def test_induced_sampling(self):
"""
An integration test for the whole routine
"""
dimension = 3
parameters = [Parameter(3, "Uniform", upper=1, lower=-1)]*dimension
basis = Basis("total-order", [3]*dimension)
induced_sampling = Induced(parameters, basis)
quadrature_points = induced_sampling.get_points()
assert quadrature_points.shape == (induced_sampling.samples_number, 3)
def test_generate_sampling_class(self):
"""
test if the method returns a function object for sampling interface
"""
parameters = [Parameter(1, "gaussian")] * 3
basis = Basis("total order")
generator_class = Sampling(parameters, basis,
('induced-sampling', {
"sampling-ratio": 2,
"subsampling-optimisation": 'qr'
}))
assert generator_class.sampling_class.__class__ == InducedSampling
def _calculate_subspace(self, S, f):
parameters = [
Parameter(distribution='uniform',
lower=np.min(S[:, i]),
upper=np.max(S[:, i]),
order=1) for i in range(0, self.n)
]
self.poly = Poly(parameters, basis=Basis('total-order'), method='least-squares', \
sampling_args={'sample-points': S, 'sample-outputs': f})
self.poly.set_model()
self.Subs = Subspaces(full_space_poly=self.poly,
method='active-subspace',
subspace_dimension=self.d)
if self.subspace_method == 'variable-projection':
U0 = self.Subs.get_subspace()[:, :self.d]
self.Subs = Subspaces(method='variable-projection', sample_points=S, sample_outputs=f, \
subspace_init=U0, subspace_dimension=self.d, polynomial_degree=2, max_iter=300)
self.U = self.Subs.get_subspace()[:, :self.d]
elif self.subspace_method == 'active-subspaces':
U0 = self.Subs.get_subspace()[:, 1].reshape(-1, 1)
U1 = null_space(U0.T)
self.U = U0
for i in range(self.d - 1):
R = []
for j in range(U1.shape[1]):
U = np.hstack((self.U, U1[:, j].reshape(-1, 1)))
Y = np.dot(S, U)
myParameters = [Parameter(distribution='uniform', lower=np.min(Y[:,k]), upper=np.max(Y[:,k]), \
order=2) for k in range(Y.shape[1])]
myBasis = Basis('total-order')
poly = Poly(myParameters, myBasis, method='least-squares', \
sampling_args={'sample-points':Y, 'sample-outputs':f})
poly.set_model()
f_eval = poly.get_polyfit(Y)
_, _, r, _, _ = linregress(f_eval.flatten(), f.flatten())
R.append(r**2)
index = np.argmax(R)
self.U = np.hstack((self.U, U1[:, index].reshape(-1, 1)))
U1 = np.delete(U1, index, 1)
def _fit_poly(X, y):
# try:
N, d = X.shape
myParameters = []
for dimension in range(d):
values = X[:,dimension]
values_min = np.amin(values)
values_max = np.amax(values)
if (values_min - values_max) ** 2 < 0.01:
values_min -= 0.01
values_max += 0.01
myParameters.append(Parameter(distribution='Uniform', lower=values_min, upper=values_max, order=self.order))
else:
if self.distribution == 'uniform':
myParameters.append(Parameter(distribution='Uniform', lower=values_min, upper=values_max, order=self.order))
elif self.distribution == 'data':
input_dist = Weight(values, support=[values_min, values_max], pdf=False)
myParameters.append(Parameter(distribution='data',weight_function=input_dist,order=self.order))
if self.basis == "hyperbolic-basis":
myBasis = Basis(self.basis, orders=[self.order for _ in range(d)], q=0.5)
else:
myBasis = Basis(self.basis, orders=[self.order for _ in range(d)])
container["index_node_global"] += 1
poly = Poly(myParameters, myBasis, method=self.poly_method, sampling_args={'sample-points':X, 'sample-outputs':y}, solver_args=self.poly_solver_args)
poly.set_model()
mse = np.linalg.norm(y - poly.get_polyfit(X).reshape(-1)) ** 2 / N
# except Exception as e:
# print("Warning fitting of Poly failed:", e)
# print(d, values_min, values_max)
# mse, poly = np.inf, None
return mse, poly
def test_sampling(self):
d = 4
order = 5
param = Parameter(distribution='uniform',
order=order,
lower=-1.0, upper=1.0)
myparameters = [param for _ in range(d)]
mybasis = Basis('total-order')
mypoly = Poly(myparameters, mybasis,
method='least-squares',
sampling_args={'mesh': 'induced',
'subsampling-algorithm': 'qr',
'sampling-ratio': 1})
assert mypoly._quadrature_points.shape == (mypoly.basis.cardinality, d)
def vandermonde(eta, p):
_, n = eta.shape
listing = []
for i in range(0, n):
listing.append(p)
Object = Basis('total-order', listing)
#Establish n Parameter objects
params = []
P = Parameter(order=p, lower=-1, upper=1, distribution='uniform')
for i in range(0, n):
params.append(P)
#Use the params list to establish the Poly object
Polybasis = Poly(params, Object, method='least-squares')
V = Polybasis.get_poly(eta)
V = V.T
return V, Polybasis
def _set_sparsegrid_quadrature_rule(self, orders=None):
"""
Generates a sparse grid quadrature rule based on the parameters in Poly.
:param Poly self:
An instance of the Poly class.
:param list orders:
A list of the highest polynomial orders along each dimension.
