# This file is part of DEAP.
#
# DEAP is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as
# published by the Free Software Foundation, either version 3 of
# the License, or (at your option) any later version.
#
# DEAP is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public
# License along with DEAP. If not, see <http://www.gnu.org/licenses/>.
# Special thanks to Nikolaus Hansen for providing major part of
# this code. The CMA-ES algorithm is provided in many other languages
# and advanced versions at http://www.lri.fr/~hansen/cmaesintro.html.
"""A module that provides support for the Covariance Matrix Adaptation
Evolution Strategy.
"""
import copy
from math import sqrt, log, exp
from itertools import cycle
import warnings
import numpy
from . import tools
[docs]class Strategy(object):
"""
A strategy that will keep track of the basic parameters of the CMA-ES
algorithm ([Hansen2001]_).
:param centroid: An iterable object that indicates where to start the
evolution.
:param sigma: The initial standard deviation of the distribution.
:param parameter: One or more parameter to pass to the strategy as
described in the following table, optional.
+----------------+---------------------------+----------------------------+
| Parameter | Default | Details |
+================+===========================+============================+
| ``lambda_`` | ``int(4 + 3 * log(N))`` | Number of children to |
| | | produce at each generation,|
| | | ``N`` is the individual's |
| | | size (integer). |
+----------------+---------------------------+----------------------------+
| ``mu`` | ``int(lambda_ / 2)`` | The number of parents to |
| | | keep from the |
| | | lambda children (integer). |
+----------------+---------------------------+----------------------------+
| ``cmatrix`` | ``identity(N)`` | The initial covariance |
| | | matrix of the distribution |
| | | that will be sampled. |
+----------------+---------------------------+----------------------------+
| ``weights`` | ``"superlinear"`` | Decrease speed, can be |
| | | ``"superlinear"``, |
| | | ``"linear"`` or |
| | | ``"equal"``. |
+----------------+---------------------------+----------------------------+
| ``cs`` | ``(mueff + 2) / | Cumulation constant for |
| | (N + mueff + 3)`` | step-size. |
+----------------+---------------------------+----------------------------+
| ``damps`` | ``1 + 2 * max(0, sqrt(( | Damping for step-size. |
| | mueff - 1) / (N + 1)) - 1)| |
| | + cs`` | |
+----------------+---------------------------+----------------------------+
| ``ccum`` | ``4 / (N + 4)`` | Cumulation constant for |
| | | covariance matrix. |
+----------------+---------------------------+----------------------------+
| ``ccov1`` | ``2 / ((N + 1.3)^2 + | Learning rate for rank-one |
| | mueff)`` | update. |
+----------------+---------------------------+----------------------------+
| ``ccovmu`` | ``2 * (mueff - 2 + 1 / | Learning rate for rank-mu |
| | mueff) / ((N + 2)^2 + | update. |
| | mueff)`` | |
+----------------+---------------------------+----------------------------+
.. [Hansen2001] Hansen and Ostermeier, 2001. Completely Derandomized
Self-Adaptation in Evolution Strategies. *Evolutionary Computation*
"""
def __init__(self, centroid, sigma, **kargs):
self.params = kargs
# Create a centroid as a numpy array
self.centroid = numpy.array(centroid)
self.dim = len(self.centroid)
self.sigma = sigma
self.pc = numpy.zeros(self.dim)
self.ps = numpy.zeros(self.dim)
self.chiN = sqrt(self.dim) * (1 - 1. / (4. * self.dim) +
1. / (21. * self.dim ** 2))
self.C = self.params.get("cmatrix", numpy.identity(self.dim))
self.diagD, self.B = numpy.linalg.eigh(self.C)
indx = numpy.argsort(self.diagD)
self.diagD = self.diagD[indx] ** 0.5
self.B = self.B[:, indx]
self.BD = self.B * self.diagD
self.cond = self.diagD[indx[-1]] / self.diagD[indx[0]]
self.lambda_ = self.params.get("lambda_", int(4 + 3 * log(self.dim)))
self.update_count = 0
self.computeParams(self.params)
[docs] def generate(self, ind_init):
r"""Generate a population of :math:`\lambda` individuals of type
*ind_init* from the current strategy.
:param ind_init: A function object that is able to initialize an
individual from a list.
:returns: A list of individuals.
"""
arz = numpy.random.standard_normal((self.lambda_, self.dim))
arz = self.centroid + self.sigma * numpy.dot(arz, self.BD.T)
return [ind_init(a) for a in arz]
[docs] def update(self, population):
"""Update the current covariance matrix strategy from the
*population*.
