import numpy as np
import numba as nb
from pypop7.optimizers.es.es import ES
@nb.jit(nopython=True)
def cholesky_update(rm, z, downdate):
# https://github.com/scipy/scipy/blob/d20f92fce9f1956bfa65365feeec39621a071932/
# scipy/linalg/_decomp_cholesky_update.py
rm, z, alpha, beta = rm.T, z, np.empty_like(z), np.empty_like(z)
alpha[-1], beta[-1] = 1.0, 1.0
sign = -1 if downdate else 1
for r in range(len(z)):
a = z[r]/rm[r, r]
alpha[r] = alpha[r - 1] + sign*np.power(a, 2)
beta[r] = np.sqrt(alpha[r])
z[r + 1:] -= a*rm[r, r + 1:]
rm[r, r:] *= beta[r]/beta[r - 1]
rm[r, r + 1:] += sign*a/(beta[r]*beta[r - 1])*z[r + 1:]
return rm.T
[docs]class OPOA2015(ES):
"""(1+1)-Active-CMA-ES 2015 (OPOA2015).
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 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.opoa2015 import OPOA2015
>>> 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
>>> opoa2015 = OPOA2015(problem, options) # initialize the optimizer class
>>> results = opoa2015.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"OPOA2015: {results['n_function_evaluations']}, {results['best_so_far_y']}")
OPOA2015: 5000, 5.955151843487958e-17
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/mrxu4suj>`_ for more details.
References
----------
Krause, O. and Igel, C., 2015, January.
A more efficient rank-one covariance matrix update for evolution strategies.
In Proceedings of ACM Conference on Foundations of Genetic Algorithms (pp. 129-136).
https://dl.acm.org/doi/abs/10.1145/2725494.2725496
"""
def __init__(self, problem, options):
options['n_individuals'] = 1 # mandatory setting
options['n_parents'] = 1 # mandatory setting
ES.__init__(self, problem, options)
if self.lr_sigma is None:
self.lr_sigma = 1.0/(1.0 + self.ndim_problem/2.0)
self.p_ts = options.get('p_ts', 2.0/11.0)
self.c_p = options.get('c_p', 1.0/12.0)
self.c_c = options.get('c_c', 2.0/(self.ndim_problem + 2.0))
self.c_cov = options.get('c_cov', 2.0/(np.power(self.ndim_problem, 2) + 6.0))
self.p_t = options.get('p_t', 0.44)
self.c_m = options.get('c_m', 0.4/(np.power(self.ndim_problem, 1.6) + 1.0))
self.k = options.get('k', 5)
self._ancestors = []
self._c_cf = 1.0 - self.c_cov + self.c_cov*self.c_c*(2.0 - self.c_c)
def initialize(self, args=None, is_restart=False):
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
y = self._evaluate_fitness(mean, args) # fitness
cf = np.diag(np.ones(self.ndim_problem,)) # Cholesky factorization
best_so_far_y, p_s = np.copy(y), self.p_ts
p_c = np.zeros((self.ndim_problem,)) # evolution path
return mean, y, cf, best_so_far_y, p_s, p_c
def _cholesky_update(self, cf=None, alpha=None, beta=None, v=None): # triangular rank-one update
assert self.ndim_problem == v.size
if beta < 0:
downdate, beta = True, -beta
else:
downdate = False
return cholesky_update(np.sqrt(alpha)*cf, np.sqrt(beta)*v, downdate)
def iterate(self, mean=None, cf=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,))
cf_z = np.dot(cf, z)
x = mean + self.sigma*cf_z
y = self._evaluate_fitness(x, args)
if y <= best_so_far_y:
self._ancestors.append(y)
mean, best_so_far_y = x, y
p_s = (1.0 - self.c_p)*p_s + self.c_p
is_better = True
else:
p_s *= 1.0 - self.c_p
is_better = False
self.sigma *= np.exp(self.lr_sigma*(p_s - self.p_ts)/(1.0 - self.p_ts))
if p_s >= self.p_t:
p_c *= 1.0 - self.c_c
cf = self._cholesky_update(cf, self._c_cf, self.c_cov, p_c)
elif is_better:
p_c = (1.0 - self.c_c)*p_c + np.sqrt(self.c_c*(2.0 - self.c_c))*cf_z
cf = self._cholesky_update(cf, 1.0 - self.c_cov, self.c_cov, p_c)
elif len(self._ancestors) >= self.k and y > self._ancestors[-self.k]:
del self._ancestors[0]
c_m = np.minimum(self.c_m, 1.0/(2.0*np.dot(z, z) - 1.0))
cf = self._cholesky_update(cf, 1.0 + c_m, -c_m, cf_z)
self._n_generations += 1
return mean, y, cf, best_so_far_y, p_s, p_c
def restart_reinitialize(self, mean=None, y=None, cf=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 self.is_restart and 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, cf, best_so_far_y, p_s, p_c = self.initialize(args, True)
self._list_fitness = [best_so_far_y]
self._ancestors = []
return mean, y, cf, 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, cf, best_so_far_y, p_s, p_c = self.initialize(args)
while not self._check_terminations():
self._print_verbose_info(fitness, y)
mean, y, cf, best_so_far_y, p_s, p_c = self.iterate(
mean, cf, best_so_far_y, p_s, p_c, args)
mean, y, cf, best_so_far_y, p_s, p_c = self.restart_reinitialize(
mean, y, cf, best_so_far_y, p_s, p_c, fitness, args)
return self._collect(fitness, y, mean)