:return:
**x**: A numpy.ndarray of sampled quadrature points with shape (number_of_samples, dimension).
**w**: A numpy.ndarray of the corresponding quadrature weights with shape (number_of_samples, 1).
"""
sparse_indices, sparse_factors, not_used = self.basis.get_basis()
rows = len(sparse_indices)
cols = len(sparse_indices[0])
# For storage we use dictionaries
points_store = {}
weights_store = {}
indices = np.zeros((rows))
self.tensor_product_list = []
for i in range(0,rows):
orders = sparse_indices[i,:]
myBasis = Basis('tensor-grid')
myTensor = Tensorgrid(parameters=self.parameters, basis=myBasis, orders=orders.astype(int) )
self.tensor_product_list.append(myTensor)
pts = myTensor.points
wts = myTensor.weights * sparse_factors[i]
points_store[i] = pts
weights_store[i] = wts
indices[i] = myTensor.basis.cardinality
del myTensor, myBasis
sum_indices = int(np.sum(indices))
points_saved = np.zeros((sum_indices, self.basis.dimensions))
weights_saved = np.zeros((sum_indices))
counter = int(0)
for i in range(0,rows):
for j in range(0, int(indices[i])):
for d in range(0, self.basis.dimensions):
points_saved[counter,d] = points_store[i][j, d]
weights_saved[counter] = weights_store[i][j]
counter = counter + 1
self.points , indices = np.unique(points_saved, axis=0, return_index=True)
self.weights = weights_saved[indices]
self.sparse_indices = sparse_indices
self.sparse_weights = sparse_factors
def __init__(self,
method,
full_space_poly=None,
sample_points=None,
sample_outputs=None,
polynomial_degree=2,
subspace_dimension=2,
bootstrap=False,
subspace_init=None,
max_iter=1000,
tol=None):
self.full_space_poly = full_space_poly
self.sample_points = sample_points
self.Y = None # for the zonotope vertices
if self.sample_points is not None:
self.sample_points = standardise(sample_points)
self.sample_outputs = sample_outputs
self.method = method
self.subspace_dimension = subspace_dimension
self.polynomial_degree = polynomial_degree
self.bootstrap = bootstrap
if self.method.lower() == 'active-subspace' or self.method.lower(
) == 'active-subspaces':
self.method == 'active-subspace'
if self.full_space_poly is None:
N, d = self.sample_points.shape
param = Parameter(distribution='uniform',
lower=-1,
upper=1.,
order=self.polynomial_degree)
myparameters = [param for _ in range(d)]
mybasis = Basis("total-order")
mypoly = Poly(myparameters, mybasis, method='least-squares', sampling_args={'sample-points':self.sample_points, \
'sample-outputs':self.sample_outputs})
mypoly.set_model()
self.full_space_poly = mypoly
self.sample_points = standardise(self.full_space_poly.get_points())
self.sample_outputs = self.full_space_poly.get_model_evaluations()
self._get_active_subspace()
elif self.method == 'variable-projection':
self._get_variable_projection(None, None, tol, max_iter,
subspace_init, False)
def test_samples(self):
"""
test if the method returns a function object for sampling interface
"""
dimension = 3
sampling_ratio = 3
parameters = [Parameter(1, "gaussian")] * dimension
basis = Basis("total order", [5] * dimension)
induced_sampling = InducedSampling(parameters, basis, sampling_ratio,
"qr")
# Mock additive mixture sampling
def func(array_):
return np.array([1] * dimension, float)
induced_sampling.additive_mixture_sampling = func
quadrature_points = induced_sampling.samples()
true_array = np.ones((dimension * sampling_ratio, dimension))
assert_array_equal(quadrature_points, true_array)
def _build_model(self, S, f, del_k):
"""
Constructs quadratic model for ``trust-region`` method
"""
myParameters = [
Parameter(distribution='uniform',
lower=S[0, i] - del_k,
upper=S[0, i] + del_k,
order=2) for i in range(S.shape[1])
]
myBasis = Basis('total-order')
my_poly = Poly(myParameters,
myBasis,
method='compressive-sensing',
sampling_args={
'sample-points': S,
'sample-outputs': f
})
my_poly.set_model()
return my_poly
def _set_statistics(self):
"""
Private method that is used withn the statistics routines.
"""
if self.statistics_object is None:
if self.method != 'numerical-integration' and self.dimensions <= 6 and self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
quad = Quadrature(parameters=self.parameters, basis=Basis('tensor-grid', orders= np.array(self.parameters_order) + 1), \
mesh='tensor-grid', points=None)
quad_pts, quad_wts = quad.get_points_and_weights()
poly_vandermonde_matrix = self.get_poly(quad_pts)
else:
poly_vandermonde_matrix = self.get_poly(self._quadrature_points)
quad_pts, quad_wts = self.get_points_and_weights()
if self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=self.highest_order)
else:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=MAXIMUM_ORDER_FOR_STATS)
def getPseudospectralCoefficients(self, function, override_orders=None):
if override_orders is None:
pts, wts = super(Polyint, self).getTensorQuadratureRule()
tensor_elements = self.basis.elements
P = super(Polyint, self).getPolynomial(pts)
else:
pts, wts = super(Polyint, self).getTensorQuadratureRule(override_orders)
tensor_basis = Basis('Tensor grid', override_orders)
tensor_elements = tensor_basis.elements
P = super(Polyint, self).getPolynomial(pts, tensor_elements)
m = len(wts)
W = np.mat( np.diag(np.sqrt(wts)))
A = np.mat(W * P.T)
if callable(function):
y = evalfunction(points=pts, function=function)
else:
y = function
b = np.dot( W , np.reshape(y, (m,1)) )
coefficients = np.dot(A.T , b)
return coefficients, tensor_elements, pts, wts
def _set_statistics(self):
"""
Private method that is used within the statistics routines.
