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xnes.py
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xnes.py
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import numpy as np # engine for numerical computing
import scipy
from pypop7.optimizers.nes.nes import NES
class XNES(NES):
"""Exponential Natural Evolution Strategies (XNES).
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`):
* 'n_individuals' - number of offspring/descendants, aka offspring population size (`int`),
* 'n_parents' - number of parents/ancestors, aka parental population size (`int`),
* 'mean' - initial (starting) point (`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']`.
* 'sigma' - initial global step-size, aka mutation strength (`float`).
Examples
--------
Use the optimizer `XNES` 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.nes.xnes import XNES
>>> 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
>>> xnes = XNES(problem, options) # initialize the optimizer class
>>> results = xnes.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"XNES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
XNES: 5000, 1.3565728021697798e-18
Attributes
----------
lr_cv : `float`
learning rate of covariance matrix adaptation.
lr_sigma : `float`
learning rate of global step-size adaptation.
mean : `array_like`
initial (starting) point, aka mean of Gaussian search/sampling/mutation distribution.
n_individuals : `int`
number of offspring/descendants, aka offspring population size.
n_parents : `int`
number of parents/ancestors, aka parental population size.
sigma : `float`
global step-size, aka mutation strength (i.e., overall std of Gaussian search distribution).
References
----------
Wierstra, D., Schaul, T., Glasmachers, T., Sun, Y., Peters, J. and Schmidhuber, J., 2014.
Natural evolution strategies.
Journal of Machine Learning Research, 15(1), pp.949-980.
https://jmlr.org/papers/v15/wierstra14a.html
Schaul, T., 2011.
Studies in continuous black-box optimization.
Doctoral Dissertation, Technische Universität München.
https://people.idsia.ch/~schaul/publications/thesis.pdf
Glasmachers, T., Schaul, T., Yi, S., Wierstra, D. and Schmidhuber, J., 2010, July.
Exponential natural evolution strategies.
In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 393-400).
https://dl.acm.org/doi/abs/10.1145/1830483.1830557
See the official Python source code from PyBrain:
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/distributionbased/xnes.py
"""
def __init__(self, problem, options):
options['sigma'] = np.Inf # not used for `SGES`
NES.__init__(self, problem, options)
self.lr_cv = 0.6*(3.0 + np.log(self.ndim_problem))/self.ndim_problem/np.sqrt(self.ndim_problem)
self.lr_sigma = 0.6*(3.0 + np.log(self.ndim_problem))/self.ndim_problem/np.sqrt(self.ndim_problem)
self._eye = np.eye(self.ndim_problem)
def initialize(self, is_restart=False):
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
a = np.eye(self.ndim_problem)
inv_a = np.eye(self.ndim_problem)
log_det = 0.0
self._w = np.maximum(0.0, np.log(self.n_individuals/2.0 + 1.0) - np.log(
self.n_individuals - np.arange(self.n_individuals)))
return x, y, mean, a, inv_a, log_det
def iterate(self, x=None, y=None, mean=None, a=None, args=None):
for k in range(self.n_individuals):
if self._check_terminations():
return x, y
x[k] = mean + 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, y=None, mean=None, a=None, inv_a=None, log_det=None):
order = np.argsort(-y)
u = np.empty((self.n_individuals,))
for i, o in enumerate(order):
u[o] = self._w[i]
u = u/np.sum(u) - 1.0/self.n_individuals
s = np.empty((self.n_individuals, self.ndim_problem))
for k in range(self.n_individuals):
s[k] = np.dot(inv_a, x[k] - mean)
d_c = np.dot(s.T, u)
g_cv = np.dot(np.array([np.outer(k, k) - self._eye for k in s]).T, u)
trace = np.trace(g_cv)
g_cv -= trace/self.ndim_problem*self._eye
d_a = 0.5*(self.lr_sigma*trace/self.ndim_problem*self._eye + self.lr_cv*g_cv)
mean += np.dot(a, d_c)
a = np.dot(a, scipy.linalg.expm(d_a))
inv_a = np.dot(scipy.linalg.expm(-d_a), inv_a)
log_det += 0.5*self.lr_sigma*trace
return mean, a, inv_a, log_det
def restart_reinitialize(self, x=None, y=None, mean=None, a=None, inv_a=None, log_det=None):
if self.is_restart and NES.restart_reinitialize(self, y):
x, y, mean, a, inv_a, log_det = self.initialize(True)
return x, y, mean, a, inv_a, log_det
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = NES.optimize(self, fitness_function)
x, y, mean, a, inv_a, log_det = self.initialize()
while True:
x, y = self.iterate(x, y, mean, a, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
mean, a, inv_a, log_det = self._update_distribution(x, y, mean, a, inv_a, log_det)
self._n_generations += 1
x, y, mean, a, inv_a, log_det = self.restart_reinitialize(x, y, mean, a, inv_a, log_det)
return self._collect(fitness, y, mean)