import numpy as np
from pypop7.optimizers.es.es import ES
[docs]class R1ES(ES):
"""Rank-One Evolution Strategy (R1ES).
.. note:: `R1ES` is a **low-rank** version of `CMA-ES` specifically designed for large-scale black-box optimization
(LSBBO) by Li and `Zhang <https://tinyurl.com/32hsbx28>`_. It often works well when there is a *dominated* search
direction embedded in a subspace. For more complex landscapes (e.g., there are multiple promising search
directions), other LSBBO variants (e.g., `RMES`, `LMCMA`, `LMMAES`) of `CMA-ES` may be more preferred.
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)`),
* 'c_cov' - learning rate of low-rank covariance matrix adaptation (`float`, default:
`1.0/(3.0*np.sqrt(problem['ndim_problem']) + 5.0)`),
* 'c' - learning rate of evolution path update (`float`, default:
`2.0/(problem['ndim_problem'] + 7.0)`),
* 'c_s' - learning rate of cumulative step-size adaptation (`float`, default: `0.3`),
* 'q_star' - baseline of cumulative step-size adaptation (`float`, default: `0.3`)，
* 'd_sigma' - delay factor of cumulative step-size adaptation (`float`, default: `1.0`).
Examples
--------
Use the optimizer to minimize the well-known test function
`Rosenbrock <http://en.wikipedia.org/wiki/Rosenbrock_function>`_:
.. code-block:: python
:linenos:
>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock # function to be minimized
>>> from pypop7.optimizers.es.r1es import R1ES
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 2,
... 'lower_boundary': -5*numpy.ones((2,)),
... 'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000, # set optimizer options
... 'seed_rng': 2022,
... 'mean': 3*numpy.ones((2,)),
... 'sigma': 0.1} # the global step-size may need to be tuned for better performance
>>> r1es = R1ES(problem, options) # initialize the optimizer class
>>> results = r1es.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"R1ES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
R1ES: 5000, 8.942371004351231e-10
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/2aywpp2p>`_ for more details.
Attributes
----------
c : `float`
learning rate of evolution path update.
c_cov : `float`
learning rate of low-rank covariance matrix adaptation.
c_s : `float`
learning rate of cumulative step-size adaptation.
d_sigma : `float`
delay factor of cumulative 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.
q_star : `float`
baseline of cumulative step-size adaptation.
sigma : `float`
final global step-size, aka mutation strength.
References
----------
Li, Z. and Zhang, Q., 2018.
`A simple yet efficient evolution strategy for large-scale black-box optimization.
<https://ieeexplore.ieee.org/abstract/document/8080257>`_
IEEE Transactions on Evolutionary Computation, 22(5), pp.637-646.
"""
def __init__(self, problem, options):
ES.__init__(self, problem, options)
self.c_cov = options.get('c_cov', 1.0/(3.0*np.sqrt(self.ndim_problem) + 5.0))
self.c = options.get('c', 2.0/(self.ndim_problem + 7.0))
self.c_s = options.get('c_s', 0.3)
self.q_star = options.get('q_star', 0.3)
self.d_sigma = options.get('d_sigma', 1.0)
self._x_1 = np.sqrt(1.0 - self.c_cov)
self._x_2 = np.sqrt(self.c_cov)
self._p_1 = 1.0 - self.c
self._p_2 = None
self._rr = None # for rank-based success rule (RSR)
def initialize(self, args=None, is_restart=False):
self._p_2 = np.sqrt(self.c*(2.0 - self.c)*self._mu_eff)
self._rr = np.arange(self.n_parents*2) + 1 # for rank-based success rule (RSR)
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
p = np.zeros((self.ndim_problem,)) # principal search direction
s = 0.0 # cumulative rank rate
y = np.tile(self._evaluate_fitness(mean, args), (self.n_individuals,)) # fitness
return x, mean, p, s, y
def iterate(self, x=None, mean=None, p=None, y=None, args=None):
for k in range(self.n_individuals):
if self._check_terminations():
return x, y
z = self.rng_optimization.standard_normal((self.ndim_problem,))
r = self.rng_optimization.standard_normal()
x[k] = mean + self.sigma*(self._x_1*z + self._x_2*r*p)
y[k] = self._evaluate_fitness(x[k], args)
return x, y
def _update_distribution(self, x=None, mean=None, p=None, s=None, y=None, y_bak=None):
order = np.argsort(y)[:self.n_parents]
y.sort()
mean_bak = np.dot(self._w[:self.n_parents], x[order])
p = self._p_1*p + self._p_2*(mean_bak - mean)/self.sigma
r = np.argsort(np.hstack((y_bak[:self.n_parents], y[:self.n_parents])))
rr = self._rr[r < self.n_parents] - self._rr[r >= self.n_parents]
q = np.dot(self._w, rr)/self.n_parents
s = (1.0 - self.c_s)*s + self.c_s*(q - self.q_star)
self.sigma *= np.exp(s/self.d_sigma)
return mean_bak, p, s
def restart_reinitialize(self, args=None, x=None, mean=None, p=None, s=None, y=None, fitness=None):
if self.is_restart and ES.restart_reinitialize(self, y):
x, mean, p, s, y = self.initialize(args, True)
self._print_verbose_info(fitness, y[0])
self.d_sigma *= 2.0
return x, mean, p, s, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
x, mean, p, s, y = self.initialize(args)
self._print_verbose_info(fitness, y[0])
while not self.termination_signal:
y_bak = np.copy(y)
# sample and evaluate offspring population
x, y = self.iterate(x, mean, p, y, args)
self._n_generations += 1
self._print_verbose_info(fitness, y)
if self._check_terminations():
break
mean, p, s = self._update_distribution(x, mean, p, s, y, y_bak)
x, mean, p, s, y = self.restart_reinitialize(args, x, mean, p, s, y, fitness)
results = self._collect(fitness, y, mean)
results['p'] = p
results['s'] = s
return results