import numpy as np
from scipy.linalg import solve_triangular
from pypop7.optimizers.es.es import ES
from pypop7.optimizers.es.opoa2015 import cholesky_update
[docs]class CCMAES2016(ES):
"""Cholesky-CMA-ES 2016 (CCMAES2016).
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' - number of offspring, aka offspring population size (`int`, default:
`4 + int(3*np.log(problem['ndim_problem']))`),
* 'n_parents' - number of parents, aka parental population size (`int`, default:
`int(options['n_individuals']/2)`).
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.ccmaes2016 import CCMAES2016
>>> 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
>>> ccmaes2016 = CCMAES2016(problem, options) # initialize the optimizer class
>>> results = ccmaes2016.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"CCMAES2016: {results['n_function_evaluations']}, {results['best_so_far_y']}")
CCMAES2016: 5000, 2.614231350522262e-21
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/3kbet9tn>`_ for more details.
References
----------
Krause, O., Arbonès, D.R. and Igel, C., 2016.
CMA-ES with optimal covariance update and storage complexity.
Advances in Neural Information Processing Systems, 29, pp.370-378.
https://proceedings.neurips.cc/paper/2016/hash/289dff07669d7a23de0ef88d2f7129e7-Abstract.html
"""
def __init__(self, problem, options):
ES.__init__(self, problem, options)
self.options = options
self.c_s, self.d = None, None
self.c_c, self.c_1, self.c_mu = None, None, None
def initialize(self, is_restart=False):
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
a = np.diag(np.ones((self.ndim_problem,))) # cholesky factor
p_s = np.zeros((self.ndim_problem,)) # evolution path for CSA
p_c = np.zeros((self.ndim_problem,)) # evolution path for CMA
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
self.c_s = self.options.get('c_s', self._mu_eff/(self.ndim_problem + self._mu_eff))
self.d = self.options.get('d', 1.0 + np.sqrt(self._mu_eff/self.ndim_problem))
self.c_c = self.options.get('c_c', (4.0 + self._mu_eff/self.ndim_problem)/(
self.ndim_problem + 4.0 + 2.0*self._mu_eff/self.ndim_problem))
self.c_1 = self.options.get('c_1', 2.0/(np.square(self.ndim_problem) + self._mu_eff))
self.c_mu = self.options.get('c_mu', self._mu_eff/(np.square(self.ndim_problem) + self._mu_eff))
return x, mean, a, p_s, p_c, y
def iterate(self, x=None, mean=None, a=None, y=None, args=None):
for k in range(self.n_individuals):
if self._check_terminations():
return x, y
x[k] = mean + self.sigma*np.dot(a, self.rng_optimization.standard_normal((self.ndim_problem,)))
y[k] = self._evaluate_fitness(x[k], args)
return x, y
def _update_distribution(self, x=None, mean=None, a=None, p_s=None, p_c=None, y=None):
order = np.argsort(y)[:self.n_parents]
mean_bak = np.dot(self._w, x[order])
mean_diff = (mean_bak - mean)/self.sigma
p_c = (1.0 - self.c_c)*p_c + np.sqrt(self.c_c*(2.0 - self.c_c)*self._mu_eff)*mean_diff
p_s = (1.0 - self.c_s)*p_s + np.sqrt(self.c_s*(2.0 - self.c_s)*self._mu_eff)*solve_triangular(
a, mean_diff, lower=True)
a *= np.sqrt(1.0 - self.c_1 - self.c_mu)
a = cholesky_update(a, np.sqrt(self.c_1)*p_c, False)
for i in range(self.n_parents):
a = cholesky_update(a, np.sqrt(self.c_mu*self._w[i])*(x[order[i]] - mean)/self.sigma, False)
self.sigma *= np.exp(self.c_s/self.d*(np.sqrt(np.dot(p_s, p_s))/self._e_chi - 1.0))
return mean_bak, a, p_s, p_c
def restart_reinitialize(self, x=None, mean=None, a=None, p_s=None, p_c=None, y=None):
if self.is_restart and ES.restart_reinitialize(self, y):
x, mean, a, p_s, p_c, y = self.initialize(True)
return x, mean, a, p_s, p_c, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
x, mean, a, p_s, p_c, y = self.initialize()
while not self.termination_signal:
# sample and evaluate offspring population
x, y = self.iterate(x, mean, a, y, args)
if self._check_terminations():
break
mean, a, p_s, p_c = self._update_distribution(x, mean, a, p_s, p_c, y)
self._print_verbose_info(fitness, y)
self._n_generations += 1
x, mean, a, p_s, p_c, y = self.restart_reinitialize(x, mean, a, p_s, p_c, y)
results = self._collect(fitness, y, mean)
results['p_s'] = p_s
results['p_c'] = p_c
return results