:param population: A list of individuals from which to update the
parameters.
"""
population.sort(key=lambda ind: ind.fitness, reverse=True)
old_centroid = self.centroid
self.centroid = numpy.dot(self.weights, population[0:self.mu])
c_diff = self.centroid - old_centroid
# Cumulation : update evolution path
self.ps = (1 - self.cs) * self.ps \
+ sqrt(self.cs * (2 - self.cs) * self.mueff) / self.sigma \
* numpy.dot(self.B, (1. / self.diagD) *
numpy.dot(self.B.T, c_diff))
hsig = float((numpy.linalg.norm(self.ps) /
sqrt(1. - (1. - self.cs) ** (2. * (self.update_count + 1.))) / self.chiN <
(1.4 + 2. / (self.dim + 1.))))
self.update_count += 1
self.pc = (1 - self.cc) * self.pc + hsig \
* sqrt(self.cc * (2 - self.cc) * self.mueff) / self.sigma \
* c_diff
# Update covariance matrix
artmp = population[0:self.mu] - old_centroid
self.C = (1 - self.ccov1 - self.ccovmu + (1 - hsig) *
self.ccov1 * self.cc * (2 - self.cc)) * self.C \
+ self.ccov1 * numpy.outer(self.pc, self.pc) \
+ self.ccovmu * numpy.dot((self.weights * artmp.T), artmp) \
/ self.sigma ** 2
self.sigma *= numpy.exp((numpy.linalg.norm(self.ps) / self.chiN - 1.) *
self.cs / self.damps)
self.diagD, self.B = numpy.linalg.eigh(self.C)
indx = numpy.argsort(self.diagD)
self.cond = self.diagD[indx[-1]] / self.diagD[indx[0]]
self.diagD = self.diagD[indx] ** 0.5
self.B = self.B[:, indx]
self.BD = self.B * self.diagD
[docs] def computeParams(self, params):
r"""Computes the parameters depending on :math:`\lambda`. It needs to
be called again if :math:`\lambda` changes during evolution.
:param params: A dictionary of the manually set parameters.
"""
self.mu = params.get("mu", int(self.lambda_ / 2))
rweights = params.get("weights", "superlinear")
if rweights == "superlinear":
self.weights = log(self.mu + 0.5) - \
numpy.log(numpy.arange(1, self.mu + 1))
elif rweights == "linear":
self.weights = self.mu + 0.5 - numpy.arange(1, self.mu + 1)
elif rweights == "equal":
self.weights = numpy.ones(self.mu)
else:
raise RuntimeError("Unknown weights : %s" % rweights)
self.weights /= sum(self.weights)
self.mueff = 1. / sum(self.weights ** 2)
self.cc = params.get("ccum", 4. / (self.dim + 4.))
self.cs = params.get("cs", (self.mueff + 2.) /
(self.dim + self.mueff + 3.))
self.ccov1 = params.get("ccov1", 2. / ((self.dim + 1.3) ** 2 +
self.mueff))
self.ccovmu = params.get("ccovmu", 2. * (self.mueff - 2. +
1. / self.mueff) /
((self.dim + 2.) ** 2 + self.mueff))
self.ccovmu = min(1 - self.ccov1, self.ccovmu)
self.damps = 1. + 2. * max(0, sqrt((self.mueff - 1.) /
(self.dim + 1.)) - 1.) + self.cs
self.damps = params.get("damps", self.damps)
[docs]class StrategyOnePlusLambda(object):
r"""
A CMA-ES strategy that uses the :math:`1 + \lambda` paradigm ([Igel2007]_).
:param parent: An iterable object that indicates where to start the
evolution. The parent requires a fitness attribute.
:param sigma: The initial standard deviation of the distribution.
:param lambda_: Number of offspring to produce from the parent.
