import numpy as np
from pypop7.optimizers.es.es import ES
from pypop7.optimizers.es.opoc2006 import OPOC2006
[docs]class OPOC2009(OPOC2006):
"""(1+1)-Cholesky-CMA-ES 2009 (OPOC2009).
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']`.
Examples
--------
Use the optimizer `OPOC2009` to minimize the well-known test function
`Rosenbrock <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.es.opoc2009 import OPOC2009
>>> 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,
... 'mean': 3*numpy.ones((2,)),
... 'sigma': 0.1} # the global step-size may need to be tuned for better performance
>>> opoc2009 = OPOC2009(problem, options) # initialize the optimizer class
>>> results = opoc2009.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"OPOC2009: {results['n_function_evaluations']}, {results['best_so_far_y']}")
OPOC2009: 5000, 2.7686802211556655e-17
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/3ba4ctny>`_ for more details.
References
----------
Suttorp, T., Hansen, N. and Igel, C., 2009.
Efficient covariance matrix update for variable metric evolution strategies.
Machine Learning, 75(2), pp.167-197.
https://link.springer.com/article/10.1007/s10994-009-5102-1
(See Algorithm 2 for details.)
"""
def __init__(self, problem, options):
OPOC2006.__init__(self, problem, options)
def initialize(self, args=None, is_restart=False):
mean, y, a, best_so_far_y, p_s = OPOC2006.initialize(self, args, is_restart)
p_c = np.zeros((self.ndim_problem,)) # evolution path
a_i = np.diag(np.ones((self.ndim_problem,))) # inverse of Cholesky factors
return mean, y, a, a_i, best_so_far_y, p_s, p_c
def iterate(self, mean=None, a=None, a_i=None, best_so_far_y=None, p_s=None, p_c=None, args=None):
# sample and evaluate only one offspring
z = self.rng_optimization.standard_normal((self.ndim_problem,))
x = mean + self.sigma*np.dot(a, z)
y = self._evaluate_fitness(x, args)
l_s = 1 if y <= best_so_far_y else 0
p_s = (1.0 - self.c_p)*p_s + self.c_p*l_s
self.sigma *= np.exp(self.lr_sigma*(p_s - self.p_ts)/(1.0 - self.p_ts))
if y <= best_so_far_y:
mean, best_so_far_y = x, y
if p_s < self.p_t:
p_c = (1.0 - self.c_c)*p_c + np.sqrt(self.c_c*(2.0 - self.c_c))*np.dot(a, z)
alpha = 1.0 - self.c_cov
else:
p_c *= 1.0 - self.c_c
alpha = 1.0 - self.c_cov + self.c_cov*self.c_c*(2.0 - self.c_c)
beta = self.c_cov
w = np.dot(a_i, p_c)
w_norm = np.power(np.linalg.norm(w), 2)
s_w_norm = np.sqrt(1.0 + beta/alpha*w_norm)
a = np.sqrt(alpha)*a + np.sqrt(alpha)/w_norm*(s_w_norm - 1.0)*np.dot(
p_c[:, np.newaxis], w[np.newaxis, :])
a_i = 1.0/np.sqrt(alpha)*a_i - 1.0/(np.sqrt(alpha)*w_norm)*(1.0 - 1.0/s_w_norm)*np.dot(
w[:, np.newaxis], np.dot(w[np.newaxis, :], a_i))
return mean, y, a, a_i, best_so_far_y, p_s, p_c
def restart_reinitialize(self, mean=None, y=None, a=None, a_i=None, best_so_far_y=None,
p_s=None, p_c=None, fitness=None, args=None):
self._list_fitness.append(best_so_far_y)
is_restart_1, is_restart_2 = self.sigma < self.sigma_threshold, False
if len(self._list_fitness) >= self.stagnation:
is_restart_2 = (self._list_fitness[-self.stagnation] - self._list_fitness[-1]) < self.fitness_diff
is_restart = bool(is_restart_1) or bool(is_restart_2)
if is_restart:
self._print_verbose_info(fitness, y, True)
if self.verbose:
print(' ....... *** restart *** .......')
self._n_restart += 1
self._list_generations.append(self._n_generations) # for each restart
self._n_generations = 0
self.sigma = np.copy(self._sigma_bak)
mean, y, a, a_i, best_so_far_y, p_s, p_c = self.initialize(args, True)
self._list_fitness = [best_so_far_y]
return mean, y, a, a_i, best_so_far_y, p_s, p_c
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
mean, y, a, a_i, best_so_far_y, p_s, p_c = self.initialize(args)
while not self._check_terminations():
self._print_verbose_info(fitness, y)
mean, y, a, a_i, best_so_far_y, p_s, p_c = self.iterate(
mean, a, a_i, best_so_far_y, p_s, p_c, args)
self._n_generations += 1
if self.is_restart:
mean, y, a, a_i, best_so_far_y, p_s, p_c = self.restart_reinitialize(
mean, y, a, a_i, best_so_far_y, p_s, p_c, fitness, args)
return self._collect(fitness, y, mean)