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enes.py
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enes.py
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import numpy as np # engine for numerical computing
from pypop7.optimizers.nes.ones import ONES # Original Natural Evolution Strategy (ONES) class
def _combine_block(ll):
ll = [list(map(np.asmatrix, row)) for row in ll]
h = [m.shape[1] for m in ll[0]]
v, v_i = [row[0].shape[0] for row in ll], 0
mat = np.zeros((sum(h), sum(v)))
for i, row in enumerate(ll):
h_i = 0
for j, m in enumerate(row):
mat[v_i:v_i + v[i], h_i:h_i + h[j]] = m
h_i += h[j]
v_i += v[i]
return mat
class ENES(ONES):
"""Exact Natural Evolution Strategy (ENES).
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`):
* 'n_individuals' - number of offspring/descendants, aka offspring population size (`int`),
* 'n_parents' - number of parents/ancestors, aka parental population size (`int`),
* 'mean' - initial (starting) point (`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_mean' - learning rate of distribution mean update (`float`, default: `1.0`),
* 'lr_sigma' - learning rate of global step-size adaptation (`float`, default: `1.0`).
Examples
--------
Use the optimizer `ENES` 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.nes.enes import ENES
>>> 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,))}
>>> enes = ENES(problem, options) # initialize the optimizer class
>>> results = enes.optimize() # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"ENES: {results['n_function_evaluations']}, {results['best_so_far_y']}")
ENES: 5000, 0.00035668252927080496
Attributes
----------
lr_mean : `float`
learning rate of distribution mean update (should `> 0.0`).
lr_sigma : `float`
learning rate of global step-size adaptation (should `> 0.0`).
mean : `array_like`
initial (starting) point, aka mean of Gaussian search/sampling/mutation distribution.
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']`, by
default.
n_individuals : `int`
number of offspring/descendants, aka offspring population size (should `> 0`).
n_parents : `int`
number of parents/ancestors, aka parental population size (should `> 0`).
References
----------
Wierstra, D., Schaul, T., Glasmachers, T., Sun, Y., Peters, J. and Schmidhuber, J., 2014.
`Natural evolution strategies.
<https://jmlr.org/papers/v15/wierstra14a.html>`_
Journal of Machine Learning Research, 15(1), pp.949-980.
Schaul, T., 2011.
`Studies in continuous black-box optimization.
<https://people.idsia.ch/~schaul/publications/thesis.pdf>`_
Doctoral Dissertation, Technische Universität München.
Yi, S., Wierstra, D., Schaul, T. and Schmidhuber, J., 2009, June.
`Stochastic search using the natural gradient.
<https://dl.acm.org/doi/abs/10.1145/1553374.1553522>`_
In International Conference on Machine Learning (pp. 1161-1168). ACM.
Please refer to the *official* Python source code from `PyBrain` (now not actively maintained):
https://github.com/pybrain/pybrain/blob/master/pybrain/optimization/distributionbased/nes.py
"""
def __init__(self, problem, options):
"""Initialize all the hyper-parameters and also auxiliary class members.
"""
ONES.__init__(self, problem, options)
def _update_distribution(self, x=None, y=None, mean=None, cv=None):
"""Update the mean and covariance matrix of Gaussian search/sampling/mutation distribution.
"""
order = np.argsort(-y)
u = np.empty((self.n_individuals,))
for i, o in enumerate(order):
u[o] = self._u[i]
inv_d, inv_cv = np.linalg.inv(self._d_cv), np.linalg.inv(cv)
dd = np.diag(np.diag(inv_d))
v = np.zeros((self._n_distribution, self.n_individuals))
for k in range(self.n_individuals):
diff = x[k] - mean
s = np.dot(inv_d.T, diff)
v[:self.ndim_problem, k] += diff
v[self.ndim_problem:, k] += self._triu2flat(np.outer(s, np.dot(inv_d, s)) - dd)
j = self._n_distribution - 1
g = 1.0/(inv_cv[-1, -1] + np.square(inv_d[-1, -1]))
d = 1.0/inv_cv[-1, -1]
v[j, :] = np.dot(g, v[j, :])
j -= 1
for k in reversed(list(range(self.ndim_problem - 1))):
w = inv_cv[k, k]
w_g = w + np.square(inv_d[k, k])
q = np.dot(d, inv_cv[k + 1:, k])
c = np.dot(inv_cv[k + 1:, k], q)
r, r_g = 1.0/(w - c), 1.0/(w_g - c)
t, t_g = -(1.0 + r*c)/w, -(1.0 + r_g*c)/w_g
g = _combine_block([[r_g, t_g*q], [np.asmatrix(t_g*q).T, d + r_g*np.outer(q, q)]])
d = _combine_block([[r, t*q], [np.asmatrix(t*q).T, d + r*np.outer(q, q)]])
v[j - (self.ndim_problem - k - 1):j + 1, :] = np.dot(g, v[j - (self.ndim_problem - k - 1):j + 1, :])
j -= self.ndim_problem - k
grad, v_2 = np.zeros((self._n_distribution,)), v*v
j = self._n_distribution - 1
for k in reversed(list(range(self.ndim_problem))):
base = np.sum(v_2[j - (self.ndim_problem - k - 1):j + 1, :], 0)
grad[j - (self.ndim_problem - k - 1):j + 1] = np.dot(
v[j - (self.ndim_problem - k - 1):j + 1, :], u - np.dot(base, u)/np.sum(base))
j -= self.ndim_problem - k
base = np.sum(v_2[:j + 1, :], 0)
grad[:j + 1] = np.dot(v[:j + 1, :], (u - np.dot(base, u)/np.sum(base)))
grad /= self.n_individuals
# update the mean of Gaussian search/sampling/mutation distribution
mean += self.lr_mean * grad[:self.ndim_problem]
# update the covariance matrix of Gaussian search/sampling/mutation distribution
self._d_cv += self.lr_sigma * self._flat2triu(grad[self.ndim_problem:])
cv = np.dot(self._d_cv.T, self._d_cv) # to recover covariance matrix
self._n_generations += 1
return x, y, mean, cv