(optional, defaults to 1)
:param parameter: One or more parameter to pass to the strategy as
described in the following table. (optional)
Other parameters can be provided as described in the next table
+----------------+---------------------------+----------------------------+
| Parameter | Default | Details |
+================+===========================+============================+
| ``d`` | ``1.0 + N / (2.0 * | Damping for step-size. |
| | lambda_)`` | |
+----------------+---------------------------+----------------------------+
| ``ptarg`` | ``1.0 / (5 + sqrt(lambda_)| Target success rate. |
| | / 2.0)`` | |
+----------------+---------------------------+----------------------------+
| ``cp`` | ``ptarg * lambda_ / (2.0 +| Step size learning rate. |
| | ptarg * lambda_)`` | |
+----------------+---------------------------+----------------------------+
| ``cc`` | ``2.0 / (N + 2.0)`` | Cumulation time horizon. |
+----------------+---------------------------+----------------------------+
| ``ccov`` | ``2.0 / (N**2 + 6.0)`` | Covariance matrix learning |
| | | rate. |
+----------------+---------------------------+----------------------------+
| ``pthresh`` | ``0.44`` | Threshold success rate. |
+----------------+---------------------------+----------------------------+
.. [Igel2007] Igel, Hansen, Roth, 2007. Covariance matrix adaptation for
multi-objective optimization. *Evolutionary Computation* Spring;15(1):1-28
"""
def __init__(self, parent, sigma, **kargs):
self.parent = parent
self.sigma = sigma
self.dim = len(self.parent)
self.C = numpy.identity(self.dim)
self.A = numpy.identity(self.dim)
self.pc = numpy.zeros(self.dim)
self.computeParams(kargs)
self.psucc = self.ptarg
[docs] def computeParams(self, params):
r"""Computes the parameters depending on :math:`\lambda`. It needs to
be called again if :math:`\lambda` changes during evolution.
:param params: A dictionary of the manually set parameters.
"""
# Selection :
self.lambda_ = params.get("lambda_", 1)
# Step size control :
self.d = params.get("d", 1.0 + self.dim / (2.0 * self.lambda_))
self.ptarg = params.get("ptarg", 1.0 / (5 + sqrt(self.lambda_) / 2.0))
self.cp = params.get("cp", self.ptarg * self.lambda_ / (2 + self.ptarg * self.lambda_))
# Covariance matrix adaptation
self.cc = params.get("cc", 2.0 / (self.dim + 2.0))
self.ccov = params.get("ccov", 2.0 / (self.dim ** 2 + 6.0))
self.pthresh = params.get("pthresh", 0.44)
[docs] def generate(self, ind_init):
r"""Generate a population of :math:`\lambda` individuals of type
*ind_init* from the current strategy.
:param ind_init: A function object that is able to initialize an
individual from a list.
:returns: A list of individuals.
"""
# self.y = numpy.dot(self.A, numpy.random.standard_normal(self.dim))
arz = numpy.random.standard_normal((self.lambda_, self.dim))
arz = self.parent + self.sigma * numpy.dot(arz, self.A.T)
return [ind_init(a) for a in arz]
[docs] def update(self, population):
"""Update the current covariance matrix strategy from the
*population*.
:param population: A list of individuals from which to update the
parameters.
"""
population.sort(key=lambda ind: ind.fitness, reverse=True)
lambda_succ = sum(self.parent.fitness <= ind.fitness for ind in population)
p_succ = float(lambda_succ) / self.lambda_
self.psucc = (1 - self.cp) * self.psucc + self.cp * p_succ
if self.parent.fitness <= population[0].fitness:
x_step = (population[0] - numpy.array(self.parent)) / self.sigma
self.parent = copy.deepcopy(population[0])
if self.psucc < self.pthresh:
self.pc = (1 - self.cc) * self.pc + sqrt(self.cc * (2 - self.cc)) * x_step
self.C = (1 - self.ccov) * self.C + self.ccov * numpy.outer(self.pc, self.pc)
else:
self.pc = (1 - self.cc) * self.pc
self.C = (1 - self.ccov) * self.C + self.ccov * (numpy.outer(self.pc, self.pc) + self.cc * (2 - self.cc) * self.C)
self.sigma = self.sigma * exp(1.0 / self.d * (self.psucc - self.ptarg) / (1.0 - self.ptarg))
# We use Cholesky since for now we have no use of eigen decomposition
# Basically, Cholesky returns a matrix A as C = A*A.T
# Eigen decomposition returns two matrix B and D^2 as C = B*D^2*B.T = B*D*D*B.T
# So A == B*D
# To compute the new individual we need to multiply each vector z by A
# as y = centroid + sigma * A*z
# So the Cholesky is more straightforward as we don't need to compute
# the squareroot of D^2, and multiply B and D in order to get A, we directly get A.
# This can't be done (without cost) with the standard CMA-ES as the eigen decomposition is used
# to compute covariance matrix inverse in the step-size evolutionary path computation.
self.A = numpy.linalg.cholesky(self.C)
[docs]class StrategyMultiObjective(object):
"""Multiobjective CMA-ES strategy based on the paper [Voss2010]_. It
is used similarly as the standard CMA-ES strategy with a generate-update
scheme.
:param population: An initial population of individual.
:param sigma: The initial step size of the complete system.
:param mu: The number of parents to use in the evolution. When not
provided it defaults to the length of *population*. (optional)
:param lambda_: The number of offspring to produce at each generation.
