import numpy as np # engine for numerical computing
from pypop7.optimizers.es.es import ES
[docs]class SAES(ES):
"""Self-Adaptation Evolution Strategy (SAES).
.. note:: `SAES` adapts only the *global* step-size on-the-fly with a *relatively small* population, often
resulting in *slow* (and even *premature*) convergence for large-scale black-box optimization (LBO),
especially on *ill-conditioned* fitness landscapes. Therefore, it is recommended to first attempt more
advanced ES variants (e.g. `LMCMA`, `LMMAES`) for LBO. Here we include `SAES` 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`):
* 'sigma' - initial global step-size, aka mutation strength (`float`),
* 'mean' - initial (starting) point, aka mean of Gaussian search distribution (`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']`.
* 'n_individuals' - number of offspring, aka offspring population size (`int`, default:
`4 + int(3*np.log(problem['ndim_problem']))`),
* 'n_parents' - number of parents, aka parental population size (`int`, default:
`int(options['n_individuals']/2)`),
* 'lr_sigma' - learning rate of global step-size (`float`, default:
`1.0/np.sqrt(2*problem['ndim_problem'])`).
Examples
--------
Use the black-box optimizer `SAES` 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.es.saes import SAES
>>> 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,)),
... 'sigma': 3.0} # global step-size may need to be tuned
>>> saes = SAES(problem, options) # to initialize the optimizer class
>>> results = saes.optimize() # to run the optimization/evolution process
>>> # to return the number of function evaluations and the best-so-far fitness
>>> print(f"SAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
SAES: 5000, 0.012622712890954227
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/mvkspst4>`_ for more details.
Attributes
----------
best_so_far_x : `array_like`
final best-so-far solution found during entire optimization.
best_so_far_y : `array_like`
final best-so-far fitness found during entire optimization.
lr_sigma : `float`
learning rate of global step-size adaptation.
mean : `array_like`
initial (starting) point, aka mean of Gaussian search distribution.
n_individuals : `int`
number of offspring, aka offspring population size.
n_parents : `int`
number of parents, aka parental population size.
sigma : `float`
final global step-size, aka mutation strength (changed during optimization).
References
----------
Beyer, H.G., 2020, July.
`Design principles for matrix adaptation evolution strategies.
<https://dl.acm.org/doi/abs/10.1145/3377929.3389870>`_
In Proceedings of ACM Conference on Genetic and Evolutionary Computation Companion (pp. 682-700). ACM.
http://www.scholarpedia.org/article/Evolution_strategies
See its official Matlab/Octave version from `Prof. Beyer <https://homepages.fhv.at/hgb/>`_:
https://homepages.fhv.at/hgb/downloads/mu_mu_I_lambda-ES.oct
"""
def __init__(self, problem, options):
ES.__init__(self, problem, options)
if self.lr_sigma is None:
self.lr_sigma = 1.0/np.sqrt(2*self.ndim_problem)
def initialize(self, is_restart=False):
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
sigmas = np.ones((self.n_individuals,)) # global step-sizes for all offspring
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
self._list_initial_mean.append(np.copy(mean))
return x, mean, sigmas, y
def iterate(self, x=None, mean=None, sigmas=None, y=None, args=None):
for k in range(self.n_individuals): # to sample offspring population
if self._check_terminations():
return x, sigmas, y
sigmas[k] = self.sigma*np.exp(self.lr_sigma*self.rng_optimization.standard_normal())
x[k] = mean + sigmas[k]*self.rng_optimization.standard_normal((self.ndim_problem,))
y[k] = self._evaluate_fitness(x[k], args)
return x, sigmas, y
def _restart_initialize(self, y):
min_y = np.min(y)
if min_y < self._list_fitness[-1]:
self._list_fitness.append(min_y)
else:
self._list_fitness.append(self._list_fitness[-1])
is_restart_1, is_restart_2 = self.sigma < self.sigma_threshold, False
if len(self._list_fitness) >= self.stagnation:
is_restart_2 = (self._list_fitness[-self.stagnation] - self._list_fitness[-1]) < self.fitness_diff
is_restart = bool(is_restart_1) or bool(is_restart_2)
if is_restart:
self._print_verbose_info([], y, True)
if self.verbose:
print(' ....... *** restart *** .......')
self._n_restart += 1
self._list_generations.append(self._n_generations) # for each restart
self._n_generations = 0
self.n_individuals *= 2
self.n_parents = int(self.n_individuals/2)
self._list_fitness = [np.Inf]
return is_restart
def restart_initialize(self, x=None, mean=None, sigmas=None, y=None):
if self._restart_initialize(y):
self.sigma = np.copy(self._sigma_bak)
x, mean, sigmas, y = self.initialize(True)
return x, mean, sigmas, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
x, mean, sigmas, y = self.initialize()
while True:
# sample and evaluate offspring population
x, sigmas, y = self.iterate(x, mean, sigmas, y, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
self._n_generations += 1
order = np.argsort(y)[:self.n_parents]
# use intermediate multi-recombination
mean = np.mean(x[order], axis=0)
self.sigma = np.mean(sigmas[order])
if self.is_restart:
x, mean, sigmas, y = self.restart_initialize(x, mean, sigmas, y)
return self._collect(fitness, y, mean)