import numpy as np
from pypop7.optimizers.es.es import ES
class EMA(object): # exponential moving average
def __init__(self, ndim):
self.m = np.zeros((ndim,))
def update(self, d):
self.m = d
[docs]class VKDCMA(ES):
"""Projection-based Covariance Matrix Adaptation (VKDCMA).
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.vkdcma import VKDCMA
>>> 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
>>> vkdcma = VKDCMA(problem, options) # initialize the optimizer class
>>> results = vkdcma.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"VKDCMA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
VKDCMA: 5000, 1.7960753151742513e-13
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/mrxvx4hk>`_ for more details.
Attributes
----------
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`
final global step-size, aka mutation strength.
References
----------
Akimoto, Y. and Hansen, N., 2016, September.
Online model selection for restricted covariance matrix adaptation.
In Parallel Problem Solving from Nature. Springer International Publishing.
https://link.springer.com/chapter/10.1007/978-3-319-45823-6_1
Akimoto, Y. and Hansen, N., 2016, July.
Projection-based restricted covariance matrix adaptation for high dimension.
In Proceedings of Annual Genetic and Evolutionary Computation Conference 2016 (pp. 197-204). ACM.
https://dl.acm.org/doi/abs/10.1145/2908812.2908863
See the official Python version from Prof. Akimoto:
https://gist.github.com/youheiakimoto/2fb26c0ace43c22b8f19c7796e69e108
"""
def _get_m(self, d_x, d, v, s):
dx_d = d_x/d
v_dxd = np.dot(v[:self.k_a], dx_d)
return np.sum(dx_d*dx_d) - np.sum((v_dxd*v_dxd)*(s[:self.k_a]/(s[:self.k_a] + 1.0)))
def _get_lr(self, k):
c_1 = 2.0/(self.ndim_problem*(k + 1.0) + self.ndim_problem + 2.0*(k + 2.0) + self._mu_eff)
c_mu = min(1.0 - c_1, 2.0*(self._mu_eff - 2.0 + 1.0/self._mu_eff)/(
self.ndim_problem*(k + 1) + 4.0*(k + 2.0) + self._mu_eff))
return c_1, c_mu, np.sqrt(c_1)
def _get_det(self, d, s):
return 2.0*np.sum(np.log(d)) + np.sum(np.log(1.0 + s[:self.k_a]))
def __init__(self, problem, options):
ES.__init__(self, problem, options)
self.options = options
self.k, self.k_a, self.k_i = 0, 0, 0
self.c_s, self.d_s = 0.3, np.sqrt(self.ndim_problem)
self._injection = False
self.c_1, self.c_mu, self.c_c = None, None, None
self.l_s, self.l_d, self.l_c = None, None, None
self.e_s, self.e_d, self.e_c = None, None, None
def initialize(self, is_restart=False):
self.k, self.k_a, self.k_i = 0, 0, 0
self.l_s = np.log(self.sigma)
self.l_d = 2.0*np.log(np.ones((self.ndim_problem,)))
self.l_c = np.zeros((self.ndim_problem,))
self.e_s = EMA(1)
self.e_d = EMA(self.ndim_problem)
self.e_c = EMA(self.ndim_problem)
self.c_1, self.c_mu, self.c_c = self._get_lr(self.k)
self.c_s, self.d_s = 0.3, np.sqrt(self.ndim_problem)
self._injection = False
d = np.ones((self.ndim_problem,))
p_s = 0
v = np.zeros((self.k, self.ndim_problem))
s = np.zeros((self.ndim_problem,))
p_c = np.zeros((self.ndim_problem,))
d_x = np.zeros((self.ndim_problem,))
u = np.zeros((self.ndim_problem, self.k + self.n_parents + 1))
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
x = np.zeros((self.n_individuals, self.ndim_problem))
y = np.zeros((self.n_individuals,))
return d, p_s, v, s, p_c, d_x, u, mean, x, y
def iterate(self, d=None, p_s=None, v=None, s=None, p_c=None, d_x=None,
u=None, mean=None, x=None, y=None, args=None):
k, k_a = self.k, self.k_a
z = (self.rng_optimization.standard_normal((self.n_individuals, self.ndim_problem)) + np.dot(
self.rng_optimization.standard_normal((self.n_individuals, k_a))*np.sqrt(
s[:k_a]), v[:k_a]))*d
if self._injection:
zz = (np.linalg.norm(self.rng_optimization.standard_normal((self.ndim_problem,)))/np.sqrt(
self._get_m(d_x, d, v, s)))*d_x
z[0], z[1] = zz, -zz
x = mean + self.sigma*z
for i in range(self.n_individuals):
if self._check_terminations():
return d, p_s, v, s, p_c, d_x, u, mean, x, y
y[i] = self._evaluate_fitness(x[i], args)
order = np.argsort(y)
s_z = z[order[:self.n_parents]]
d_x = np.dot(self._w, s_z)
mean += self.sigma*d_x
if self._injection:
p_s += self.c_s*((np.where(order == 1)[0][0] - np.where(order == 0)[0][0])/(
