swh:1:snp:62c7372827268787861b973733a16c7a7c4ae05b
Tip revision: 0312f5da2287cb50f37202832c94b9a1b3d021d8 authored by Artem Artemev on 24 February 2020, 12:31:30 UTC
Structure
Structure
Tip revision: 0312f5d
test_method_equivalence.py
# Copyright 2019 the GPflow authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import numpy as np
import pytest
import tensorflow as tf
from numpy.testing import assert_allclose
import gpflow
from gpflow.config import default_jitter
from gpflow.mean_functions import Constant
rng = np.random.RandomState(0)
class Datum:
X = rng.rand(20, 1) * 10
Y = np.sin(X) + 0.9 * np.cos(X * 1.6) + rng.randn(*X.shape) * 0.8
Y = np.tile(Y, 2) # two identical columns
Xtest = rng.rand(10, 1) * 10
data = (X, Y)
class DatumVGP:
N, Ns, DX, DY = 100, 10, 2, 2
np.random.seed(1)
X = np.random.randn(N, DX)
Xs = np.random.randn(Ns, DX)
Y = np.random.randn(N, DY)
q_mu = np.random.randn(N, DY)
q_sqrt = np.random.randn(DY, N, N)
q_alpha = np.random.randn(N, DX)
q_lambda = np.random.randn(N, DX) ** 2
data = (X, Y)
class DatumUpper:
X = np.random.rand(100, 1)
Y = np.sin(1.5 * 2 * np.pi * X) + np.random.randn(*X.shape) * 0.1
data = (X, Y)
def _create_full_gp_model():
"""
GP Regression
"""
full_gp_model = gpflow.models.GPR((Datum.X, Datum.Y),
kernel=gpflow.kernels.SquaredExponential(),
mean_function=gpflow.mean_functions.Constant(),
)
opt = gpflow.optimizers.Scipy()
@tf.function(autograph=False)
def full_gp_model_closure():
return - full_gp_model.log_marginal_likelihood()
opt.minimize(full_gp_model_closure, variables=full_gp_model.trainable_variables, options=dict(maxiter=300))
return full_gp_model
def _create_approximate_models():
"""
1) Variational GP (with the likelihood set to Gaussian)
2) Sparse variational GP (likelihood is Gaussian, inducing points
at the data)
3) Sparse variational GP (as above, but with the whitening rotation
of the inducing variables)
4) Sparse variational GP Regression (as above, but there the inducing
variables are 'collapsed' out, as in Titsias 2009)
5) FITC Sparse GP Regression
"""
model_1 = gpflow.models.VGP((Datum.X, Datum.Y),
gpflow.kernels.SquaredExponential(),
likelihood=gpflow.likelihoods.Gaussian(),
mean_function=gpflow.mean_functions.Constant())
model_2 = gpflow.models.SVGP(gpflow.kernels.SquaredExponential(),
gpflow.likelihoods.Gaussian(),
inducing_variable=Datum.X.copy(),
q_diag=False,
mean_function=gpflow.mean_functions.Constant(),
num_latent=Datum.Y.shape[1])
gpflow.utilities.set_trainable(model_2.inducing_variable, False)
model_3 = gpflow.models.SVGP(kernel=gpflow.kernels.SquaredExponential(),
likelihood=gpflow.likelihoods.Gaussian(),
inducing_variable=Datum.X.copy(), q_diag=False, whiten=True,
mean_function=gpflow.mean_functions.Constant(),
num_latent=Datum.Y.shape[1])
gpflow.utilities.set_trainable(model_3.inducing_variable, False)
model_4 = gpflow.models.GPRFITC((Datum.X, Datum.Y),
kernel=gpflow.kernels.SquaredExponential(),
inducing_variable=Datum.X.copy(),
mean_function=Constant())
gpflow.utilities.set_trainable(model_4.inducing_variable, False)
model_5 = gpflow.models.SGPR((Datum.X, Datum.Y), gpflow.kernels.SquaredExponential(),
inducing_variable=Datum.X.copy(),
mean_function=Constant())
gpflow.utilities.set_trainable(model_5.inducing_variable, False)
# Train models
opt = gpflow.optimizers.Scipy()
@tf.function(autograph=False)
def model_1_closure():
return - model_1.log_marginal_likelihood()
@tf.function(autograph=False)
def model_2_closure():
return - model_2.elbo(Datum.data)
@tf.function(autograph=False)
def model_3_closure():
return - model_3.elbo(Datum.data)
@tf.function(autograph=False)
def model_4_closure():
return - model_4.log_marginal_likelihood()
@tf.function(autograph=False)
def model_5_closure():
