import numpy as np # engine for numerical computing
from pypop7.optimizers.es.es import ES # abstract class of all evolution strategies (ES)
[docs]class RES(ES):
"""Rechenberg's (1+1)-Evolution Strategy with 1/5th success rule (RES).
.. note:: `RES` is the first evolution strategy with self-adaptation of the *global* step-size (designed
by Rechenberg, one recipient of `IEEE Evolutionary Computation Pioneer Award 2002
<https://tinyurl.com/456as566>`_). As theoretically investigated in his *seminal* Ph.D. dissertation
at Technical University of Berlin, the existence of narrow **evolution window** explains the necessarity
of *global* step-size adaptation to maximize the local convergence progress, if possible. Note that
a similar theoretical study was independently conducted in the automatic control community (that is,
`[Schumer&Steiglitz, 1968, IEEE-TAC] <https://ieeexplore.ieee.org/abstract/document/1098903>`_).
Since there is only one parent and only one offspring for each generation (iteration), `RES` generally
shows limited *exploration* ability 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 `RES` (AKA two-membered ES) mainly for *benchmarking* and *theoretical* purposes. Interestingly,
owing to its popularity, sometimes `RES` is still used now, such as, `[Williams&Li, 2024, NeurIPS]
<https://tinyurl.com/4vdphufe>`_.
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']`.
* 'lr_sigma' - learning rate of global step-size self-adaptation (`float`, default:
`1.0/np.sqrt(problem['ndim_problem'] + 1.0)`).
Examples
--------
Use the black-box optimizer `RES` 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.res import RES
>>> 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
>>> res = RES(problem, options) # to initialize the black-box optimizer class
>>> results = res.optimize() # to run its optimization/evolution process
>>> # to return the used number of function evaluations and the found best-so-far fitness
>>> print(f"RES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
RES: 5000, 0.00011689296624022443
For its correctness checking of Python-based coding, refer to `this code-based repeatability report
<https://tinyurl.com/5n6ndrn7>`_ 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.
sigma : `float`
final global step-size, aka mutation strength (updated during optimization).
References
----------
Auger, A., Hansen, N., López-Ibáñez, M. and Rudolph, G., 2022.
`Tributes to Ingo Rechenberg (1934--2021).
<https://dl.acm.org/doi/10.1145/3511282.3511283>`_
ACM SIGEVOlution, 14(4), pp.1-4.
Agapie, A., Solomon, O. and Giuclea, M., 2021.
`Theory of (1+1) ES on the RIDGE.
<https://ieeexplore.ieee.org/abstract/document/9531957>`_
IEEE Transactions on Evolutionary Computation, 26(3), pp.501-511.
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.
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.
Rechenberg, I., 2000.
`Case studies in evolutionary experimentation and computation.
<https://www.sciencedirect.com/science/article/pii/S0045782599003813>`_
Computer Methods in Applied Mechanics and Engineering, 186(2-4), pp.125-140.
Rechenberg, I., 1989.
`Evolution strategy: Nature’s way of optimization.
<https://link.springer.com/chapter/10.1007/978-3-642-83814-9_6>`_
In Optimization: Methods and Applications, Possibilities and Limitations (pp. 106-126).
Springer, Berlin, Heidelberg.
Rechenberg, I., 1984.
`The evolution strategy. A mathematical model of Darwinian evolution.
<https://link.springer.com/chapter/10.1007/978-3-642-69540-7_13>`_
In Synergetics—from Microscopic to Macroscopic Order (pp. 122-132). Springer, Berlin, Heidelberg.
Schumer, M.A. and `Steiglitz, K. <https://www.cs.princeton.edu/~ken/>`_, 1968.
`Adaptive step size random search.
<https://ieeexplore.ieee.org/abstract/document/1098903>`_
IEEE Transactions on Automatic Control, 13(3), pp.270-276.
"""
def __init__(self, problem, options):
options['n_parents'] = 1 # mandatory setting
options['n_individuals'] = 1 # mandatory setting
ES.__init__(self, problem, options)
if self.lr_sigma is None:
self.lr_sigma = 1.0/np.sqrt(self.ndim_problem + 1.0)
assert self.lr_sigma > 0, f'`self.lr_sigma` = {self.lr_sigma}, but should > 0.'
def initialize(self, args=None, is_restart=False):
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
y = self._evaluate_fitness(mean, args) # fitness
best_so_far_y = np.copy(y)
self._list_initial_mean.append(np.copy(mean))
return mean, y, best_so_far_y
def iterate(self, args=None, mean=None): # to sample and evaluate only one offspring
x = mean + self.sigma*self.rng_optimization.standard_normal((self.ndim_problem,))
y = self._evaluate_fitness(x, args)
return x, y
def restart_reinitialize(self, args=None, mean=None, y=None, best_so_far_y=None, fitness=None):
if not self.is_restart:
return mean, y, best_so_far_y
self._list_fitness.append(best_so_far_y)
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(fitness, 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.sigma = np.copy(self._sigma_bak)
mean, y, best_so_far_y = self.initialize(args, True)
self._list_fitness = [best_so_far_y]
return mean, y, best_so_far_y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
mean, y, best_so_far_y = self.initialize(args)
while not self.termination_signal:
self._print_verbose_info(fitness, y)
x, y = self.iterate(args, mean)
self._n_generations += 1
if self._check_terminations():
break
self.sigma *= np.power(np.exp(float(y < best_so_far_y) - 0.2), self.lr_sigma)
if y <= best_so_far_y:
mean, best_so_far_y = x, y
mean, y, best_so_far_y = self.restart_reinitialize(args, mean, y, best_so_far_y, fitness)
return self._collect(fitness, y, mean)