import numpy as np # engine for numerical computing
from scipy.stats import norm
from pypop7.optimizers.es.es import ES
[docs]class MMES(ES):
"""Mixture Model-based Evolution Strategy (MMES).
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']`.
* 'm' - number of candidate direction vectors (`int`, default:
`2*int(np.ceil(np.sqrt(problem['ndim_problem'])))`),
* 'c_c' - learning rate of evolution path update (`float`, default:
`0.4/np.sqrt(problem['ndim_problem'])`),
* 'ms' - mixing strength (`int`, default: `4`),
* 'c_s' - learning rate of global step-size adaptation (`float`, default: `0.3`),
* 'a_z' - target significance level (`float`, default: `0.05`),
* 'distance' - minimal distance of updating evolution paths (`int`, default:
`int(np.ceil(1.0/options['c_c']))`),
* '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 # engine for numerical computing
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.es.mmes import MMES
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 200,
... 'lower_boundary': -5*numpy.ones((200,)),
... 'upper_boundary': 5*numpy.ones((200,))}
>>> options = {'max_function_evaluations': 500000, # set optimizer options
... 'seed_rng': 2022,
... 'mean': 3*numpy.ones((200,)),
... 'sigma': 0.1} # the global step-size may need to be tuned for better performance
>>> mmes = MMES(problem, options) # initialize the optimizer class
>>> results = mmes.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"MMES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
MMES: 500000, 7.350414979801825
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/3ym72w5m>`_ for more details.
Attributes
----------
a_z : `float`
target significance level.
c_c : `float`
learning rate of evolution path update.
c_s : `float`
learning rate of global step-size adaptation.
distance : `int`
minimal distance of updating evolution paths.
m : `int`
number of candidate direction vectors.
mean : `array_like`
initial (starting) point, aka mean of Gaussian search distribution.
ms : `int`
mixing strength.
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
----------
He, X., Zheng, Z. and Zhou, Y., 2021.
MMES: Mixture model-based evolution strategy for large-scale optimization.
IEEE Transactions on Evolutionary Computation, 25(2), pp.320-333.
https://ieeexplore.ieee.org/abstract/document/9244595
See the official Matlab version from He:
https://github.com/hxyokokok/MMES
"""
def __init__(self, problem, options):
ES.__init__(self, problem, options)
# set number of candidate direction vectors
self.m = options.get('m', 2*int(np.ceil(np.sqrt(self.ndim_problem))))
assert self.m > 0
# set learning rate of evolution path
self.c_c = options.get('c_c', 0.4/np.sqrt(self.ndim_problem))
self.ms = options.get('ms', 4) # mixing strength (l)
assert self.ms > 0
# set for paired test adaptation (PTA)
self.c_s = options.get('c_s', 0.3) # learning rate of global step-size adaptation
self.a_z = options.get('a_z', 0.05) # target significance level
# set minimal distance of updating evolution paths (T)
self.distance = options.get('distance', int(np.ceil(1.0/self.c_c)))
# set success probability of geometric distribution (different from 4/n in the original paper)
self.c_a = options.get('c_a', 3.8/self.ndim_problem) # same as the official Matlab code
self.gamma = options.get('gamma', 1.0 - np.power(1.0 - self.c_a, self.m))
self._n_mirror_sampling = None
self._z_1 = np.sqrt(1.0 - self.gamma)
self._z_2 = np.sqrt(self.gamma/self.ms)
self._p_1 = 1.0 - self.c_c
self._p_2 = np.sqrt(self.c_c*(2.0 - self.c_c))
self._w_1 = 1.0 - self.c_s
self._w_2 = np.sqrt(self.c_s*(2.0 - self.c_s))
def initialize(self, args=None, is_restart=False):
self._n_mirror_sampling = int(np.ceil(self.n_individuals/2))
x = np.zeros((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
p = np.zeros((self.ndim_problem,)) # evolution path
w = 0.0
q = np.zeros((self.m, self.ndim_problem)) # candidate direction vectors
t = np.zeros((self.m,)) # recorded generations
v = np.arange(self.m) # indexes to evolution paths
y = np.tile(self._evaluate_fitness(mean, args), (self.n_individuals,)) # fitness
return x, mean, p, w, q, t, v, y
def iterate(self, x=None, mean=None, q=None, v=None, y=None, args=None):
for k in range(self._n_mirror_sampling): # mirror sampling
zq = np.zeros((self.ndim_problem,))
for _ in range(self.ms):
j_k = v[(self.m - self.rng_optimization.geometric(self.c_a) % self.m) - 1]
zq += self.rng_optimization.standard_normal()*q[j_k]
z = self._z_1*self.rng_optimization.standard_normal((self.ndim_problem,))
z += self._z_2*zq
x[k] = mean + self.sigma*z
if (self._n_mirror_sampling + k) < self.n_individuals:
x[self._n_mirror_sampling + k] = mean - self.sigma*z
for k in range(self.n_individuals):
if self._check_terminations():
return x, y
y[k] = self._evaluate_fitness(x[k], args)
return x, y
def _update_distribution(self, x=None, mean=None, p=None, w=None, q=None,
t=None, v=None, y=None, y_bak=None):
order = np.argsort(y)[:self.n_parents]
y.sort()
mean_w = np.dot(self._w[:self.n_parents], x[order])
p = self._p_1*p + self._p_2*np.sqrt(self._mu_eff)*(mean_w - mean)/self.sigma
mean = mean_w
if self._n_generations < self.m:
q[self._n_generations] = p
else:
k_star = np.argmin(t[v[1:]] - t[v[:(self.m - 1)]])
k_star += 1
if t[v[k_star]] - t[v[k_star - 1]] > self.distance:
k_star = 0
v = np.append(np.append(v[:k_star], v[(k_star + 1):]), v[k_star])
t[v[-1]], q[v[-1]] = self._n_generations, p
# conduct success-based mutation strength adaptation
l_w = np.dot(self._w, y_bak[:self.n_parents] > y[:self.n_parents])
w = self._w_1*w + self._w_2*np.sqrt(self._mu_eff)*(2*l_w - 1)
self.sigma *= np.exp(norm.cdf(w) - 1.0 + self.a_z)
return mean, p, w, q, t, v
def restart_reinitialize(self, args=None, x=None, mean=None, p=None, w=None, q=None,
t=None, v=None, y=None, fitness=None):
if self.is_restart and ES.restart_reinitialize(self, y):
x, mean, p, w, q, t, v, y = self.initialize(args, True)
self._print_verbose_info(fitness, y[0])
return x, mean, p, w, q, t, v, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
x, mean, p, w, q, t, v, y = self.initialize(args)
self._print_verbose_info(fitness, y[0])
while not self.termination_signal:
y_bak = np.copy(y)
# sample and evaluate offspring population
x, y = self.iterate(x, mean, q, v, y, args)
if self._check_terminations():
break
mean, p, w, q, t, v = self._update_distribution(x, mean, p, w, q, t, v, y, y_bak)
self._n_generations += 1
self._print_verbose_info(fitness, y)
x, mean, p, w, q, t, v, y = self.restart_reinitialize(
args, x, mean, p, w, q, t, v, y, fitness)
results = self._collect(fitness, y, mean)
results['p'] = p
results['w'] = w
return results