(optional, defaults to 1)
:param indicator: The indicator function to use. (optional, default to
:func:`~deap.tools.hypervolume`)
Other parameters can be provided as described in the next table
+----------------+---------------------------+----------------------------+
| Parameter | Default | Details |
+================+===========================+============================+
| ``d`` | ``1.0 + N / 2.0`` | Damping for step-size. |
+----------------+---------------------------+----------------------------+
| ``ptarg`` | ``1.0 / (5 + 1.0 / 2.0)`` | Target success rate. |
+----------------+---------------------------+----------------------------+
| ``cp`` | ``ptarg / (2.0 + ptarg)`` | Step size learning rate. |
+----------------+---------------------------+----------------------------+
| ``cc`` | ``2.0 / (N + 2.0)`` | Cumulation time horizon. |
+----------------+---------------------------+----------------------------+
| ``ccov`` | ``2.0 / (N**2 + 6.0)`` | Covariance matrix learning |
| | | rate. |
+----------------+---------------------------+----------------------------+
| ``pthresh`` | ``0.44`` | Threshold success rate. |
+----------------+---------------------------+----------------------------+
.. [Voss2010] Voss, Hansen, Igel, "Improved Step Size Adaptation
for the MO-CMA-ES", 2010.
"""
def __init__(self, population, sigma, **params):
self.parents = population
self.dim = len(self.parents[0])
# Selection
self.mu = params.get("mu", len(self.parents))
self.lambda_ = params.get("lambda_", 1)
# Step size control
self.d = params.get("d", 1.0 + self.dim / 2.0)
self.ptarg = params.get("ptarg", 1.0 / (5.0 + 0.5))
self.cp = params.get("cp", self.ptarg / (2.0 + self.ptarg))
# Covariance matrix adaptation
self.cc = params.get("cc", 2.0 / (self.dim + 2.0))
self.ccov = params.get("ccov", 2.0 / (self.dim ** 2 + 6.0))
self.pthresh = params.get("pthresh", 0.44)
# Internal parameters associated to the mu parent
self.sigmas = [sigma] * len(population)
# Lower Cholesky matrix (Sampling matrix)
self.A = [numpy.identity(self.dim) for _ in range(len(population))]
# Inverse Cholesky matrix (Used in the update of A)
self.invCholesky = [numpy.identity(self.dim) for _ in range(len(population))]
self.pc = [numpy.zeros(self.dim) for _ in range(len(population))]
self.psucc = [self.ptarg] * len(population)
self.indicator = params.get("indicator", tools.hypervolume)
[docs] def generate(self, ind_init):
r"""Generate a population of :math:`\lambda` individuals of type
*ind_init* from the current strategy.
:param ind_init: A function object that is able to initialize an
individual from a list.
:returns: A list of individuals with a private attribute :attr:`_ps`.
This last attribute is essential to the update function, it
indicates that the individual is an offspring and the index
of its parent.
"""
arz = numpy.random.randn(self.lambda_, self.dim)
individuals = list()
# Make sure every parent has a parent tag and index
for i, p in enumerate(self.parents):
p._ps = "p", i
# Each parent produce an offspring
if self.lambda_ == self.mu:
for i in range(self.lambda_):
# print "Z", list(arz[i])
individuals.append(ind_init(self.parents[i] + self.sigmas[i] * numpy.dot(self.A[i], arz[i])))
individuals[-1]._ps = "o", i
# Parents producing an offspring are chosen at random from the first front
else:
ndom = tools.sortLogNondominated(self.parents, len(self.parents), first_front_only=True)
for i in range(self.lambda_):
j = numpy.random.randint(0, len(ndom))
_, p_idx = ndom[j]._ps
individuals.append(ind_init(self.parents[p_idx] + self.sigmas[p_idx] * numpy.dot(self.A[p_idx], arz[i])))
individuals[-1]._ps = "o", p_idx
return individuals
def _select(self, candidates):
if len(candidates) <= self.mu:
return candidates, []
pareto_fronts = tools.sortLogNondominated(candidates, len(candidates))
chosen = list()
mid_front = None
not_chosen = list()
# Fill the next population (chosen) with the fronts until there is not enough space
# When an entire front does not fit in the space left we rely on the hypervolume
# for this front
# The remaining fronts are explicitly not chosen
full = False
for front in pareto_fronts:
if len(chosen) + len(front) <= self.mu and not full:
chosen += front
elif mid_front is None and len(chosen) < self.mu:
mid_front = front
# With this front, we selected enough individuals
full = True
else:
not_chosen += front
# Separate the mid front to accept only k individuals
k = self.mu - len(chosen)
if k > 0:
# reference point is chosen in the complete population
# as the worst in each dimension +1
ref = numpy.array([ind.fitness.wvalues for ind in candidates]) * -1
ref = numpy.max(ref, axis=0) + 1
for _ in range(len(mid_front) - k):
idx = self.indicator(mid_front, ref=ref)
not_chosen.append(mid_front.pop(idx))
chosen += mid_front
return chosen, not_chosen
def _rankOneUpdate(self, invCholesky, A, alpha, beta, v):
w = numpy.dot(invCholesky, v)
# Under this threshold, the update is mostly noise
if w.max() > 1e-20:
w_inv = numpy.dot(w, invCholesky)
norm_w2 = numpy.sum(w ** 2)
a = sqrt(alpha)
root = numpy.sqrt(1 + beta / alpha * norm_w2)
b = a / norm_w2 * (root - 1)
A = a * A + b * numpy.outer(v, w)
invCholesky = 1.0 / a * invCholesky - b / (a ** 2 + a * b * norm_w2) * numpy.outer(w, w_inv)
return invCholesky, A
[docs] def update(self, population):
"""Update the current covariance matrix strategies from the
*population*.
