import numpy as np # engine for numerical computing
from pypop7.optimizers.nes.nes import NES
from pypop7.optimizers.nes.sges import SGES
[docs]class ONES(SGES):
"""Original Natural Evolution Strategy (ONES).
.. note:: `NES` constitutes a **well-principled** approach to real-valued black box function optimization with
a relatively clean derivation **from first principles**. Here we include `ONES` **mainly** for *benchmarking*
and *theoretical* purpose.
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`),
* 'lr_mean' - learning rate of distribution mean update (`float`, default: `1.0`),
* 'lr_sigma' - learning rate of global step-size adaptation (`float`, default: `1.0`).
Examples
--------
Use the optimizer `ONES` 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.ones import ONES
>>> 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
>>> ones = ONES(problem, options) # initialize the optimizer class
>>> results = ones.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"ONES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
ONES: 5000, 4.08973753355584e-05
Attributes
----------
lr_mean : `float`
learning rate of distribution mean update.
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
See the official Python source code from PyBrain:
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/distributionbased/nes.py
"""
def __init__(self, problem, options):
SGES.__init__(self, problem, options)
if options.get('lr_mean') is None:
self.lr_mean = 1.0
if options.get('lr_sigma') is None:
self.lr_sigma = 1.0
def _update_distribution(self, x=None, y=None, mean=None, cv=None):
order = np.argsort(-y)
u = np.empty((self.n_individuals,))
for i, o in enumerate(order):
u[o] = self._u[i]
inv_cv = np.linalg.inv(cv)
phi = np.ones((self.n_individuals, self._n_distribution + 1))
for k in range(self.n_individuals):
diff = x[k] - mean
phi[k, :self.ndim_problem] = np.dot(inv_cv, diff)
_grad_cv = 0.5*(np.dot(np.dot(inv_cv, np.outer(diff, diff)), inv_cv) - inv_cv)
phi[k, self.ndim_problem:-1] = self._triu2flat(np.dot(self._d_cv, _grad_cv + _grad_cv.T))
grad = np.dot(np.linalg.pinv(phi), u)[:-1]
mean += self.lr_mean*grad[:self.ndim_problem]
self._d_cv += self.lr_sigma*self._flat2triu(grad[self.ndim_problem:])
cv = np.dot(self._d_cv.T, self._d_cv)
self._n_generations += 1
return x, y, mean, cv
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = NES.optimize(self, fitness_function)
x, y, mean, cv = self.initialize()
while True:
x, y = self.iterate(x, y, mean, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
x, y, mean, cv = self._update_distribution(x, y, mean, cv)
x, y, mean, cv = self.restart_reinitialize(x, y, mean, cv)
return self._collect(fitness, y, mean)