import numpy as np # engine for numerical computing
from pypop7.optimizers.ep.ep import EP
[docs]class CEP(EP):
"""Classical Evolutionary Programming with self-adaptive mutation (CEP).
.. note:: To obtain satisfactory performance for large-scale black-box optimization, the number of
offspring (`n_individuals`) and also initial global step-size (`sigma`) may need to be **carefully**
tuned (e.g. via manual trial-and-error or automatical hyper-parameter optimization).
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 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.cep import CEP
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 2,
... 'lower_boundary': -5*numpy.ones((2,)),
... 'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000, # set optimizer options
... 'seed_rng': 2022,
... 'sigma': 0.1}
>>> cep = CEP(problem, options) # initialize the optimizer class
>>> results = cep.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"CEP: {results['n_function_evaluations']}, {results['best_so_far_y']}")
CEP: 5000, 0.3544823323771589
For its correctness checking, refer to `this code-based repeatability report
<https://tinyurl.com/b9vpmfdv>`_ for more details.
Attributes
----------
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`
learning rate of individual step-sizes self-adaptation.
tau_apostrophe : `float`
learning rate of individual step-sizes self-adaptation.
References
----------
Yao, X., Liu, Y. and Lin, G., 1999.
Evolutionary programming made faster.
IEEE Transactions on Evolutionary Computation, 3(2), pp.82-102.
https://ieeexplore.ieee.org/abstract/document/771163
Bäck, T. and Schwefel, H.P., 1993.
An overview of evolutionary algorithms for parameter optimization.
Evolutionary Computation, 1(1), pp.1-23.
https://direct.mit.edu/evco/article-abstract/1/1/1/1092/An-Overview-of-Evolutionary-Algorithms-for
"""
def __init__(self, problem, options):
EP.__init__(self, problem, options)
self.q = options.get('q', 10) # number of opponents for pairwise comparisons
# set two learning-rates of individual step-sizes adaptation
self.tau = options.get('tau', 1.0/np.sqrt(2.0*np.sqrt(self.ndim_problem)))
self.tau_apostrophe = options.get('tau_apostrophe', 1.0/np.sqrt(2.0*self.ndim_problem))
def initialize(self, args=None):
x = self.rng_initialization.uniform(self.initial_lower_boundary, self.initial_upper_boundary,
size=(self.n_individuals, self.ndim_problem))
sigmas = self.sigma*np.ones((self.n_individuals, self.ndim_problem)) # eta
y = np.empty((self.n_individuals,))
for i in range(self.n_individuals):
if self._check_terminations():
break
y[i] = self._evaluate_fitness(x[i], args)
xx = np.empty((self.n_individuals, self.ndim_problem))
ss = np.empty((self.n_individuals, self.ndim_problem)) # eta
yy = np.copy(y)
return x, sigmas, y, xx, ss, yy
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
# base = self.rng_optimization.standard_normal()
# ss[i] = sigmas[i]*np.exp(self.tau_apostrophe*base + self.tau*self.rng_optimization.standard_normal(
# size=(self.ndim_problem,)))
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_normal(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
def optimize(self, fitness_function=None, args=None):
fitness = EP.optimize(self, fitness_function)
x, sigmas, y, xx, ss, yy = self.initialize(args)
while True:
self._print_verbose_info(fitness, yy)
x, sigmas, y, xx, ss, yy = self.iterate(x, sigmas, y, xx, ss, yy, args)
if self._check_terminations():
break
return self._collect(fitness, yy)