import numpy as np
from pypop7.optimizers.es.es import ES
[docs]class LMCMAES(ES):
"""Limited-Memory Covariance Matrix Adaptation Evolution Strategy (LMCMAES).
.. note:: For perhaps better performance, please use its lateset version called `LMCMA
<https://tinyurl.com/mry5dw36>`_. Here we include it mainly for *benchmarking* 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']`.
* 'm' - number of direction vectors (`int`, default:
`4 + int(3*np.log(problem['ndim_problem']))`),
* 'n_steps' - target number of generations between vectors (`int`, default: `options['m']`),
* 'c_c' - learning rate for evolution path update (`float`, default: `1.0/options['m']`).
* 'c_1' - learning rate for covariance matrix adaptation (`float`, default:
`1.0/(10.0*np.log(problem['ndim_problem'] + 1.0))`),
* 'c_s' - learning rate for population success rule (`float`, default: `0.3`),
* 'd_s' - delay rate for population success rule (`float`, default: `1.0`),
* 'z_star' - target success rate for population success rule (`float`, default: `0.25`),
* '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)`).
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.lmcmaes import LMCMAES
>>> 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
>>> lmcmaes = LMCMAES(problem, options) # initialize the optimizer class
>>> results = lmcmaes.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"LMCMAES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
LMCMAES: 5000, 4.590739937885748e-16
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/yc7wyew4>`_ for more details.
Attributes
----------
c_c : `float`
learning rate for evolution path update.
c_s : `float`
learning rate for population success rule.
c_1 : `float`
learning rate for covariance matrix adaptation.
d_s : `float`
delay rate for population success rule.
m : `int`
number of direction vectors.
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.
n_steps : `int`
target number of generations between vectors.
sigma : `float`
initial global step-size, aka mutation strength.
z_star : `float`
target success rate for population success rule.
References
----------
Loshchilov, I., 2014, July.
A computationally efficient limited memory CMA-ES for large scale optimization.
In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 397-404). ACM.
https://dl.acm.org/doi/abs/10.1145/2576768.2598294
See the official C++ version from Loshchilov:
https://sites.google.com/site/lmcmaeses/
"""
def __init__(self, problem, options):
ES.__init__(self, problem, options)
self.m = options.get('m', 4 + int(3*np.log(self.ndim_problem))) # number of direction vectors
self.n_steps = options.get('n_steps', self.m) # target number of generations between vectors
self.c_c = options.get('c_c', 1.0/self.m) # learning rate for evolution path update
self.c_1 = options.get('c_1', 1.0/(10.0*np.log(self.ndim_problem + 1.0)))
self.c_s = options.get('c_s', 0.3) # learning rate for population success rule (PSR)
self.d_s = options.get('d_s', 1.0) # damping parameter for PSR
self.z_star = options.get('z_star', 0.25) # target success rate for PSR
self._a = np.sqrt(1.0 - self.c_1)
self._c = 1.0/np.sqrt(1.0 - self.c_1)
self._bd_1 = np.sqrt(1.0 - self.c_1)
self._bd_2 = self.c_1/(1.0 - self.c_1)
self._p_c_1 = 1.0 - self.c_c
self._p_c_2 = None
self._j = None
self._l = None
self._it = None
self._rr = None # for PSR
def initialize(self, is_restart=False):
mean = self._initialize_mean(is_restart) # mean of Gaussian search distribution
x = np.empty((self.n_individuals, self.ndim_problem)) # offspring population
p_c = np.zeros((self.ndim_problem,)) # evolution path
s = 0.0 # for PSR of global step-size adaptation
vm = np.empty((self.m, self.ndim_problem))
pm = np.empty((self.m, self.ndim_problem))
b = np.empty((self.m,))
d = np.empty((self.m,))
y = np.empty((self.n_individuals,)) # fitness (no evaluation)
