

float

mean

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

Type:

array_like

sigma

final global step-size, aka mutation strength (updated during optimization).

Type:

float

References

Auger, A., Hansen, N., López-Ibáñez, M. and Rudolph, G., 2022. Tributes to Ingo Rechenberg (1934–2021). ACM SIGEVOlution, 14(4), pp.1-4.

Agapie, A., Solomon, O. and Giuclea, M., 2021. Theory of (1+1) ES on the RIDGE. IEEE Transactions on Evolutionary Computation, 26(3), pp.501-511.

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.

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

Rechenberg, I., 2000. Case studies in evolutionary experimentation and computation. Computer Methods in Applied Mechanics and Engineering, 186(2-4), pp.125-140.

Rechenberg, I., 1989. Evolution strategy: Nature’s way of optimization. In Optimization: Methods and Applications, Possibilities and Limitations (pp. 106-126). Springer, Berlin, Heidelberg.

Rechenberg, I., 1984. The evolution strategy. A mathematical model of darwinian evolution. In Synergetics—from Microscopic to Macroscopic Order (pp. 122-132). Springer, Berlin, Heidelberg.

Schumer, M.A. and Steiglitz, K., 1968. Adaptive step size random search. IEEE Transactions on Automatic Control, 13(3), pp.270-276.