:param population: A list of individuals from which to update the
parameters.
"""
chosen, not_chosen = self._select(population + self.parents)
cp, cc, ccov = self.cp, self.cc, self.ccov
d, ptarg, pthresh = self.d, self.ptarg, self.pthresh
# Make copies for chosen offspring only
last_steps = [self.sigmas[ind._ps[1]] if ind._ps[0] == "o" else None for ind in chosen]
sigmas = [self.sigmas[ind._ps[1]] if ind._ps[0] == "o" else None for ind in chosen]
invCholesky = [self.invCholesky[ind._ps[1]].copy() if ind._ps[0] == "o" else None for ind in chosen]
A = [self.A[ind._ps[1]].copy() if ind._ps[0] == "o" else None for ind in chosen]
pc = [self.pc[ind._ps[1]].copy() if ind._ps[0] == "o" else None for ind in chosen]
psucc = [self.psucc[ind._ps[1]] if ind._ps[0] == "o" else None for ind in chosen]
# Update the internal parameters for successful offspring
for i, ind in enumerate(chosen):
t, p_idx = ind._ps
# Only the offspring update the parameter set
if t == "o":
# Update (Success = 1 since it is chosen)
psucc[i] = (1.0 - cp) * psucc[i] + cp
sigmas[i] = sigmas[i] * exp((psucc[i] - ptarg) / (d * (1.0 - ptarg)))
if psucc[i] < pthresh:
xp = numpy.array(ind)
x = numpy.array(self.parents[p_idx])
pc[i] = (1.0 - cc) * pc[i] + sqrt(cc * (2.0 - cc)) * (xp - x) / last_steps[i]
invCholesky[i], A[i] = self._rankOneUpdate(invCholesky[i], A[i], 1 - ccov, ccov, pc[i])
else:
pc[i] = (1.0 - cc) * pc[i]
pc_weight = cc * (2.0 - cc)
invCholesky[i], A[i] = self._rankOneUpdate(invCholesky[i], A[i], 1 - ccov + pc_weight, ccov, pc[i])
self.psucc[p_idx] = (1.0 - cp) * self.psucc[p_idx] + cp
self.sigmas[p_idx] = self.sigmas[p_idx] * exp((self.psucc[p_idx] - ptarg) / (d * (1.0 - ptarg)))
# It is unnecessary to update the entire parameter set for not chosen individuals
# Their parameters will not make it to the next generation
for ind in not_chosen:
t, p_idx = ind._ps
# Only the offspring update the parameter set
if t == "o":
self.psucc[p_idx] = (1.0 - cp) * self.psucc[p_idx]
self.sigmas[p_idx] = self.sigmas[p_idx] * exp((self.psucc[p_idx] - ptarg) / (d * (1.0 - ptarg)))
# Make a copy of the internal parameters
# The parameter is in the temporary variable for offspring and in the original one for parents
self.parents = chosen
self.sigmas = [sigmas[i] if ind._ps[0] == "o" else self.sigmas[ind._ps[1]] for i, ind in enumerate(chosen)]
self.invCholesky = [invCholesky[i] if ind._ps[0] == "o" else self.invCholesky[ind._ps[1]] for i, ind in enumerate(chosen)]
self.A = [A[i] if ind._ps[0] == "o" else self.A[ind._ps[1]] for i, ind in enumerate(chosen)]
self.pc = [pc[i] if ind._ps[0] == "o" else self.pc[ind._ps[1]] for i, ind in enumerate(chosen)]
self.psucc = [psucc[i] if ind._ps[0] == "o" else self.psucc[ind._ps[1]] for i, ind in enumerate(chosen)]
class StrategyActiveOnePlusLambda(object):
"""A CMA-ES strategy that combines the :math:`(1 + \\lambda)` paradigm
[Igel2007]_, the mixed integer modification [Hansen2011]_, active
covariance update [Arnold2010]_ and constraint handling [Arnold2012]_.
