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ssaes.py
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ssaes.py
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import numpy as np # engine for numerical computing
from pypop7.optimizers.es.es import ES # abstract class of all evolution strategies (ES)
class SSAES(ES):
"""Schwefel's Self-Adaptation Evolution Strategy (SSAES).
.. note:: `SSAES` adapts all the **individual** step-sizes (aka coordinate-wise standard deviations)
on-the-fly, proposed by Schwefel (one recipient of `IEEE Evolutionary Computation Pioneer Award
2002 <https://tinyurl.com/456as566>`_ and `IEEE Frank Rosenblatt Award 2011
<https://en.wikipedia.org/wiki/IEEE_Frank_Rosenblatt_Award>`_). Since it often needs a *relatively
large* population (e.g., larger than number of dimensionality) for reliable self-adaptation, `SSAES`
suffers easily from *slow* convergence for large-scale black-box optimization. Therefore, it is
recommended to first attempt more advanced ES variants (e.g., `LMCMA`, `LMMAES`) for large-scale
black-box optimization. Here we include `SSAES` mainly for *benchmarking* and *theoretical* purpose.
Currently the `restart` process is not implemented owing to its typically slow convergence.
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:
`5*problem['ndim_problem']`),
* 'n_parents' - number of parents, aka parental population size (`int`, default:
`int(options['n_individuals']/4)`),
* 'lr_sigma' - learning rate of global step-size self-adaptation (`float`, default:
`1.0/np.sqrt(problem['ndim_problem'])`),
* 'lr_axis_sigmas' - learning rate of individual step-sizes self-adaptation (`float`, default:
`1.0/np.power(problem['ndim_problem'], 1.0/4.0)`).
Examples
--------
Use the black-box optimizer `SSAES` 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.ssaes import SSAES
>>> 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 for optimality
>>> ssaes = SSAES(problem, options) # to initialize the black-box optimizer class
>>> results = ssaes.optimize() # to run the optimization/evolution process
>>> print(f"SSAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
SSAES: 5000, 0.0002
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/zsmsyh2x>`_ 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_axis_sigmas : `float`
learning rate of individual step-sizes self-adaptation.
lr_sigma : `float`
learning rate of global step-size self-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`
initial global step-size, aka mutation strength.
_axis_sigmas : `array_like`
final individuals step-sizes (updated during optimization).
References
----------
Hansen, N., Arnold, D.V. and Auger, A., 2015.
`Evolution strategies.
<https://link.springer.com/chapter/10.1007%2F978-3-662-43505-2_44>`_
In Springer Handbook of Computational Intelligence (pp. 871-898). Springer, Berlin, Heidelberg.
Beyer, H.G. and Schwefel, H.P., 2002.
`Evolution strategies–A comprehensive introduction.
<https://link.springer.com/article/10.1023/A:1015059928466>`_
Natural Computing, 1(1), pp.3-52.
Schwefel, H.P., 1988.
`Collective intelligence in evolving systems.
<https://link.springer.com/chapter/10.1007/978-3-642-73953-8_8>`_
In Ecodynamics (pp. 95-100). Springer, Berlin, Heidelberg.
Schwefel, H.P., 1984.
`Evolution strategies: A family of non-linear optimization techniques based on imitating
some principles of organic evolution.
<https://link.springer.com/article/10.1007/BF01876146>`_
Annals of Operations Research, 1(2), pp.165-167.
"""
def __init__(self, problem, options):
if options.get('n_individuals') is None:
options['n_individuals'] = 5*problem.get('ndim_problem')
if options.get('n_parents') is None:
options['n_parents'] = int(options['n_individuals']/4)
ES.__init__(self, problem, options)
if self.lr_sigma is None:
self.lr_sigma = 1.0/np.sqrt(self.ndim_problem) # learning rate of global step-size self-adaptation
assert self.lr_sigma > 0, f'`self.lr_sigma` = {self.lr_sigma}, but should > 0.'
# set learning rate of individual step-sizes self-adaptation
self.lr_axis_sigmas = options.get('lr_axis_sigmas', 1.0/np.power(self.ndim_problem, 1.0/4.0))
assert self.lr_axis_sigmas > 0, f'`self.lr_axis_sigmas` = {self.lr_axis_sigmas}, but should > 0.'
self._axis_sigmas = self.sigma*np.ones((self.ndim_problem,)) # individual step-sizes
def initialize(self):
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean() # mean of Gaussian search distribution
sigmas = np.empty((self.n_individuals, self.ndim_problem)) # individual step-sizes for all offspring
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
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): # sample offspring population
if self._check_terminations():
return x, sigmas, y
sigma = self.lr_sigma*self.rng_optimization.standard_normal()
axis_sigmas = self.lr_axis_sigmas*self.rng_optimization.standard_normal((self.ndim_problem,))
sigmas[k] = self._axis_sigmas*np.exp(axis_sigmas)*np.exp(sigma)
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 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:
x, sigmas, y = self.iterate(x, mean, sigmas, y, args)
if self._check_terminations():
break
order = np.argsort(y)[:self.n_parents]
self._axis_sigmas = np.mean(sigmas[order], axis=0)
mean = np.mean(x[order], axis=0)
self._print_verbose_info(fitness, y)
self._n_generations += 1
results = self._collect(fitness, y, mean)
results['_axis_sigmas'] = self._axis_sigmas
return results