"""
if self.statistics_object is None:
if hasattr(self, 'inv_R_Psi'):
# quad_pts, quad_wts = self.quadrature.get_points_and_weights()
N_quad = 20000
quad_pts = self.corr.get_correlated_samples(N=N_quad)
quad_wts = 1.0 / N_quad * np.ones(N_quad)
poly_vandermonde_matrix = self.get_poly(quad_pts)
elif self.method != 'numerical-integration' and self.dimensions <= 6 and self.highest_order <= MAXIMUM_ORDER_FOR_STATS:
quad = Quadrature(parameters=self.parameters, basis=Basis('tensor-grid', orders= np.array(self.parameters_order) + 1), \
mesh='tensor-grid', points=None)
quad_pts, quad_wts = quad.get_points_and_weights()
poly_vandermonde_matrix = self.get_poly(quad_pts)
elif self.mesh == 'monte-carlo':
quad = Quadrature(parameters=self.parameters,
basis=self.basis,
mesh=self.mesh,
points=None,
oversampling=10.0)
quad_pts, quad_wts = quad.get_points_and_weights()
N_quad = len(quad_wts)
quad_wts = 1.0 / N_quad * np.ones(N_quad)
poly_vandermonde_matrix = self.get_poly(quad_pts)
else:
poly_vandermonde_matrix = self.get_poly(
self._quadrature_points)
quad_pts, quad_wts = self.get_points_and_weights()
if self.highest_order <= MAXIMUM_ORDER_FOR_STATS and (
self.basis.basis_type.lower() == 'total-order'
or self.basis.basis_type.lower() == 'hyperbolic-basis'):
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=self.highest_order)
else:
self.statistics_object = Statistics(self.parameters, self.basis, self.coefficients, quad_pts, \
quad_wts, poly_vandermonde_matrix, max_sobol_order=MAXIMUM_ORDER_FOR_STATS)
def test_induced_jacobi_evaluation(self):
dimension = 3
parameters = [Parameter(1, "Uniform", upper=1, lower=-1)] * dimension
basis = Basis("total-order")
induced_sampling = Induced(parameters, basis)
parameter = parameters[0]
parameter.order = 3
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 0, parameter)
np.testing.assert_allclose(cdf_value, 0.5, atol=0.00001)
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 1, parameter)
assert cdf_value == 1
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, -1, parameter)
assert cdf_value == 0
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 0.6, parameter)
np.testing.assert_allclose(cdf_value, 0.7462, atol=0.00005)
cdf_value = induced_sampling.induced_jacobi_evaluation(
0, 0, 0.999, parameter)
np.testing.assert_allclose(cdf_value, 0.99652, atol=0.000005)
def get_subspace_polynomial(self):
"""
Returns a polynomial defined over the dimension reducing subspace.
:param Subspaces self:
An instance of the Subspaces object.
:return:
**subspacepoly**: A Poly object that defines a polynomial over the subspace. The distribution of parameters is
assumed to be uniform and the maximum and minimum bounds for each parameter are defined by the maximum and minimum values
of the project samples.
"""
active_subspace = self._subspace[:, 0:self.subspace_dimension]
projected_points = np.dot(self.sample_points, active_subspace)
myparameters = []
for i in range(0, self.subspace_dimension):
param = Parameter(distribution='uniform', lower=np.min(projected_points[:,i]), upper=np.max(projected_points[:,i]), \
order=self.polynomial_degree)
myparameters.append(param)
mybasis = Basis("total-order")
subspacepoly = Poly(myparameters, mybasis, method='least-squares', sampling_args={'sample-points':projected_points, \
'sample-outputs':self.sample_outputs})
subspacepoly.set_model()
return subspacepoly
def _fit_poly(X, y):
N, d = X.shape
myParameters = []
for dimension in range(d):
values = [X[i,dimension] for i in range(N)]
values_min = min(values)
values_max = max(values)
if (values_min - values_max) ** 2 < 0.01:
myParameters.append(Parameter(distribution='Uniform', lower=values_min-0.01, upper=values_max+0.01, order=self.order))
else:
myParameters.append(Parameter(distribution='Uniform', lower=values_min, upper=values_max, order=self.order))
myBasis = Basis('total-order')
y = np.reshape(y, (y.shape[0], 1))
poly = Poly(myParameters, myBasis, method='least-squares', sampling_args={'sample-points':X, 'sample-outputs':y})
poly.set_model()
mse = ((y-poly.get_polyfit(X))**2).mean()
return mse, poly
def __init__(self, D=None, R=None):
if D is None:
raise(ValueError, 'Distributions must be given')
else:
self.D = D
if R is None:
raise(ValueError, 'Correlation matrix must be specified')
else:
self.R = R
self.std = Parameter(order=5, distribution='normal',shape_parameter_A = 0.0, shape_parameter_B = 1.0)
#
# R0 = fictive matrix of correlated normal intermediate variables
#
# 1) Check the type of correlated marginals
# 2) Use Effective Quadrature for solving Legendre
# 3) Calculate the fictive matrix
inf_lim = -8.0
sup_lim = - inf_lim
p1 = Parameter(distribution = 'uniform', lower = inf_lim, upper = sup_lim, order = 31)
myBasis = Basis('Tensor grid')
Pols = Polyint([p1, p1], myBasis)
p = Pols.quadraturePoints
w = Pols.quadratureWeights * (sup_lim - inf_lim)**2
p1 = p[:,0]
p2 = p[:,1]
R0 = np.eye((len(self.D)))
for i in range(len(self.D)):
for j in range(i+1, len(self.D), 1):
if self.R[i,j] == 0:
R0[i,j] = 0.0
else:
tp11 = -(np.array(self.D[i].getiCDF(self.std.getCDF(points=p1))) - self.D[i].mean ) / np.sqrt( self.D[i].variance )
tp22 = -(np.array(self.D[j].getiCDF(self.std.getCDF(points=p2))) - self.D[j].mean)/np.sqrt( self.D[j].variance )
rho_ij = self.R[i,j]
bivariateNormalPDF = (1.0 / (2.0 * np.pi * np.sqrt(1.0-rho_ij**2)) * np.exp(-1.0/(2.0*(1.0 - rho_ij**2)) * (p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2 )))
coefficientsIntegral = np.flipud(tp11*tp22 * w)
def check_difference(rho_ij):
bivariateNormalPDF = (1.0 / (2.0 * np.pi * np.sqrt(1.0-rho_ij**2)) * np.exp(-1.0/(2.0*(1.0 - rho_ij**2)) * (p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2 )))
diff = np.dot(coefficientsIntegral, bivariateNormalPDF)
return diff - self.R[i,j]
if (self.D[i].name!='custom') or (self.D[j].name!='custom'):
rho = optimize.newton(check_difference, self.R[i,j], maxiter=50)
else:
res = optimize.least_squares(check_difference, R[i,j], bounds=(-0.999,0.999), ftol=1.e-03)
rho = res.x
print('A Custom Marginal is present')
R0[i,j] = rho
R0[j,i] = R0[i,j]
self.A = np.linalg.cholesky(R0)
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
def _well_poised_LU(self, S, f, S_hat, f_hat):
"""
Ensures the regression set is well-poised using the LU algorithm (proposed by Andrew Conn) for ``trust-region`` method
"""
# Poised constant of algorithm
psi = 1.0
# Generate natural monomial basis
Base = Basis('total-order', orders=np.tile([1], self.n))
basis = Base.get_basis()[:, range(self.n - 1, -1, -1)]
def natural_basis_function(x, basis):
phi = np.zeros(basis.shape[0])
for j in range(basis.shape[0]):
phi[j] = 1.0
for k in range(basis.shape[1]):
phi[j] *= (x[k]**basis[j, k]) / factorial(basis[j, k])
return phi
phi_function = lambda x: natural_basis_function(x, basis)
