import numpy as np # engine for numerical computing
from pypop7.optimizers.cem.cem import CEM
[docs]class SCEM(CEM):
"""Standard Cross-Entropy Method (SCEM).
.. note:: `SCEM` uses the *fixed* smoothing strategy to update the *mean* and *std* of Gaussian search
(mutation/sampling) distribution in an online fashion.
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']`.
* 'n_individuals' - offspring population size (`int`, default: `1000`),
* 'n_parents' - parent population size (`int`, default: `200`),
* 'alpha' - smoothing factor (`float`, default: `0.8`).
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.cem.scem import SCEM
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 100,
... 'lower_boundary': -5*numpy.ones((100,)),
... 'upper_boundary': 5*numpy.ones((100,))}
>>> options = {'max_function_evaluations': 1000000, # set optimizer options
... 'seed_rng': 2022,
... 'sigma': 0.3} # the global step-size may need to be tuned for better performance
>>> scem = SCEM(problem, options) # initialize the optimizer class
>>> results = scem.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"SCEM: {results['n_function_evaluations']}, {results['best_so_far_y']}")
SCEM: 1000000, 45712.10913791263
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/yc67n5bd>`_ for more details.
Attributes
----------
alpha : `float`
smoothing factor.
mean : `array_like`
initial (starting) point, aka mean of Gaussian search distribution.
n_individuals : `int`
number of offspring, aka offspring population size.
n_parents : `int`
number of parents, aka parental population size.
sigma : `float`
initial global step-size, aka mutation strength.
References
----------
Kroese, D.P., Porotsky, S. and Rubinstein, R.Y., 2006.
The cross-entropy method for continuous multi-extremal optimization.
Methodology and Computing in Applied Probability, 8(3), pp.383-407.
https://link.springer.com/article/10.1007/s11009-006-9753-0
(See [Appendix B Main CE Program] for the official Matlab code.)
De Boer, P.T., Kroese, D.P., Mannor, S. and Rubinstein, R.Y., 2005.
A tutorial on the cross-entropy method.
Annals of Operations Research, 134(1), pp.19-67.
https://link.springer.com/article/10.1007/s10479-005-5724-z
"""
def __init__(self, problem, options):
CEM.__init__(self, problem, options)
self.alpha = options.get('alpha', 0.8) # smoothing factor
assert 0.0 <= self.alpha <= 1.0
def initialize(self, is_restart=False):
mean = self._initialize_mean(is_restart)
x = np.empty((self.n_individuals, self.ndim_problem)) # samples (population)
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
return mean, x, y
def iterate(self, mean=None, x=None, y=None, args=None):
for i in range(self.n_individuals):
if self._check_terminations():
return x, y
x[i] = mean + self._sigmas*self.rng_optimization.standard_normal((self.ndim_problem,))
y[i] = self._evaluate_fitness(x[i], args)
return x, y
def _update_parameters(self, mean=None, x=None, y=None):
xx = x[np.argsort(y)[:self.n_parents]]
mean = self.alpha*np.mean(xx, axis=0) + (1.0-self.alpha)*mean
self._sigmas = self.alpha*np.std(xx, axis=0) + (1.0-self.alpha)*self._sigmas
return mean
def optimize(self, fitness_function=None, args=None):
fitness = CEM.optimize(self, fitness_function)
mean, x, y = self.initialize()
while True:
x, y = self.iterate(mean, x, y, args)
self._print_verbose_info(fitness, y)
if self._check_terminations():
break
self._n_generations += 1
mean = self._update_parameters(mean, x, y)
return self._collect(fitness, y, mean)