import numpy as np # engine for numerical computing
from pypop7.optimizers.es.es import ES
[docs]class DSAES(ES):
"""Derandomized Self-Adaptation Evolution Strategy (DSAES).
.. note:: `DSAES` adapts all the *individual* step-sizes on-the-fly with a *relatively small* population.
The default setting (i.e., using a `small` population) may result in *relatively fast* (local) convergence,
but perhaps with the risk of getting trapped in suboptima on multi-modal fitness landscape. Therefore, it is
recommended to first attempt more advanced ES variants (e.g., `LMCMA`, `LMMAES`) for large-scale black-box
optimization. Here we include `DSAES` 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: `10`),
* 'lr_sigma' - learning rate of global step-size self-adaptation (`float`, default: `1.0/3.0`).
Examples
--------
Use the black-box optimizer `DSAES` 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.dsaes import DSAES
>>> 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
>>> dsaes = DSAES(problem, options) # to initialize the optimizer class
>>> results = dsaes.optimize() # to run the optimization/evolution process
>>> # to return the number of function evaluations and the best-so-far fitness
>>> print(f"DSAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
DSAES: 5000, 1.9916050765897666e-07
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/2c8e89kj>`_ 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 self-adaptation.
mean : `array_like`
initial (starting) point, aka mean of Gaussian search distribution.
n_individuals : `int`
number of offspring, aka offspring population size.
sigma : `float`
initial global step-size, aka mutation strength.
_axis_sigmas : `array_like`
final individuals step-sizes from the elitist.
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.
Ostermeier, A., Gawelczyk, A. and Hansen, N., 1994.
`A derandomized approach to self-adaptation of evolution strategies.
<https://direct.mit.edu/evco/article-abstract/2/4/369/1407/A-Derandomized-Approach-to-Self-Adaptation-of>`_
Evolutionary Computation, 2(4), pp.369-380.
"""
def __init__(self, problem, options):
if options.get('n_individuals') is None:
options['n_individuals'] = 10
ES.__init__(self, problem, options)
if self.lr_sigma is None: # learning rate of global step-size adaptation
self.lr_sigma = 1.0/3.0
assert self.lr_sigma > 0, f'`self.lr_sigma` = {self.lr_sigma}, but should > 0.'
self._axis_sigmas = None
self._e_hnd = np.sqrt(2.0/np.pi) # E[|N(0,1)|]: expectation of half-normal distribution
def initialize(self, is_restart=False):
self._axis_sigmas = self._sigma_bak*np.ones((self.ndim_problem,))
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
# set individual step-sizes for all offspring
sigmas = np.ones((self.n_individuals, self.ndim_problem))
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): # sample offspring population
if self._check_terminations():
return x, sigmas, y
sigma = self.lr_sigma*self.rng_optimization.standard_normal()
z = self.rng_optimization.standard_normal((self.ndim_problem,))
x[k] = mean + np.exp(sigma)*self._axis_sigmas*z
# mimick the effect of intermediate recombination
sigmas_1 = np.power(np.exp(np.abs(z)/self._e_hnd - 1.0), 1.0/self.ndim_problem)
sigmas_2 = np.power(np.exp(sigma), 1.0/np.sqrt(self.ndim_problem))
sigmas[k] = self._axis_sigmas*sigmas_1*sigmas_2
y[k] = self._evaluate_fitness(x[k], args)
return x, sigmas, y
def restart_reinitialize(self, x=None, mean=None, sigmas=None, y=None):
if not self.is_restart:
return x, mean, sigmas, 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 = np.all(self._axis_sigmas < 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
x, mean, sigmas, y = self.initialize(True)
self._list_fitness = [np.Inf]
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:
x, sigmas, y = self.iterate(x, mean, sigmas, y, args)
if self._check_terminations():
break
order = np.argsort(y)[0]
self._axis_sigmas = np.copy(sigmas[order])
mean = np.copy(x[order])
self._print_verbose_info(fitness, y)
self._n_generations += 1
x, mean, sigmas, y = self.restart_reinitialize(x, mean, sigmas, y)
results = self._collect(fitness, y, mean)
results['_axis_sigmas'] = self._axis_sigmas
return results