# Initialise U matrix of LU factorisation of M matrix (see Conn et al.)
U = np.zeros((self.p, self.p))
# Initialise the first row of U to the e1 basis vector which corresponds to solution with all zeros
U[0, 0] = 1.0
# Perform the LU factorisation algorithm for the rest of the points
for k in range(1, self.p):
v = np.zeros(self.p)
for j in range(k):
v[j] = -U[j, k] / U[j, j]
v[k] = 1.0
# If there are still points to choose from, find if points meet criterion. If so, use the index to choose
# point with given index to be next point in regression/interpolation set
if S_hat.size != 0:
M = self._natural_basis_matrix(S_hat, v, phi_function)
index2 = np.argmax(M)
if M[index2] < psi:
index2 = None
else:
index2 = None
# If index exists, choose the point with that index and delete it from possible choices
if index2 is not None:
s = S_hat[index2, :].flatten()
S = np.vstack((S, s))
f = np.vstack((f, f_hat[index2].flatten()))
S_hat = np.delete(S_hat, index2, 0)
f_hat = np.delete(f_hat, index2, 0)
phi = phi_function(s.flatten())
# If index doesn't exist, solve an optimisation point to find the point in the range which best satisfies criterion
else:
s = optimize.minimize(
lambda x: -abs(np.dot(v, phi_function(x.flatten()))),
np.zeros(self.n),
method='COBYLA',
constraints=[{
'type': 'ineq',
'fun': lambda x: 1.0 - x
}, {
'type': 'ineq',
'fun': lambda x: 1.0 + x
}],
options={'disp': False})['x'].flatten()
S = np.vstack((S, s))
f = np.vstack((f, np.array([np.inf])))
phi = phi_function(s.flatten())
# Update U factorisation in LU algorithm
U[k, k] = np.dot(v, phi)
for i in range(k + 1, self.p):
U[k, i] += phi[i]
for j in range(k):
U[k, i] -= (phi[j] * U[j, i]) / U[j, j]
return S, f, S_hat, f_hat
def __init__(self, poly, correlation_matrix, verbose=False):
self.poly = poly
D = self.poly.get_parameters()
self.D = D
self.R = correlation_matrix
self.std = Parameter(order=5,
distribution='normal',
shape_parameter_A=0.0,
shape_parameter_B=1.0)
inf_lim = -8.0
sup_lim = -inf_lim
p1 = Parameter(distribution='uniform',
lower=inf_lim,
upper=sup_lim,
order=31)
myBasis = Basis('tensor-grid')
Pols = Poly([p1, p1], myBasis, method='numerical-integration')
p = Pols.get_points()
w = Pols.get_weights() * (sup_lim - inf_lim)**2
p1 = p[:, 0]
p2 = p[:, 1]
R0 = np.eye((len(self.D)))
for i in range(len(self.D)):
for j in range(i + 1, len(self.D), 1):
if self.R[i, j] == 0:
R0[i, j] = 0.0
else:
tp11 = -(np.array(self.D[i].get_icdf(
self.std.get_cdf(points=p1))) -
self.D[i].mean) / np.sqrt(self.D[i].variance)
tp22 = -(np.array(self.D[j].get_icdf(
self.std.get_cdf(points=p2))) -
self.D[j].mean) / np.sqrt(self.D[j].variance)
rho_ij = self.R[i, j]
bivariateNormalPDF = (
1.0 / (2.0 * np.pi * np.sqrt(1.0 - rho_ij**2)) *
np.exp(-1.0 / (2.0 * (1.0 - rho_ij**2)) *
(p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2)))
coefficientsIntegral = np.flipud(tp11 * tp22 * w)
def check_difference(rho_ij):
bivariateNormalPDF = (
1.0 / (2.0 * np.pi * np.sqrt(1.0 - rho_ij**2)) *
np.exp(-1.0 / (2.0 * (1.0 - rho_ij**2)) *
(p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2)))
diff = np.dot(coefficientsIntegral, bivariateNormalPDF)
return diff - self.R[i, j]
if (self.D[i].name != 'custom') or (self.D[j].name !=
'custom'):
rho = optimize.newton(check_difference,
self.R[i, j],
maxiter=50)
else:
res = optimize.least_squares(check_difference,
R[i, j],
bounds=(-0.999, 0.999),
ftol=1.e-03)
rho = res.x
print('A Custom Marginal is present')
R0[i, j] = rho
R0[j, i] = R0[i, j]
self.A = np.linalg.cholesky(R0)
if verbose is True:
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
list_of_parameters = []
for i in range(0, len(self.D)):
standard_parameter = Parameter(order=self.D[i].order,
distribution='gaussian',
shape_parameter_A=0.,
shape_parameter_B=1.)
list_of_parameters.append(standard_parameter)
self.polystandard = deepcopy(self.poly)
self.polystandard._set_parameters(list_of_parameters)
self.standard_samples = self.polystandard.get_points()
self._points = self.get_correlated_from_uncorrelated(
self.standard_samples)
def __init__(self,
sample_points,
sample_outputs,
num_ridges,
max_iters=1,
learning_rate=0.001,
W=None,
coeffs=None,
momentum_rate=.001,
opt='sd',
poly_deg=2,
verbose=False):
self.sample_points = sample_points
self.sample_outputs = sample_outputs
self.verbose = verbose
# network architecture params
if isinstance(num_ridges, int):
self.num_ridges = [num_ridges]
else:
self.num_ridges = num_ridges
# num_ridges is the number of hidden units at each hidden layer. Does not count the input layer
self.num_layers = len(self.num_ridges)
self.dims = sample_points.shape[1]
# initialize network data structures
max_layer_size = max(self.num_ridges)
self.poly_array = np.empty((self.num_layers, max_layer_size),
dtype=object)
#TODO: not hardcode poly type? Have different ridges at different nodes?
