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.
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
Schaul, T., Glasmachers, T. and Schmidhuber, J., 2011, July.
High dimensions and heavy tails for natural evolution strategies.
In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 845-852). ACM.
https://dl.acm.org/doi/abs/10.1145/2001576.2001692
See the official Python source code from PyBrain:
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)