import numpy as np # engine for numerical computing
from pypop7.optimizers.ep.cep import CEP
[docs]class FEP(CEP):
"""Fast Evolutionary Programming with self-adaptive mutation of individual step-sizes (FEP).
.. note:: `FEP` was proposed mainly by Yao et al. in 1999 (the recipient of `IEEE Evolutionary Computation Pioneer
Award 2013 <https://tinyurl.com/456as566>`_ and `IEEE Frank Rosenblatt Award 2020 <https://tinyurl.com/yj28zxfa>`_
), where the classical Gaussian sampling distribution is replaced by the heavy-tailed Cachy distribution for
better exploration on multi-modal black-box optimization problems.
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`),
* 'n_individuals' - number of offspring, aka offspring population size (`int`, default: `100`),
* 'q' - number of opponents for pairwise comparisons (`int`, default: `10`),
* 'tau' - learning rate of individual step-sizes self-adaptation (`float`, default:
`1.0/np.sqrt(2.0*np.sqrt(problem['ndim_problem']))`),
* 'tau_apostrophe' - learning rate of individual step-sizes self-adaptation (`float`, default:
`1.0/np.sqrt(2.0*problem['ndim_problem'])`.
Examples
--------
Use the optimizer `FEP` to minimize the well-known test function
`Rosenbrock <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy # engine for numerical computing
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.ep.fep import FEP
>>> 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,
... 'sigma': 3.0} # global step-size may need to be tuned
>>> fep = FEP(problem, options) # to initialize the optimizer class
>>> results = fep.optimize() # to run its optimization/evolution process
>>> # to return the number of function evaluations and the best-so-far fitness
>>> print(f"FEP: {results['n_function_evaluations']}, {results['best_so_far_y']}")
FEP: 5000, 0.005781004466936902
For its correctness checking, refer to `this code-based repeatability report
<https://tinyurl.com/bdh7epah>`_ for more details.
Attributes
----------
best_so_far_x : `array_like`
final best-so-far solution found during entire optimization.
best_so_far_y : `array_like`
final best-so-far fitness found during entire optimization.
n_individuals : `int`
number of offspring, aka offspring population size.
q : `int`
number of opponents for pairwise comparisons.
sigma : `float`
initial global step-size, aka mutation strength.
tau : `float`
self-adaptation learning rate of individual step-sizes.
tau_apostrophe : `float`
self-adaptation learning rate of individual step-sizes.
References
----------
Yao, X., Liu, Y. and Lin, G., 1999.
`Evolutionary programming made faster.
<https://ieeexplore.ieee.org/abstract/document/771163>`_
IEEE Transactions on Evolutionary Computation, 3(2), pp.82-102.
Chellapilla, K. and Fogel, D.B., 1999.
`Evolution, neural networks, games, and intelligence.
<https://ieeexplore.ieee.org/abstract/document/784222>`_
Proceedings of the IEEE, 87(9), pp.1471-1496.
Bäck, T. and Schwefel, H.P., 1993.
`An overview of evolutionary algorithms for parameter optimization.
<https://direct.mit.edu/evco/article-abstract/1/1/1/1092/An-Overview-of-Evolutionary-Algorithms-for>`_
Evolutionary Computation, 1(1), pp.1-23.
"""
def __init__(self, problem, options):
CEP.__init__(self, problem, options)
def iterate(self, x=None, sigmas=None, y=None, xx=None, ss=None, yy=None, args=None):
for i in range(self.n_individuals):
if self._check_terminations():
return x, sigmas, y, xx, ss, yy
ss[i] = sigmas[i]*np.exp(self.tau_apostrophe*self.rng_optimization.standard_normal(
size=(self.ndim_problem,)) + self.tau*self.rng_optimization.standard_normal(
size=(self.ndim_problem,)))
xx[i] = x[i] + ss[i]*self.rng_optimization.standard_cauchy(size=(self.ndim_problem,))
yy[i] = self._evaluate_fitness(xx[i], args)
new_x = np.vstack((xx, x))
new_sigmas = np.vstack((ss, sigmas))
new_y = np.hstack((yy, y))
n_win = np.zeros((2*self.n_individuals,)) # number of win
for i in range(2*self.n_individuals):
for j in self.rng_optimization.choice([k for k in range(2*self.n_individuals) if k != i],
size=self.q, replace=False):
if new_y[i] < new_y[j]:
n_win[i] += 1
order = np.argsort(-n_win)[:self.n_individuals]
x[:self.n_individuals] = new_x[order]
sigmas[:self.n_individuals] = new_sigmas[order]
y[:self.n_individuals] = new_y[order]
self._n_generations += 1
return x, sigmas, y, xx, ss, yy