for k in range(self.num_layers):
for j in range(self.num_ridges[k]):
self.poly_array[k, j] = Poly(
Parameter(order=poly_deg,
distribution='uniform',
lower=-3,
upper=3), Basis("total-order"))
self.poly_card = self.poly_array[0, 0].basis.cardinality
layer_sizes = [self.dims] + self.num_ridges
if W is None:
self.W = [
np.random.randn(layer_sizes[k + 1], layer_sizes[k])
for k in range(self.num_layers)
]
else:
self.W = W
if coeffs is None:
self.coeffs = [
np.random.randn(self.num_ridges[k], self.poly_card)
for k in range(self.num_layers)
]
else:
self.coeffs = coeffs
self.update_coeffs()
# Note: We will keep data for every input point in one array.
n_points = self.sample_points.shape[0]
self.delta = []
for k in range(self.num_layers):
self.delta.append(np.zeros((self.num_ridges[k], n_points)))
self.act_mat = [] # Lambda
for k in range(self.num_layers):
self.act_mat.append(np.zeros((self.num_ridges[k], n_points)))
self.Z = [] # node value before activation
for k in range(self.num_layers):
self.Z.append(np.zeros((self.num_ridges[k], n_points)))
self.Y = [] # After activation
for k in range(self.num_layers):
self.Y.append(np.zeros((self.num_ridges[k], n_points)))
self.phi = [] # basis fn evaluations
for k in range(self.num_layers):
self.phi.append(np.zeros((self.num_ridges[k], n_points)))
self.evaluate_fit(self.sample_points, train=True)
# optimization params
self.max_iters = max_iters
self.opt = opt
self.learning_rate = learning_rate
self.momentum_rate = momentum_rate
def _omorf(self, s_old, del_k, del_min, eta1, eta2, gam1, gam2, omega_s,
max_evals, random_initial, epsilon, d, subspace_method):
"""
Computes optimum using the ``omorf`` method
"""
self.n = s_old.size
self.s_old = self._apply_scaling(s_old)
if del_k is None:
if self.bounds is None:
self.del_k = 0.1 * max(np.linalg.norm(self.s_old, ord=np.inf),
1.0)
else:
self.del_k = 0.1
else:
self.del_k = del_k
self._update_bounds()
self.f_old = self._blackbox_evaluation(self.s_old)
self.d = d
self.q = int(comb(self.d + 2, 2))
self.p = self.n + 1
self.random_initial = random_initial
self.subspace_method = subspace_method
self.epsilon = epsilon
Base = Basis('total-order', orders=np.tile([2], self.d))
self.basis = Base.get_basis()[:, range(self.d - 1, -1, -1)]
itermax = 10000
# Construct the sample set
S_full, f_full = self._generate_initial_set()
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
for i in range(itermax):
# self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
my_poly = self._build_model(S_red, f_red)
m_old = np.asscalar(my_poly.get_polyfit(np.dot(self.s_old,
self.U)))
s_new, m_new = self._compute_step(my_poly)
# Safety step implemented in BOBYQA
if np.linalg.norm(s_new - self.s_old,
ord=np.inf) < omega_s * self.del_k:
if max(np.linalg.norm(
S_full - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
self.del_k *= gam1
elif max(np.linalg.norm(
S_red - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
else:
S_red, f_red = self._sample_set('improve', S_red, f_red)
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
continue
if self.S.shape == np.unique(np.vstack((self.S, s_new)),
axis=0).shape:
ind_repeat = np.argmin(
np.linalg.norm(self.S - s_new, ord=np.inf, axis=1))
f_new = self.f[ind_repeat]
else:
f_new = self._blackbox_evaluation(s_new)
S_red = np.vstack((S_red, s_new))
f_red = np.vstack((f_red, f_new))
S_full = np.vstack((S_full, s_new))
f_full = np.vstack((f_full, f_new))
# Calculate trust-region factor
rho_k = (self.f_old - f_new) / (m_old - m_new)
self._choose_best(self.S, self.f)
self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
if rho_k >= eta2:
S_red, f_red = self._sample_set('replace', S_red, f_red)
S_full, f_full = self._sample_set('replace', S_full, f_full)
self.del_k *= gam2
elif rho_k >= eta1:
S_red, f_red = self._sample_set('replace', S_red, f_red)
S_full, f_full = self._sample_set('replace', S_full, f_full)
else:
if max(np.linalg.norm(
S_full - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
self.del_k *= gam1
elif max(np.linalg.norm(
S_red - self.s_old, axis=1,
ord=np.inf)) <= self.epsilon * self.del_k:
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
self._calculate_subspace(S_full, f_full)
S_red, f_red = self._sample_set('new')
else:
S_red, f_red = self._sample_set('improve', S_red, f_red)
S_full, f_full = self._sample_set('improve',
S_full,
f_full,
full_space=True)
self.S = self._remove_scaling(self.S)
self._choose_best(self.S, self.f)
return self.s_old, self.f_old
def __init__(self,
correlation_matrix,
poly=None,
parameters=None,
method=None,
verbose=False):
if (poly is None) and (method is not None):
raise ValueError('Need to specify poly for probability transform.')
if poly is not None:
self.poly = poly
D = self.poly.get_parameters()
elif parameters is not None:
D = parameters
else:
raise ValueError('Need to specify either poly or parameters.')
