import numpy as np # engine for numerical computing
from pypop7.optimizers.nes.nes import NES # abstract class of Natural Evolution Strategies (NES) classes
[docs]class SGES(NES):
"""Search Gradient-based Evolution Strategy (SGES).
.. note:: Here we include `SGES` (also called **vanilla version** of `NES`) **only** for *theoretical*
and *educational* purposes, since in practice advanced versions (e.g., `ENES`, `XNES`, `SNES`, and
`R1NES`) are more preferred than `SGES` in most cases.
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']`.
* 'lr_mean' - learning rate of distribution mean update (`float`, default: `0.01`),
* 'lr_sigma' - learning rate of global step-size adaptation (`float`, default: `0.01`).
Examples
--------
Use the black-box optimizer `SGES` to minimize the well-known test function
`Rosenbrock <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.nes.sges import SGES
>>> problem = {'fitness_function': rosenbrock, # to define problem arguments
... 'ndim_problem': 2,
... 'lower_boundary': -5.0*numpy.ones((2,)),
... 'upper_boundary': 5.0*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000, # to set optimizer options
... 'seed_rng': 2022,
... 'mean': 3.0*numpy.ones((2,))}
>>> sges = SGES(problem, options) # to initialize the optimizer class
>>> results = sges.optimize() # to run the optimization process
>>> print(f"SGES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
SGES: 5000, 0.0190
Attributes
----------
lr_mean : `float`
learning rate of distribution mean update (should `> 0.0`).
lr_sigma : `float`
learning rate of global step-size adaptation (should `> 0.0`).
mean : `array_like`
initial (starting) point, aka mean of Gaussian search/sampling/mutation distribution.
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']`, by
default.
n_individuals : `int`
number of offspring/descendants, aka offspring population size (should `> 0`).
n_parents : `int`
number of parents/ancestors, aka parental population size (should `> 0`).
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.
Please refer to the *official* Python source code from `PyBrain` (now not actively maintained):
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/distributionbased/ves.py
"""
def __init__(self, problem, options):
"""Initialize all the hyper-parameters and also auxiliary class members.
"""
options['n_individuals'] = options.get('n_individuals', 100)
options['sigma'] = np.inf # but not used here (only to avoid raise an error in superclass `ES`)
NES.__init__(self, problem, options)
if self.lr_mean is None:
self.lr_mean = 0.01
assert self.lr_mean > 0.0, f'`self.lr_mean` = {self.lr_mean}, but should > 0.0.'
if self.lr_sigma is None:
self.lr_sigma = 0.01
assert self.lr_sigma > 0.0, f'`self.lr_sigma` = {self.lr_sigma}, but should > 0.0.'
# set parameter number of Gaussian search/sampling/mutation distribution
self._n_distribution = int(self.ndim_problem + self.ndim_problem * (self.ndim_problem + 1) / 2)
self._d_cv = None # all derivatives w.r.t. covariance matrix
def initialize(self, is_restart=False):
"""Initialize the offspring population, their fitness, mean and covariance matrix of Gaussian
search/sampling/mutation distribution.
"""
NES.initialize(self)
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
y = np.empty((self.n_individuals,)) # fitness (no evaluation when initialization)
mean = self._initialize_mean(is_restart) # mean of Gaussian search/sampling/mutation distribution
cv = np.eye(self.ndim_problem) # covariance matrix of Gaussian search/sampling/mutation distribution
self._d_cv = np.eye(self.ndim_problem) # all derivatives w.r.t. covariance matrix
return x, y, mean, cv
def iterate(self, x=None, y=None, mean=None, args=None):
"""Iterate the generation and fitness evaluation process of the offspring population.
"""
for k in range(self.n_individuals): # for each offspring individual
if self._check_terminations():
return x, y
# generate each offspring individual according to Gaussian search/sampling/mutation distribution
x[k] = mean + np.dot(self._d_cv.T, self.rng_optimization.standard_normal((self.ndim_problem,)))
# evaluate the fitness of each offspring individual according to objective function
y[k] = self._evaluate_fitness(x[k], args)
return x, y
def _triu2flat(self, cv):
"""Convert the upper-triangular matrix to an entirely flat vector.
"""
v = np.zeros((self._n_distribution - self.ndim_problem,))
s, e = 0, self.ndim_problem # starting and ending index
for r in range(self.ndim_problem):
v[s:e] = cv[r, r:]
s, e = e, e + (self.ndim_problem - (r + 1))
return v
def _flat2triu(self, g):
"""Convert the entirely flat vector to an upper-triangular matrix.
"""
cv = np.zeros((self.ndim_problem, self.ndim_problem))
s, e = 0, self.ndim_problem # starting and ending index
for r in range(self.ndim_problem):
cv[r, r:] = g[s:e]
s, e = e, e + (self.ndim_problem - (r + 1))
return cv
def _update_distribution(self, x=None, y=None, mean=None, cv=None):
"""Update the mean and covariance matrix of Gaussian search/sampling/mutation distribution.
"""
# sort the offspring population for *maximization* (`-y`) rather than *minimization*
order = np.argsort(-y)
# ensure that the better an offspring, the larger its weight
u = np.empty((self.n_individuals,))
for i, o in enumerate(order):
u[o] = self._u[i]
# calculate the inverse of covariance matrix
inv_cv = np.linalg.inv(cv)
# calculate all derivatives w.r.t. both mean and covariance matrix
phi = np.zeros((self.n_individuals, self._n_distribution))
# calculate all derivatives w.r.t. mean for all offspring
phi[:, :self.ndim_problem] = np.dot(inv_cv, (x - mean).T).T
# calculate all derivatives w.r.t. covariance matrix for all offspring
grad_cv = np.empty((self.n_individuals, self._n_distribution - self.ndim_problem))
for k in range(self.n_individuals): # for each offspring individual
diff = x[k] - mean
_grad_cv = 0.5 * (np.dot(np.dot(inv_cv, np.outer(diff, diff)), inv_cv) - inv_cv)
grad_cv[k] = self._triu2flat(np.dot(self._d_cv, (_grad_cv + _grad_cv.T)))
phi[:, self.ndim_problem:] = grad_cv
# use *fitness baseline* to reduce estimation variance rather than directly using
# grad = np.sum(phi * (np.outer(u, np.ones((self._n_distribution,)))), 0)
phi_square = phi * phi # dynamic base
grad = np.sum(phi * (np.outer(u, np.ones((self._n_distribution,))) - np.dot(
u, phi_square) / np.dot(np.ones((self.n_individuals,)), phi_square)), 0)
# update the mean of Gaussian search/sampling/mutation distribution
mean += self.lr_mean * grad[:self.ndim_problem]
# update the covariance matrix of Gaussian search/sampling/mutation distribution
self._d_cv += self.lr_sigma * self._flat2triu(grad[self.ndim_problem:])
cv = np.dot(self._d_cv.T, self._d_cv) # to recover covariance matrix
self._n_generations += 1
return mean, cv
def restart_reinitialize(self, x=None, y=None, mean=None, cv=None):
"""Restart and re-initialize the optimization/evolution process, if needed.
"""
if self.is_restart and NES.restart_reinitialize(self, y):
x, y, mean, cv = self.initialize(True)
return x, y, mean, cv
def optimize(self, fitness_function=None, args=None):
"""Run the optimization/evolution process for all generations (iterations).
"""
fitness = NES.optimize(self, fitness_function) # to store all fitness generated during optimization
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)
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)