self.n_individuals - 1.0) - p_s)
self.sigma *= np.exp(p_s/self.d_s)
h_s = p_s < 0.5
else:
self._injection, h_s = True, True
p_c = (1.0 - self.c_c)*p_c + h_s*np.sqrt(self.c_c*(2.0 - self.c_c)*self._mu_eff)*d_x
if self.c_mu == 0.0:
r_u = k_a + 1
alpha = np.sqrt(abs(1.0 - self.c_mu - self.c_1 + self.c_1*(
1.0 - h_s)*self.c_c*(2.0 - self.c_c)))
u[:, :k_a] = (v[:k_a].T*(np.sqrt(s[:k_a])*alpha))
u[:, r_u - 1] = np.sqrt(self.c_1)*(p_c/d)
elif self.c_1 == 0.0:
r_u = k_a + self.n_parents
alpha = np.sqrt(abs(1.0 - self.c_mu - self.c_1 + self.c_1*(
1.0 - h_s)*self.c_c*(2.0 - self.c_c)))
u[:, :k_a] = (v[:k_a].T*(np.sqrt(s[:k_a])*alpha))
u[:, k_a:r_u] = np.sqrt(self.c_mu)*np.sqrt(self._w)*(s_z/d).T
else:
r_u = k_a + self.n_parents + 1
alpha = np.sqrt(abs(1.0 - self.c_mu - self.c_1 + self.c_1*(
1.0 - h_s)*self.c_c*(2.0 - self.c_c)))
u[:, :k_a] = (v[:k_a].T*(np.sqrt(s[:k_a])*alpha))
u[:, k_a:r_u - 1] = np.sqrt(self.c_mu)*np.sqrt(self._w)*(s_z/d).T
u[:, r_u - 1] = np.sqrt(self.c_1)*(p_c/d)
if self.ndim_problem > r_u:
dd, right = np.linalg.eigh(np.dot(u[:, :r_u].T, u[:, :r_u]))
i_e = np.argsort(dd)[::-1]
gamma = 0 if r_u <= k else dd[i_e[k:]].sum()/(self.ndim_problem - k)
self.k_a = k_a = min(int(np.sum(dd >= 0)), k)
s[:k_a] = (dd[i_e[:k_a]] - gamma)/(alpha*alpha + gamma)
v[:k_a] = (np.dot(u[:, :r_u], right[:, i_e[:k_a]])/np.sqrt(dd[i_e[:k_a]])).T
else:
dd, left = np.linalg.eigh(np.dot(u[:, :r_u], u[:, :r_u].T))
i_e = np.argsort(dd)[::-1]
gamma = 0 if r_u <= k else dd[i_e[k:]].sum()/(self.ndim_problem - k)
self.k_a = k_a = min(int(np.sum(dd >= 0)), k)
s[:k_a] = (dd[i_e[:k_a]] - gamma)/(alpha*alpha + gamma)
v[:k_a] = left[:, i_e[:k_a]].T
d *= np.sqrt((alpha*alpha + np.sum(u[:, :r_u]*u[:, :r_u], axis=1))/(
1.0 + np.dot(s[:k_a], v[:k_a]*v[:k_a])))
e = np.exp(self._get_det(d, s)/self.ndim_problem/2.0)
d, p_c = d/e, p_c/e
self.k_i += 1
self.e_s.update(np.log(self.sigma) - self.l_s)
l_s_c = self.e_s.m/(0.5*min(1.0, self.n_individuals/self.ndim_problem)/3.0)
self.l_s = np.log(self.sigma)
self.e_d.update(2.0*np.log(d) + np.log(1.0 + np.dot(s[:self.k], v[:self.k]**2)) - self.l_d)
l_d_c = self.e_d.m/(self.c_mu + self.c_1)
self.l_d = 2.0*np.log(d) + np.log(1.0 + np.dot(s[:self.k], v[:self.k]**2))
self.e_c.update(np.log(1.0 + s) - self.l_c)
l_l_c = self.e_c.m/(self.c_mu + self.c_1)
self.l_c = np.log(1.0 + s)
k_i = (self.k_i > (2.0*self.ndim_problem - 1.0))*(self.k < (self.ndim_problem - 1.0))*np.all(
(1.0 + s[:self.k]) > 30.0)*(np.abs(l_s_c) < 0.1)*np.all(np.abs(l_d_c) < 1.0)
k_d = (self.k > 0)*(1.0 + s[:self.k] < 30.0)*(l_l_c[:self.k] < 0.0)
if (self.k_i > (2*self.ndim_problem - 1)) and k_i:
self.k_a = k
self.k = new_k = min(max(int(np.ceil(self.k*1.414)), self.k + 1), (self.ndim_problem - 1))
v = np.vstack((v, np.zeros((new_k - k, self.ndim_problem))))
u = np.empty((self.ndim_problem, new_k + self.n_parents + 1))
(self.c_1, self.c_mu, self.c_c) = self._get_lr(self.k)
self.k_i = 0
elif self.k_i > k*(2*self.ndim_problem - 1) and np.any(k_d):
keep = np.logical_not(k_d)
new_k = max(np.count_nonzero(keep), 0)
v = v[keep]
s[:new_k] = (s[:keep.shape[0]])[keep]
s[new_k:] = 0
self.k = self.k_a = new_k
(self.c_1, self.c_mu, self.c_c) = self._get_lr(self.k)
e = np.exp(self._get_det(d, s)/self.ndim_problem/2.0)
d, p_c = d/e, p_c/e
return d, p_s, v, s, p_c, d_x, u, mean, x, y
def restart_reinitialize(self, d=None, p_s=None, v=None, s=None, p_c=None,
d_x=None, u=None, mean=None, x=None, y=None):
if self.is_restart and ES.restart_reinitialize(self, y):
d, p_s, v, s, p_c, d_x, u, mean, x, y = self.initialize(True)
return d, p_s, v, s, p_c, d_x, u, mean, x, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
d, p_s, v, s, p_c, d_x, u, mean, x, y = self.initialize()
while not self.termination_signal:
d, p_s, v, s, p_c, d_x, u, mean, x, y = self.iterate(d, p_s, v, s, p_c, d_x, u, mean, x, y, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
self._n_generations += 1
d, p_s, v, s, p_c, d_x, u, mean, x, y = self.restart_reinitialize(d, p_s, v, s, p_c, d_x, u, mean, x, y)
results = self._collect(fitness, y, mean)
results['d'] = d
return results