self.D = D
self.R = correlation_matrix
self.std = Parameter(order=5,
distribution='normal',
shape_parameter_A=0.0,
shape_parameter_B=1.0)
inf_lim = -8.0
sup_lim = -inf_lim
p1 = Parameter(distribution='uniform',
lower=inf_lim,
upper=sup_lim,
order=31)
myBasis = Basis('tensor-grid')
self.Pols = Poly([p1, p1], myBasis, method='numerical-integration')
Pols = self.Pols
p = Pols.get_points()
# w = Pols.get_weights()
w = Pols.get_weights() * (sup_lim - inf_lim)**2
p1 = p[:, 0]
p2 = p[:, 1]
R0 = np.eye((len(self.D)))
for i in range(len(self.D)):
for j in range(i + 1, len(self.D), 1):
if self.R[i, j] == 0:
R0[i, j] = 0.0
else:
z1 = np.array(self.D[i].get_icdf(
self.std.get_cdf(points=p1)))
z2 = np.array(self.D[j].get_icdf(
self.std.get_cdf(points=p2)))
tp11 = (z1 - self.D[i].mean) / np.sqrt(self.D[i].variance)
tp22 = (z2 - self.D[j].mean) / np.sqrt(self.D[j].variance)
coefficientsIntegral = np.flipud(tp11 * tp22 * w)
def check_difference(rho_ij):
bivariateNormalPDF = (
1.0 / (2.0 * np.pi * np.sqrt(1.0 - rho_ij**2)) *
np.exp(-1.0 / (2.0 * (1.0 - rho_ij**2)) *
(p1**2 - 2.0 * rho_ij * p1 * p2 + p2**2)))
diff = np.dot(coefficientsIntegral, bivariateNormalPDF)
return diff - self.R[i, j]
# if (self.D[i].name!='custom') or (self.D[j].name!='custom'):
rho = optimize.newton(check_difference,
self.R[i, j],
maxiter=50)
# else:
# # ???
# res = optimize.least_squares(check_difference, self.R[i,j], bounds=(-0.999,0.999), ftol=1.e-03)
# rho = res.x
# print('A Custom Marginal is present')
R0[i, j] = rho
R0[j, i] = R0[i, j]
self.R0 = R0.copy()
self.A = np.linalg.cholesky(R0)
if verbose:
print('The Cholesky decomposition of fictive matrix R0 is:')
print(self.A)
print('The fictive matrix is:')
print(R0)
if method is None:
pass
elif method.lower() == 'nataf-transform':
list_of_parameters = []
for i in range(0, len(self.D)):
standard_parameter = Parameter(order=self.D[i].order,
distribution='gaussian',
shape_parameter_A=0.,
shape_parameter_B=1.)
list_of_parameters.append(standard_parameter)
# have option so that we don't need to obtain
self.corrected_poly = deepcopy(self.poly)
if hasattr(self.corrected_poly, '_quadrature_points'):
self.corrected_poly._set_parameters(list_of_parameters)
self.standard_samples = self.corrected_poly._quadrature_points
self._points = self.get_correlated_samples(
X=self.standard_samples)
# self.corrected_poly._quadrature_points = self._points.copy()
elif method.lower() == 'gram-schmidt':
basis_card = poly.basis.cardinality
oversampling = 10
N_Psi = oversampling * basis_card
S_samples = self.get_correlated_samples(N=N_Psi)
w_weights = 1.0 / N_Psi * np.ones(N_Psi)
Psi = poly.get_poly(S_samples).T
WPsi = np.diag(np.sqrt(w_weights)) @ Psi
self.WPsi = WPsi
R_Psi = np.linalg.qr(WPsi)[1]
self.R_Psi = R_Psi
self.R_Psi[0, :] *= np.sign(self.R_Psi[0, 0])
self.corrected_poly = deepcopy(poly)
self.corrected_poly.inv_R_Psi = np.linalg.inv(self.R_Psi)
self.corrected_poly.corr = self
self.corrected_poly._set_points_and_weights()
P = self.corrected_poly.get_poly(
self.corrected_poly._quadrature_points)
W = np.mat(
np.diag(np.sqrt(self.corrected_poly._quadrature_weights)))
A = W * P.T
self.corrected_poly.A = A
self.corrected_poly.P = P
if hasattr(self.corrected_poly, '_quadrature_points'):
# TODO: Correlated quadrature points?
self._points = self.corrected_poly._quadrature_points
else:
raise ValueError('Invalid method for correlations.')
def fit(self, X, y):
"""
Fits the tree to the provided data
:param PolyTree self:
An instance of the PolyTree class.
:param numpy.ndarray X:
Training input data
:param numpy.ndarray y:
Training output data
"""
def _build_tree():
global index_node_global
def _splitter(node):
# Extract data
X, y = node["data"]
depth = node["depth"]
N, d = X.shape
# Find feature splits that might improve loss
did_split = False
if self.splitting_criterion == "model_aware":
loss_best = node["loss"]
elif self.splitting_criterion == "model_agnostic" or self.splitting_criterion == "loss_gradient":
loss_best = np.inf
else:
raise Exception("invalid splitting_criterion")
data_best = None
polys_best = None
j_feature_best = None
threshold_best = None
if self.verbose:
polys_fit = 0
# Perform threshold split search only if node has not hit max depth
if (depth >= 0) and (depth < self.max_depth):
if self.splitting_criterion != "loss_gradient":
for j_feature in range(d):
last_threshold = np.inf
if self.search == 'exhaustive':
threshold_search = X[:, j_feature]
elif self.search == 'grid':
if self.samples > N:
samples = N
else:
samples = self.samples
threshold_search = np.linspace(
np.min(X[:, j_feature]),
np.max(X[:, j_feature]),
num=samples)
else:
raise Exception(
'Incorrect search type! Must be \'exhaustive\' or \'grid\''
)
# Perform threshold split search on j_feature
for threshold in np.unique(
np.sort(threshold_search)):
# Split data based on threshold
(X_left, y_left), (X_right,
y_right) = self._split_data(
j_feature, threshold, X,
y)
#print(j_feature, threshold, X_left, X_right)
N_left, N_right = len(X_left), len(X_right)
# Do not attempt to split if split conditions not satisfied
if not (N_left >= self.min_samples_leaf
and N_right >= self.min_samples_leaf):
continue
# Compute weight loss function
if self.splitting_criterion == "model_aware":
loss_left, poly_left = _fit_poly(
X_left, y_left)
loss_right, poly_right = _fit_poly(
X_right, y_right)
loss_split = (N_left * loss_left +
N_right * loss_right) / N
if self.verbose: polys_fit += 2
elif self.splitting_criterion == "model_agnostic":