This version of CMA-ES requires the random vector and the mutation
that created each individual. The vector and mutation are stored in each
individual as :attr:`_z` and :attr:`_y` respectively. Updating with
individuals not containing these attributes will result in an
:class:`AttributeError`.
Notes:
When using this strategy (especially when using constraints) you should
monitor the strategy :attr:`condition_number`. If it goes above a given
threshold (say :math:`10^{12}`), you should think of restarting the
optimization as the covariance matrix is going degenerate. See the
constrained active CMA-ES example for a simple example of restart.
:param parent: An iterable object that indicates where to start the
evolution. The parent requires a fitness attribute.
:param sigma: The initial standard deviation of the distribution.
:param step: The minimal step size for each dimension. Use 0 for
continuous dimensions.
:param lambda_: Number of offspring to produce from the parent.
(optional, defaults to 1)
:param **kwargs: One or more parameter to pass to the strategy as
described in the following table. (optional)
+----------------+---------------------------+------------------------------+
| Parameter | Default | Details |
+================+===========================+==============================+
| ``d`` | ``1.0 + N / (2.0 * | Damping for step-size. |
| | lambda_)`` | |
+----------------+---------------------------+------------------------------+
| ``ptarg`` | ``1.0 / (5 + sqrt(lambda_)| Taget success rate |
| | / 2.0)`` | (from 1 + lambda algorithm). |
+----------------+---------------------------+------------------------------+
| ``cp`` | ``ptarg * lambda_ / (2.0 +| Step size learning rate. |
| | ptarg * lambda_)`` | |
+----------------+---------------------------+------------------------------+
| ``cc`` | ``2.0 / (N + 2.0)`` | Cumulation time horizon. |
+----------------+---------------------------+------------------------------+
| ``ccov`` | ``2.0 / (N**2 + 6.0)`` | Covariance matrix learning |
| | | rate. |
+----------------+---------------------------+------------------------------+
| ``ccovn`` | ``0.4 / (N**1.6 + 1.0)`` | Covariance matrix negative |
| | | learning rate. |
+----------------+---------------------------+------------------------------+
| ``cconst`` | ``1.0 / (N + 2.0)`` | Constraint vectors learning |
| | | rate. |
+----------------+---------------------------+------------------------------+
| ``beta`` | ``0.1 / (lambda_ * (N + | Covariance matrix learning |
| | 2.0))`` | rate for constraints. |
| | | |
+----------------+---------------------------+------------------------------+
| ``pthresh`` | ``0.44`` | Threshold success rate. |
+----------------+---------------------------+------------------------------+
.. [Igel2007] Igel, Hansen and Roth. Covariance matrix adaptation for
multi-objective optimization. 2007
.. [Arnold2010] Arnold and Hansen. Active covariance matrix adaptation for
the (1+1)-CMA-ES. 2010.
.. [Hansen2011] Hansen. A CMA-ES for Mixed-Integer Nonlinear Optimization.
Research Report] RR-7751, INRIA. 2011
.. [Arnold2012] Arnold and Hansen. A (1+1)-CMA-ES for Constrained Optimisation.
2012
"""
def __init__(self, parent, sigma, steps, **kargs):
self.parent = parent
self.sigma = sigma
self.dim = len(self.parent)
self.A = numpy.identity(self.dim)
self.invA = numpy.identity(self.dim)
self.condition_number = numpy.linalg.cond(self.A)
self.pc = numpy.zeros(self.dim)
# Save parameters
self.params = kargs.copy()
# Covariance matrix adaptation
self.cc = self.params.get("cc", 2.0 / (self.dim + 2.0))
self.ccovp = self.params.get("ccovp", 2.0 / (self.dim ** 2 + 6.0))
self.ccovn = self.params.get("ccovn", 0.4 / (self.dim ** 1.6 + 1.0))
self.cconst = self.params.get("cconst", 1.0 / (self.dim + 2.0))
self.pthresh = self.params.get("pthresh", 0.44)
self.lambda_ = self.params.get("lambda_", 1)
self.psucc = self.ptarg
self.S_int = numpy.array(steps)
self.i_I_R = numpy.flatnonzero(2 * self.sigma * numpy.diag(self.A)**0.5
< self.S_int)
self.constraint_vecs = None
self.ancestors_fitness = list()
@property
def lambda_(self):
return self._lambda
@lambda_.setter
def lambda_(self, value):
self._lambda = value
self._compute_lambda_parameters()
def _compute_lambda_parameters(self):
"""Computes the parameters depending on :math:`\lambda`. It needs to
be called again if :math:`\lambda` changes during evolution.
