import numpy as np
from pypop7.optimizers.es.es import ES
[docs]class LMCMA(ES):
"""Limited-Memory Covariance Matrix Adaptation (LMCMA).
.. note:: Currently `LMCMA` is a **State-Of-The-Art (SOTA)** variant of `CMA-ES` designed especially for large-scale
black-box optimization. Inspired by `L-BFGS` (a well-established *second-order* gradient-based optimizer),
it stores only *m* direction vectors to reconstruct the covariance matirx on-the-fly, resulting in **O(mn)**
time complexity w.r.t. each sampling, where *m=O(log(n))* and *n* is the dimensionality of objective function.
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']))`),
* 'base_m' - base number of direction vectors (`int`, default: `4`),
* 'period' - update period (`int`, default:
`int(np.maximum(1, np.log(problem['ndim_problem'])))`),
* 'n_steps' - target number of generations between vectors (`int`, default:
`problem['ndim_problem']`),
* 'c_c' - learning rate for evolution path update (`float`, default:
`0.5/np.sqrt(problem['ndim_problem'])`),
* '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' - changing rate for population success rule (`float`, default: `1.0`),
* 'z_star' - target success rate for population success rule (`float`, default: `0.3`),
* '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 `LMCMA` 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.lmcma import LMCMA
>>> problem = {'fitness_function': rosenbrock, # define problem arguments
... 'ndim_problem': 1000,
... 'lower_boundary': -5*numpy.ones((1000,)),
... 'upper_boundary': 5*numpy.ones((1000,))}
>>> options = {'max_function_evaluations': 1e5*1000, # set optimizer options
... 'fitness_threshold': 1e-8,
... 'seed_rng': 2022,
... 'mean': 3*numpy.ones((1000,)),
... 'sigma': 0.1} # the global step-size may need to be tuned for better performance
>>> lmcma = LMCMA(problem, options) # initialize the optimizer class
>>> results = lmcma.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"LMCMA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
LMCMA: 2482983, 9.998653039116044e-09
For its correctness checking of coding, refer to `this code-based repeatability report
<https://tinyurl.com/24jnfhbs>`_ for more details.
Attributes
----------
base_m : `int`
base number of direction vectors.
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`
changing 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.
period : `int`
update period.
sigma : `float`
final global step-size, aka mutation strength.
z_star : `float`
target success rate for population success rule.
References
----------
Loshchilov, I., 2017.
LM-CMA: An alternative to L-BFGS for large-scale black box optimization.
Evolutionary Computation, 25(1), pp.143-171.
https://direct.mit.edu/evco/article-abstract/25/1/143/1041/LM-CMA-An-Alternative-to-L-BFGS-for-Large-Scale
(See Algorithm 7 for details.)
See the official C++ version from Loshchilov, which provides an interface for Matlab users:
https://sites.google.com/site/ecjlmcma/
(Unfortunately, this website link appears to be not available now.)
"""
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.base_m = options.get('base_m', 4) # base number of direction vectors
self.period = options.get('period', int(np.maximum(1, np.log(self.ndim_problem)))) # update period
self.n_steps = options.get('n_steps', self.ndim_problem) # target number of generations between vectors
self.c_c = options.get('c_c', 0.5/np.sqrt(self.ndim_problem)) # learning rate for evolution path
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) # changing rate for PSR
self.z_star = options.get('z_star', 0.3) # 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
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._j = [None]*self.m
self._l = [None]*self.m
self._it = 0
self._rr = np.arange(self.n_individuals*2, 0, -1) - 1
return mean, x, p_c, s, vm, pm, b, d, y
def _rademacher(self):
"""Sampling from Rademacher distribution."""
random = self.rng_optimization.integers(2, size=(self.ndim_problem,))
random[random == 0] = -1
return np.double(random)
def _a_z(self, z=None, pm=None, vm=None, b=None, start=None, it=None):
"""Az(): Cholesky factor-vector update."""
x = np.copy(z)
for t in range(start, 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: # SelectSubst(): direction vectors selection
base_m = (10.0*self.base_m if k == 0 else self.base_m)*np.abs(
self.rng_optimization.standard_normal())
base_m = float(self._it if base_m > self._it else base_m)
a_z = self._a_z(self._rademacher(), pm, vm, b,
int(self._it - base_m) if self._it > 1 else 0, self._it)
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):
"""Ainvz(): inverse Cholesky factor-vector update."""
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
# select and store direction vectors - to preserve a certain temporal distance in terms of
# number of generations between the stored direction vectors
if self._n_generations % self.period == 0: # temporal distance
_n_generations = int(self._n_generations/self.period) # temporal distance
i_min = 1 # index of the first vector that will be replaced by the new one
# the higher is the index of `self._j`, the more recent is the corresponding direction vector
if _n_generations < self.m:
self._j[_n_generations] = _n_generations
else:
if self.m > 1:
# find a pair of consecutively saved vectors with the distance between them
# closest to a target distance
d_min = (self._l[self._j[1]] - self._l[self._j[0]]) - self.n_steps
for j in range(2, self.m):
d_cur = (self._l[self._j[j]] - self._l[self._j[j - 1]]) - self.n_steps
if d_cur < d_min:
d_min, i_min = d_cur, j
i_min = 0 if d_min >= 0 else i_min
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 = int(np.minimum(self.m, _n_generations + 1))
self._l[self._j[self._it - 1]] = _n_generations*self.period
pm[self._j[self._it - 1]] = p_c
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._a/v_n*(bd_3 - 1.0)
d[self._j[i]] = self._c/v_n*(1.0 - 1.0/bd_3)
if self._n_generations > 0: # for 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)
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