loss_split = np.std(
y) - (N_left * np.std(y_left) +
N_right * np.std(y_right)) / N
# Update best parameters if loss is lower
if loss_split < loss_best:
did_split = True
loss_best = loss_split
if self.splitting_criterion == "model_aware":
polys_best = [poly_left, poly_right]
data_best = [(X_left, y_left),
(X_right, y_right)]
j_feature_best = j_feature
threshold_best = threshold
# Gradient based splitting criterion from ref. [2]
else:
# Fit a single poly to parent node
loss, poly = _fit_poly(X, y)
# Now run the splitting algo using gradients from this poly
did_split, j_feature_best, threshold_best = self._find_split_from_grad(
poly, X, y.reshape(-1, 1))
# If model_agnostic or gradient based, fit poly's to children now we have split
if self.splitting_criterion != "model_aware" and did_split:
(X_left, y_left), (X_right, y_right) = self._split_data(
j_feature_best, threshold_best, X, y)
loss_left, poly_left = _fit_poly(X_left, y_left)
loss_right, poly_right = _fit_poly(X_right, y_right)
N_left, N_right = len(X_left), len(X_right)
loss_best = (N_left * loss_left + N_right * loss_right) / N
polys_best = [poly_left, poly_right]
if self.splitting_criterion == "loss_gradient":
data_best = [(X_left, y_left), (X_right, y_right)]
if self.verbose: polys_fit += 2
if self.verbose and did_split:
print(
"Node (X.shape = {}) fitted with {} polynomials generated"
.format(X.shape, polys_fit))
elif self.verbose:
print(
"Node (X.shape = {}) failed to fit after {} polynomials generated"
.format(X.shape, polys_fit))
if did_split and depth > self.actual_max_depth:
self.actual_max_depth = depth
# Return the best result
result = {
"did_split": did_split,
"loss": loss_best,
"polys": polys_best,
"data": data_best,
"j_feature": j_feature_best,
"threshold": threshold_best,
"N": N
}
return result
def _fit_poly(X, y):
# try:
N, d = X.shape
myParameters = []
for dimension in range(d):
values = X[:, dimension]
values_min = np.amin(values)
values_max = np.amax(values)
if (values_min - values_max)**2 < 0.01:
myParameters.append(
Parameter(distribution='Uniform',
lower=values_min - 0.01,
upper=values_max + 0.01,
order=self.order))
else:
myParameters.append(
Parameter(distribution='Uniform',
lower=values_min,
upper=values_max,
order=self.order))
if self.basis == "hyperbolic-basis":
myBasis = Basis(self.basis,
orders=[self.order for _ in range(d)],
q=0.5)
else:
myBasis = Basis(self.basis,
orders=[self.order for _ in range(d)])
container["index_node_global"] += 1
poly = Poly(myParameters,
myBasis,
method=self.poly_method,
sampling_args={
'sample-points': X,
'sample-outputs': y
},
solver_args=self.poly_solver_args)
poly.set_model()
mse = np.linalg.norm(y -
poly.get_polyfit(X).reshape(-1))**2 / N
# except Exception as e:
# print("Warning fitting of Poly failed:", e)
# print(d, values_min, values_max)
# mse, poly = np.inf, None
return mse, poly
def _create_node(X, y, depth, container):
poly_loss, poly = _fit_poly(X, y)
node = {
"name": "node",
"index": container["index_node_global"],
"loss": poly_loss,
"poly": poly,
"data": (X, y),
"n_samples": len(X),
"j_feature": None,
"threshold": None,
"children": {
"left": None,
"right": None
},
"depth": depth,
"flag": False
}
container["index_node_global"] += 1
return node
def _split_traverse_node(node, container):
result = _splitter(node)
if not result["did_split"]:
return
node["j_feature"] = result["j_feature"]
node["threshold"] = result["threshold"]
del node["data"]
(X_left, y_left), (X_right, y_right) = result["data"]
poly_left, poly_right = result["polys"]
node["children"]["left"] = _create_node(
X_left, y_left, node["depth"] + 1, container)
node["children"]["right"] = _create_node(
X_right, y_right, node["depth"] + 1, container)
node["children"]["left"]["poly"] = poly_left
node["children"]["right"]["poly"] = poly_right
# Split nodes
_split_traverse_node(node["children"]["left"], container)
_split_traverse_node(node["children"]["right"], container)
container = {"index_node_global": 0}
root = _create_node(X, y, 0, container)
_split_traverse_node(root, container)
return root
N, d = X.shape
if self.basis == "hyperbolic-basis":
self.cardinality = Basis(self.basis,
orders=[self.order for _ in range(d)],
q=0.5).get_cardinality()
else:
self.cardinality = Basis(self.basis,
orders=[self.order for _ in range(d)
]).get_cardinality()
if self.min_samples_leaf == None or self.min_samples_leaf == self.cardinality:
self.min_samples_leaf = int(np.ceil(self.cardinality * 1.25))
elif self.cardinality > self.min_samples_leaf:
print(
"WARNING: Basis cardinality ({}) greater than the minimum samples per leaf ({}). This may cause reduced performance."
.format(self.cardinality, self.min_samples_leaf))
self.tree = _build_tree()
def __init__(self,
method,
full_space_poly=None,
sample_points=None,
sample_outputs=None,
subspace_dimension=2,
polynomial_degree=2,
param_args=None,
poly_args=None,
dr_args=None):
self.full_space_poly = full_space_poly
self.sample_points = sample_points
self.Y = None # for the zonotope vertices
self.sample_outputs = sample_outputs
self.method = method
self.subspace_dimension = subspace_dimension
self.polynomial_degree = polynomial_degree
my_poly_args = {'method': 'least-squares', 'solver_args': {}}
if poly_args is not None:
my_poly_args.update(poly_args)
self.poly_args = my_poly_args
my_param_args = {
'distribution': 'uniform',
'order': self.polynomial_degree,
'lower': -1,
'upper': 1
}
if param_args is not None:
my_param_args.update(param_args)