"""
# Step size control :
self.d = self.params.get("d", 1.0 + self.dim / (2.0 * self.lambda_))
self.ptarg = self.params.get("ptarg", 1.0 / (5 + numpy.sqrt(self.lambda_)
/ 2.0))
self.cp = self.params.get("cp", (self.ptarg * self.lambda_
/ (2 + self.ptarg * self.lambda_)))
self.beta = self.params.get("beta", 0.1 / (self.lambda_ * (self.dim + 2.0)))
def generate(self, ind_init):
"""Generate a population of :math:`\lambda` individuals of type
*ind_init* from the current strategy.
:param ind_init: A function object that is able to initialize an
individual from a list.
:returns: A list of individuals.
"""
# Generate individuals
z = numpy.random.standard_normal((self.lambda_, self.dim))
y = numpy.dot(self.A, z.T).T
x = self.parent + self.sigma * y + self.S_int * self._integer_mutation()
if any(self.S_int > 0):
# Bring values to the integer steps
round_values = numpy.tile(self.S_int > 0, (self.lambda_, 1))
steps = numpy.tile(self.S_int, (self.lambda_, 1))
x[round_values] = steps[round_values] * numpy.around(x[round_values]
/ steps[round_values])
# The update method requires to remember the y of each individual
population = list(map(ind_init, x))
for ind, yi, zi in zip(population, y, z):
ind._y = yi
ind._z = zi
return population
def _integer_mutation(self):
n_I_R = self.i_I_R.shape[0]
# Mixed integer CMA-ES is developped for (mu/mu , lambda)
# We have a (1 + lambda) setting, thus we make the integer mutation
# probabilistic. The integer mutation is lambda / 2 if all dimensions
# are integers or min(lambda / 2 - 1, lambda / 10 + n_I_R + 1). The minus
# 1 accounts for the last new candidate getting its integer mutation from
# the last best solution. We skip this last best solution part.
if n_I_R == 0:
return numpy.zeros((self.lambda_, self.dim))
elif n_I_R == self.dim:
p = self.lambda_ / 2.0 / self.lambda_
# lambda_int = int(numpy.floor(self.lambda_ / 2))
else:
p = (min(self.lambda_ / 2.0, self.lambda_ / 10.0 + n_I_R / self.dim)
/ self.lambda_)
# lambda_int = int(min(numpy.floor(self.lambda_ / 10) + n_I_R + 1,
# numpy.floor(self.lambda_ / 2) - 1))
Rp = numpy.zeros((self.lambda_, self.dim))
Rpp = numpy.zeros((self.lambda_, self.dim))
# Ri' has exactly one of its components set to one.
# The Ri' are dependent in that the number of mutations for each coordinate
# differs at most by one
for i, j in zip(range(self.lambda_), cycle(self.i_I_R)):
# Probabilistically choose lambda_int individuals
if numpy.random.rand() < p:
Rp[i, j] = 1
Rpp[i, j] = numpy.random.geometric(p=0.7**(1.0 / n_I_R)) - 1
I_pm1 = (-1)**numpy.random.randint(0, 2, (self.lambda_, self.dim))
R_int = I_pm1 * (Rp + Rpp)
