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csaes.py
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csaes.py
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import numpy as np # engine for numerical computing
from pypop7.optimizers.es.es import ES
from pypop7.optimizers.es.dsaes import DSAES
class CSAES(DSAES):
"""Cumulative Step-size self-Adaptation Evolution Strategy (CSAES).
.. note:: `CSAES` adapts all the *individual* step-sizes on-the-fly with a *relatively small* population,
according to the well-known `CSA <http://link.springer.com/chapter/10.1007/3-540-58484-6_263>`_ rule
(a.k.a. cumulative (evolution) path-length control) from the Evolutionary Computation community. The
default setting (i.e., using a `small` population) can 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 `CSAES` 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(np.floor(3*np.log(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 adaptation (`float`, default:
`np.sqrt(options['n_parents']/(problem['ndim_problem'] + options['n_parents']))`).
Examples
--------
Use the black-box optimizer `CSAES` 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.csaes import CSAES
>>> 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': 0.1} # global step-size may need to be tuned
>>> csaes = CSAES(problem, options) # to initialize the optimizer class
>>> results = csaes.optimize() # to run the optimization/evolution process
>>> # to return the number of function evaluations and best-so-far fitness
>>> print(f"CSAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
CSAES: 5000, 0.010143683086819875
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/2s4ctvdw>`_ 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`
initial global step-size, aka mutation strength.
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.
Kern, S., Müller, S.D., Hansen, N., Büche, D., Ocenasek, J. and Koumoutsakos, P., 2004.
`Learning probability distributions in continuous evolutionary algorithms–a comparative review.
<https://link.springer.com/article/10.1023/B:NACO.0000023416.59689.4e>`_
Natural Computing, 3, pp.77-112.
Ostermeier, A., Gawelczyk, A. and Hansen, N., 1994, October.
`Step-size adaptation based on non-local use of selection information
<http://link.springer.com/chapter/10.1007/3-540-58484-6_263>`_
In International Conference on Parallel Problem Solving from Nature (pp. 189-198).
Springer, Berlin, Heidelberg.
"""
def __init__(self, problem, options):
if options.get('n_individuals') is None: # number of offspring, aka offspring population size
options['n_individuals'] = 4 + int(np.floor(3*np.log(problem['ndim_problem'])))
if options.get('n_parents') is None: # number of parents, aka parental population size
options['n_parents'] = int(options['n_individuals']/4)
if options.get('lr_sigma') is None: # learning rate of global step-size adaptation
options['lr_sigma'] = np.sqrt(options['n_parents']/(
problem['ndim_problem'] + options['n_parents']))
DSAES.__init__(self, problem, options)
self._s_1 = None
self._s_2 = None
# set E[||N(0,I)||]: expectation of chi distribution
self._e_chi = np.sqrt(self.ndim_problem)*(
1.0 - 1.0/(4.0*self.ndim_problem) + 1.0/(21.0*np.power(self.ndim_problem, 2)))
def initialize(self, is_restart=False):
self._s_1 = 1.0 - self.lr_sigma
self._s_2 = np.sqrt(self.lr_sigma*(2.0 - self.lr_sigma)*self.n_parents)
self._axis_sigmas = self._sigma_bak*np.ones((self.ndim_problem,))
z = np.empty((self.n_individuals, self.ndim_problem)) # noise for offspring population
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
s = np.zeros((self.ndim_problem,)) # evolution path
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
self._list_initial_mean.append(np.copy(mean))
return z, x, mean, s, y
def iterate(self, z=None, x=None, mean=None, y=None, args=None):
for k in range(self.n_individuals): # to sample offspring population
if self._check_terminations():
return z, x, y
z[k] = self.rng_optimization.standard_normal((self.ndim_problem,))
x[k] = mean + self._axis_sigmas*z[k]
y[k] = self._evaluate_fitness(x[k], args)
return z, x, y
def _update_distribution(self, z=None, x=None, s=None, y=None):
order = np.argsort(y)[:self.n_parents]
s = self._s_1*s + self._s_2*np.mean(z[order], axis=0)
sigmas_1 = np.power(np.exp(np.abs(s)/self._e_hnd - 1.0), 1.0/(3.0*self.ndim_problem))
sigmas_2 = np.power(np.exp(np.linalg.norm(s)/self._e_chi - 1.0),
self.lr_sigma/(1.0 + np.sqrt(self.n_parents/self.ndim_problem)))
self._axis_sigmas *= (sigmas_1*sigmas_2)
mean = np.mean(x[order], axis=0)
return s, mean
def restart_reinitialize(self, z=None, x=None, mean=None, s=None, y=None):
if not self.is_restart:
return z, x, mean, s, 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
self.n_parents = int(self.n_individuals/4)
self.lr_sigma = np.sqrt(self.n_parents/(self.ndim_problem + self.n_parents))
z, x, mean, s, y = self.initialize(True)
self._list_fitness = [np.Inf]
return z, x, mean, s, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
z, x, mean, s, y = self.initialize()
while True:
# sample and evaluate offspring population
z, x, y = self.iterate(z, x, mean, y, args)
if self._check_terminations():
break
self._print_verbose_info(fitness, y)
self._n_generations += 1
s, mean = self._update_distribution(z, x, s, y)
z, x, mean, s, y = self.restart_reinitialize(z, x, mean, s, y)
results = self._collect(fitness, y, mean)
results['s'] = s
results['_axis_sigmas'] = self._axis_sigmas
return results