# I suppose we can detect if lower and upper is present to decide between these categories?
bounded_distrs = [
'analytical', 'beta', 'chebyshev', 'arcsine', 'truncated-gaussian',
'uniform'
]
unbounded_distrs = [
'gaussian', 'normal', 'gumbel', 'logistic', 'students-t',
'studentst'
]
semi_bounded_distrs = [
'chi', 'chi-squared', 'exponential', 'gamma', 'lognormal',
'log-normal', 'pareto', 'rayleigh', 'weibull'
]
if dr_args is not None:
if 'standardize' in dr_args:
dr_args['standardise'] = dr_args['standardize']
if self.method.lower() == 'active-subspace' or self.method.lower(
) == 'active-subspaces':
self.method = 'active-subspace'
if dr_args is not None:
self.standardise = getattr(dr_args, 'standardise', True)
else:
self.standardise = True
if self.full_space_poly is None:
# user provided input/output data
N, d = self.sample_points.shape
if self.standardise:
self.data_scaler = scaler_minmax()
self.data_scaler.fit(self.sample_points)
self.std_sample_points = self.data_scaler.transform(
self.sample_points)
else:
self.std_sample_points = self.sample_points.copy()
param = Parameter(**my_param_args)
if param_args is not None:
if (hasattr(dr_args, 'lower')
or hasattr(dr_args, 'upper')) and self.standardise:
warnings.warn(
'Points standardised but parameter range provided. Overriding default ([-1,1])...',
UserWarning)
myparameters = [param for _ in range(d)]
mybasis = Basis("total-order")
mypoly = Poly(myparameters,
mybasis,
sampling_args={
'sample-points': self.std_sample_points,
'sample-outputs': self.sample_outputs
},
**my_poly_args)
mypoly.set_model()
self.full_space_poly = mypoly
else:
# User provided polynomial
# Standardise according to distribution specified. Only care about the scaling (not shift)
# TODO: user provided callable with parameters?
user_params = self.full_space_poly.parameters
d = len(user_params)
self.sample_points = self.full_space_poly.get_points()
if self.standardise:
scale_factors = np.zeros(d)
centers = np.zeros(d)
for dd, p in enumerate(user_params):
if p.name.lower() in bounded_distrs:
scale_factors[dd] = (p.upper - p.lower) / 2.0
centers[dd] = (p.upper + p.lower) / 2.0
elif p.name.lower() in unbounded_distrs:
scale_factors[dd] = np.sqrt(p.variance)
centers[dd] = p.mean
else:
scale_factors[dd] = np.sqrt(p.variance)
centers[dd] = 0.0
self.param_scaler = scaler_custom(centers, scale_factors)
self.std_sample_points = self.param_scaler.transform(
self.sample_points)
else:
self.std_sample_points = self.sample_points.copy()
if not hasattr(self.full_space_poly, 'coefficients'):
raise ValueError('Please call set_model() first on poly.')
self.sample_outputs = self.full_space_poly.get_model_evaluations()
# TODO: use dr_args for resampling of gradient points
as_args = {'grad_points': None}
if dr_args is not None:
as_args.update(dr_args)
self._get_active_subspace(**as_args)
elif self.method == 'variable-projection':
self.data_scaler = scaler_minmax()
self.data_scaler.fit(self.sample_points)
self.std_sample_points = self.data_scaler.transform(
self.sample_points)
if dr_args is not None:
vp_args = {
'gamma': 0.1,
'beta': 1e-4,
'tol': 1e-7,
'maxiter': 1000,
'U0': None,
'verbose': False
}
vp_args.update(dr_args)
self._get_variable_projection(**vp_args)
else:
self._get_variable_projection()
def _trust_region(self, s_old, del_k, del_min, eta1, eta2, gam1, gam2, omega_s, max_evals, random_initial, scale_bounds, epsilon):
"""
Computes optimum using the ``trust-region`` method
"""
itermax = 10000
self.n = s_old.size
self.q = int(comb(self.n+2, 2))
self.p = int(comb(self.n+2, 2))
self.random_initial = random_initial
self.scale_bounds = scale_bounds
self.epsilon = epsilon
Base = Basis('total-order', orders=np.tile([2], self.n))
self.basis = Base.get_basis()[:,range(self.n-1, -1, -1)]
self.s_old = self._apply_scaling(s_old)
self.f_old = self._blackbox_evaluation(self.s_old)
if del_k is None:
if self.bounds is None:
self.del_k = 0.1*max(np.linalg.norm(self.s_old, ord=np.inf), 1.0)
else:
self.del_k = 0.1
else:
self.del_k = del_k
self._update_bounds()
# Construct the sample set
S, f = self._generate_initial_set()
for i in range(itermax):
# print(self.s_old)
# print('-------------')
self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
my_poly = self._build_model(S, f)
m_old = np.asscalar(my_poly.get_polyfit(self.s_old))
s_new, m_new = self._compute_step(my_poly)
# Safety step implemented in BOBYQA
if np.linalg.norm(s_new - self.s_old, ord=np.inf) < omega_s*self.del_k:
S, f = self._sample_set('improve', S, f)
if max(np.linalg.norm(S-self.s_old, axis=1, ord=np.inf)) <= self.epsilon*self.del_k:
self.del_k *= gam1
continue
elif self.S.shape == np.unique(np.vstack((self.S, s_new)), axis=0).shape:
ind_repeat = np.argmin(np.linalg.norm(self.S - s_new, ord=np.inf, axis=1))
f_new = self.f[ind_repeat]
else:
f_new = self._blackbox_evaluation(s_new)
S = np.vstack((S, s_new))
f = np.vstack((f, f_new))
# Calculate trust-region factor
rho_k = (self.f_old - f_new) / (m_old - m_new)
self._choose_best(self.S, self.f)
self._update_bounds()
if len(self.f) >= max_evals or self.del_k < del_min:
break
if rho_k >= eta2:
S, f = self._sample_set('replace', S, f)
self.del_k *= gam2
elif rho_k >= eta1:
S, f = self._sample_set('replace', S, f)
else:
if max(np.linalg.norm(S-self.s_old, axis=1, ord=np.inf)) <= self.epsilon*self.del_k:
S, f = self._sample_set('improve', S, f)
self.del_k *= gam1
else:
S, f = self._sample_set('improve', S, f)
self.S = self._remove_scaling(self.S)
self._choose_best(self.S, self.f)
return self.s_old, self.f_old