# Usually in mu/mu, lambda the last individual is set to the step taken.
# We don't use this sheme in the 1 + lambda scheme
# if self.update_count > 0:
# R_int[-1, :] = (numpy.floor(-self.S_int - self.last_best)
# - numpy.floor(-self.S_int - self.centroid))
return R_int
def _rank1update(self, individual, p_succ):
update_cov = False
self.psucc = (1 - self.cp) * self.psucc + self.cp * p_succ
if not hasattr(self.parent, "fitness") \
or self.parent.fitness <= individual.fitness:
self.parent = copy.deepcopy(individual)
self.ancestors_fitness.append(copy.deepcopy(individual.fitness))
if len(self.ancestors_fitness) > 5:
self.ancestors_fitness.pop()
# Must guard if pc is all 0 to prevent w_norm_sqrd to be 0
if self.psucc < self.pthresh or numpy.allclose(self.pc, 0):
self.pc = (1 - self.cc) * self.pc + (numpy.sqrt(self.cc * (2 - self.cc))
* individual._y)
a = numpy.sqrt(1 - self.ccovp)
w = numpy.dot(self.invA, self.pc)
w_norm_sqrd = numpy.linalg.norm(w) ** 2
b = numpy.sqrt(1 - self.ccovp) / w_norm_sqrd \
* (numpy.sqrt(1 + self.ccovp / (1 - self.ccovp) * w_norm_sqrd)
- 1)
else:
self.pc = (1 - self.cc) * self.pc
d = self.ccovp * (1 + self.cc * (2 - self.cc))
a = numpy.sqrt(1 - d)
w = numpy.dot(self.invA, self.pc)
w_norm_sqrd = numpy.linalg.norm(w) ** 2
b = numpy.sqrt(1 - d) \
* (numpy.sqrt(1 + self.ccovp * w_norm_sqrd / (1 - d)) - 1) \
/ w_norm_sqrd
update_cov = True
elif len(self.ancestors_fitness) >= 5 \
and individual.fitness < self.ancestors_fitness[0] \
and self.psucc < self.pthresh:
# Active covariance update requires w = z and not w = inv(A)s
w = individual._z
w_norm_sqrd = numpy.linalg.norm(w) ** 2
if 1 < self.ccovn * (2 * w_norm_sqrd - 1):
ccovn = 1 / (2 * w_norm_sqrd - 1)
else:
ccovn = self.ccovn
a = numpy.sqrt(1 + ccovn)
b = numpy.sqrt(1 + ccovn) / w_norm_sqrd \
* (numpy.sqrt(1 - ccovn / (1 + ccovn) * w_norm_sqrd) - 1)
update_cov = True
if update_cov:
self.A = self.A * a + b * numpy.outer(numpy.dot(self.A, w), w)
self.invA = (1 / a * self.invA
- b / (a ** 2 + a * b * w_norm_sqrd)
* numpy.dot(self.invA, numpy.outer(w, w)))
# TODO: Add integer mutation i_I_R component
self.sigma = self.sigma * numpy.exp(1.0 / self.d
* ((self.psucc - self.ptarg)
/ (1.0 - self.ptarg)))
def _infeasible_update(self, individual):
if not hasattr(individual.fitness, "constraint_violation"):
return
if self.constraint_vecs is None:
shape = len(individual.fitness.constraint_violation), self.dim
self.constraint_vecs = numpy.zeros(shape)
for i in range(self.constraint_vecs.shape[0]):
if individual.fitness.constraint_violation[i]:
self.constraint_vecs[i] = (1 - self.cconst) * self.constraint_vecs[i] \
+ self.cconst * individual._y
W = numpy.dot(self.invA, self.constraint_vecs.T).T # M x N
constraint_violation = numpy.sum(individual.fitness.constraint_violation)
A_prime = (
self.A - self.beta / constraint_violation
* numpy.sum(
list(
numpy.outer(self.constraint_vecs[i], W[i])
/ numpy.dot(W[i], W[i])
for i in range(self.constraint_vecs.shape[0])
if individual.fitness.constraint_violation[i]
),
axis=0
)
)
try:
self.invA = numpy.linalg.inv(A_prime)
except numpy.linalg.LinAlgError:
warnings.warn("Singular matrix inversion, "
"invalid update in CMA-ES ignored", RuntimeWarning)
else:
self.A = A_prime
def update(self, population):
"""Update the current covariance matrix strategy from the *population*.
:param population: A list of individuals from which to update the
parameters.
"""
valid_population = [ind for ind in population if ind.fitness.valid]
invalid_population = [ind for ind in population if not ind.fitness.valid]
if len(valid_population) > 0:
# Rank 1 update
valid_population.sort(key=lambda ind: ind.fitness, reverse=True)
if not hasattr(self.parent, "fitness"):
lambda_succ = len(valid_population)
else:
lambda_succ = sum(self.parent.fitness <= ind.fitness
for ind in valid_population)
# Use len(valid) to not account for individuals violating constraints
self._rank1update(valid_population[0],
float(lambda_succ) / len(valid_population))
if len(invalid_population) > 0:
# Learn constraint from all invalid individuals
for ind in invalid_population:
self._infeasible_update(ind)
# Used to monitor the convariance matrix conditioning
self.condition_number = numpy.linalg.cond(self.A)
C = numpy.dot(self.A, self.A.T)
self.i_I_R = numpy.flatnonzero(2 * self.sigma * numpy.diag(C)**0.5
< self.S_int)
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