import numpy as np # engine for numerical computing
from pypop7.optimizers.nes.nes import NES
[docs]class R1NES(NES):
"""Rank-One Natural Evolution Strategies (R1NES).
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 `R1NES` 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.r1nes import R1NES
>>> 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
>>> r1nes = R1NES(problem, options) # initialize the optimizer class
>>> results = r1nes.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"R1NES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
R1NES: 5000, 0.005172532562628031
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.
<https://jmlr.org/papers/v15/wierstra14a.html>`_
Journal of Machine Learning Research, 15(1), pp.949-980.
Schaul, T., 2011.
`Studies in continuous black-box optimization.
<https://people.idsia.ch/~schaul/publications/thesis.pdf>`_
Doctoral Dissertation, Technische Universität München.
Schaul, T., Glasmachers, T. and Schmidhuber, J., 2011, July.
`High dimensions and heavy tails for natural evolution strategies.
<https://dl.acm.org/doi/abs/10.1145/2001576.2001692>`_
In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 845-852). ACM.
Please refer to the *official* Python source code from `PyBrain` (now not actively maintained):
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/distributionbased/rank1.py
"""
def __init__(self, problem, options):
options['sigma'] = np.inf # not used for `R1NES`
NES.__init__(self, problem, options)
self.n_individuals = int(max(5, max(4*np.log2(self.ndim_problem), 0.2*self.ndim_problem)))
self.lr_sigma = 0.1
self.lr_cv = 0.1
def initialize(self, is_restart=False):
s = np.empty((self.n_individuals, self.ndim_problem)) # noise of offspring population
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
p_v = self.rng_initialization.standard_normal((self.ndim_problem,))
p_v /= np.sqrt(np.dot(p_v, p_v))
l_d = np.log(1.0)/2.0
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 s, y, mean, p_v, l_d
def iterate(self, s=None, y=None, mean=None, p_v=None, l_d=None, args=None):
for k in range(self.n_individuals):
if self._check_terminations():
return s, y
s[k] = (self.rng_optimization.standard_normal((self.ndim_problem,)) +
p_v*self.rng_optimization.standard_normal())
y[k] = self._evaluate_fitness(mean + np.exp(l_d)*s[k], args)
return s, y
def _update_distribution(self, s=None, y=None, mean=None, p_v=None, l_d=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)
ww = [w for i, w in enumerate(s) if u[i] != 0]
u = [k for k in u if k != 0]
r = np.sqrt(np.dot(p_v, p_v))
v, c = p_v/r, np.log(r)
w_2 = np.array([np.dot(w, w) for w in ww])
v_w = np.array([np.dot(v, w) for w in ww])
wv_2 = np.array([np.square(vw) for vw in v_w])
mean += np.exp(l_d)*np.dot(u, ww)
k = ((np.square(r) - self.ndim_problem + 2.0)*wv_2 - (np.square(r) + 1.0)*w_2)/(
2.0*r*(self.ndim_problem - 1.0))
d_u = np.dot(k, u)*v + np.dot(v_w/r*u, ww)
d_c = np.dot(d_u, v)/r
e = min(self.lr_cv, 2.0*np.sqrt(np.square(r)/np.dot(d_u, d_u)))
if d_c > 0.0:
p_v += e*d_u
else:
c += e*d_c
v += e*(d_u/r - d_c*v)
v /= np.sqrt(np.dot(v, v))
p_v = np.exp(c)*v
l_d += self.lr_sigma*(1.0/(2.0*(self.ndim_problem - 1.0))*np.dot(
(w_2 - self.ndim_problem) - (wv_2 - 1.0), u))
return mean, p_v, l_d
def restart_reinitialize(self, s=None, y=None, mean=None, p_v=None, l_d=None):
if self.is_restart and NES.restart_reinitialize(self, y):
s, y, mean, p_v, l_d = self.initialize(True)
return s, y, mean, p_v, l_d
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = NES.optimize(self, fitness_function)
s, y, mean, p_v, l_d = self.initialize()
while True:
s, y = self.iterate(s, y, mean, p_v, l_d, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
mean, p_v, l_d = self._update_distribution(s, y, mean, p_v, l_d)
self._n_generations += 1
s, y, mean, p_v, l_d = self.restart_reinitialize(s, y, mean, p_v, l_d)
return self._collect(fitness, y, mean)