self._p_c_2 = np.sqrt(self.c_c*(2.0 - self.c_c)*self._mu_eff)
self._rr = np.arange(self.n_individuals*2, 0, -1) - 1
self._j = [None]*self.m
self._l = [None]*self.m
self._it = 0
return mean, x, p_c, s, vm, pm, b, d, y
def _a_z(self, z=None, pm=None, vm=None, b=None): # Algorithm 3 Az()
x = np.copy(z)
for t in range(self._it):
x = self._a*x + b[self._j[t]]*np.dot(vm[self._j[t]], z)*pm[self._j[t]]
return x
def iterate(self, mean=None, x=None, pm=None, vm=None, y=None, b=None, args=None):
sign, a_z = 1, np.empty((self.ndim_problem,)) # for mirrored sampling
for k in range(self.n_individuals):
if self._check_terminations():
return x, y
if sign == 1:
z = self.rng_optimization.standard_normal((self.ndim_problem,))
a_z = self._a_z(z, pm, vm, b)
x[k] = mean + sign*self.sigma*a_z
y[k] = self._evaluate_fitness(x[k], args)
sign *= -1 # sampling in the opposite direction for mirrored sampling
return x, y
def _a_inv_z(self, v=None, vm=None, d=None, i=None): # Algorithm 4 Ainvz()
x = np.copy(v)
for t in range(0, i):
x = self._c*x - d[self._j[t]]*np.dot(vm[self._j[t]], x)*vm[self._j[t]]
return x
def _update_distribution(self, mean=None, x=None, p_c=None, s=None, vm=None,
pm=None, b=None, d=None, y=None, y_bak=None):
mean_bak = np.dot(self._w, x[np.argsort(y)[:self.n_parents]])
p_c = self._p_c_1*p_c + self._p_c_2*(mean_bak - mean)/self.sigma
i_min = 1
if self._n_generations < self.m:
self._j[self._n_generations] = self._n_generations
else:
d_min = self._l[self._j[i_min]] - self._l[self._j[i_min - 1]]
for j in range(2, self.m):
d_cur = self._l[self._j[j]] - self._l[self._j[j - 1]]
if d_cur < d_min:
d_min, i_min = d_cur, j
# start from 0 if all pairwise distances exceed `self.n_steps`
i_min = 0 if d_min >= self.n_steps else i_min
# update indexes of evolution paths (`self._j[i_min]` is index of evolution path needed to delete)
updated = self._j[i_min]
for j in range(i_min, self.m - 1):
self._j[j] = self._j[j + 1]
self._j[self.m - 1] = updated
self._it = np.minimum(self._n_generations + 1, self.m)
self._l[self._j[self._it - 1]] = self._n_generations # to update its generation
pm[self._j[self._it - 1]] = p_c # to add the latest evolution path
# since `self._j[i_min]` is deleted, all vectors (from vm) depending on it need to be computed again
for i in range(0 if i_min == 1 else i_min, self._it):
vm[self._j[i]] = self._a_inv_z(pm[self._j[i]], vm, d, i)
v_n = np.dot(vm[self._j[i]], vm[self._j[i]])
bd_3 = np.sqrt(1.0 + self._bd_2*v_n)
b[self._j[i]] = self._bd_1/v_n*(bd_3 - 1.0)
d[self._j[i]] = 1.0/(self._bd_1*v_n)*(1.0 - 1.0/bd_3)
if self._n_generations > 0: # for population success rule (PSR)
r = np.argsort(np.hstack((y, y_bak)))
z_psr = np.sum(self._rr[r < self.n_individuals] - self._rr[r >= self.n_individuals])
z_psr = z_psr/np.power(self.n_individuals, 2) - self.z_star
s = (1.0 - self.c_s)*s + self.c_s*z_psr
self.sigma *= np.exp(s/self.d_s)
return mean_bak, p_c, s, vm, pm, b, d
def restart_reinitialize(self, mean=None, x=None, p_c=None, s=None,
vm=None, pm=None, b=None, d=None, y=None):
if self.is_restart and ES.restart_reinitialize(self, y):
mean, x, p_c, s, vm, pm, b, d, y = self.initialize(True)
self.d_s *= 2.0
return mean, x, p_c, s, vm, pm, b, d, y
def optimize(self, fitness_function=None, args=None): # for all generations (iterations)
fitness = ES.optimize(self, fitness_function)
mean, x, p_c, s, vm, pm, b, d, y = self.initialize()
while not self.termination_signal:
y_bak = np.copy(y)
# sample and evaluate offspring population
x, y = self.iterate(mean, x, pm, vm, y, b, args)
if self._check_terminations():
break
mean, p_c, s, vm, pm, b, d = self._update_distribution(
mean, x, p_c, s, vm, pm, b, d, y, y_bak)
self._print_verbose_info(fitness, y)
self._n_generations += 1
mean, x, p_c, s, vm, pm, b, d, y = self.restart_reinitialize(
mean, x, p_c, s, vm, pm, b, d, y)
results = self._collect(fitness, y, mean)
results['p_c'] = p_c
results['s'] = s
return results