return - model_5.log_marginal_likelihood()
opt.minimize(model_1_closure, variables=model_1.trainable_variables, options=dict(maxiter=300))
opt.minimize(model_2_closure, variables=model_2.trainable_variables, options=dict(maxiter=300))
opt.minimize(model_3_closure, variables=model_3.trainable_variables, options=dict(maxiter=300))
opt.minimize(model_4_closure, variables=model_4.trainable_variables, options=dict(maxiter=300))
opt.minimize(model_5_closure, variables=model_5.trainable_variables, options=dict(maxiter=300))
return model_1, model_2, model_3, model_4, model_5
def _create_vgp_model(kernel, likelihood, q_mu=None, q_sqrt=None):
model_vgp = gpflow.models.VGP((DatumVGP.X, DatumVGP.Y), kernel, likelihood)
if q_mu is not None and q_sqrt is not None:
model_vgp.q_mu.assign(q_mu)
model_vgp.q_sqrt.assign(q_sqrt)
return model_vgp
def _create_vgpao_model(kernel, likelihood, q_alpha, q_lambda):
model_vgpoa = gpflow.models.VGPOpperArchambeau((DatumVGP.X, DatumVGP.Y), kernel, likelihood, num_latent=DatumVGP.DY)
model_vgpoa.q_alpha.assign(q_alpha)
model_vgpoa.q_lambda.assign(q_lambda)
return model_vgpoa
def _create_svgp_model(kernel, likelihood, q_mu, q_sqrt, whiten):
model_svgp = gpflow.models.SVGP(kernel, likelihood, DatumVGP.X.copy(), whiten=whiten, q_diag=False,
num_latent=DatumVGP.DY)
model_svgp.q_mu.assign(q_mu)
model_svgp.q_sqrt.assign(q_sqrt)
return model_svgp
@pytest.mark.parametrize('approximate_model', _create_approximate_models())
def test_equivalence(approximate_model):
"""
With a Gaussian likelihood, and inducing points (where appropriate)
positioned at the data, many of the gpflow methods are equivalent (perhaps
subject to some optimization).
"""
gpr_model = _create_full_gp_model()
gpr_likelihood = - gpr_model.log_likelihood()
if isinstance(approximate_model, gpflow.models.SVGP):
approximate_likelihood = - approximate_model.log_likelihood(Datum.data)
else:
approximate_likelihood = - approximate_model.log_likelihood()
assert_allclose(gpr_likelihood, approximate_likelihood, rtol=1e-6)
gpr_kernel_ls = gpr_model.kernel.lengthscale.read_value()
gpr_kernel_var = gpr_model.kernel.variance.read_value()
approximate_kernel_ls = approximate_model.kernel.lengthscale.read_value()
approximate_kernel_var = approximate_model.kernel.variance.read_value()
assert_allclose(gpr_kernel_ls, approximate_kernel_ls, 1e-4)
assert_allclose(gpr_kernel_var, approximate_kernel_var, 1e-3)
gpr_mu, gpr_var = gpr_model.predict_y(Datum.Xtest)
approximate_mu, approximate_var = approximate_model.predict_y(Datum.Xtest)
assert_allclose(gpr_mu, approximate_mu, 1e-3)
assert_allclose(gpr_var, approximate_var, 1e-4)
def test_equivalence_vgp_and_svgp():
kernel = gpflow.kernels.Matern52()
likelihood = gpflow.likelihoods.StudentT()
svgp_model = _create_svgp_model(kernel, likelihood, DatumVGP.q_mu, DatumVGP.q_sqrt, whiten=True)
vgp_model = _create_vgp_model(kernel, likelihood, DatumVGP.q_mu, DatumVGP.q_sqrt)
likelihood_svgp = svgp_model.log_likelihood(DatumVGP.data)
likelihood_vgp = vgp_model.log_likelihood()
assert_allclose(likelihood_svgp, likelihood_vgp, rtol=1e-2)
svgp_mu, svgp_var = svgp_model.predict_f(DatumVGP.Xs)
vgp_mu, vgp_var = vgp_model.predict_f(DatumVGP.Xs)
assert_allclose(svgp_mu, vgp_mu)
assert_allclose(svgp_var, vgp_var)
def test_equivalence_vgp_and_opper_archambeau():
kernel = gpflow.kernels.Matern52()
likelihood = gpflow.likelihoods.StudentT()
vgp_oa_model = _create_vgpao_model(kernel, likelihood, DatumVGP.q_alpha, DatumVGP.q_lambda)
K = kernel(DatumVGP.X) + np.eye(DatumVGP.N) * default_jitter()
L = np.linalg.cholesky(K)
L_inv = np.linalg.inv(L)
K_inv = np.linalg.inv(K)
mean = K @ DatumVGP.q_alpha
prec_dnn = K_inv[None, :, :] + np.array([np.diag(l ** 2) for l in DatumVGP.q_lambda.T])
var_dnn = np.linalg.inv(prec_dnn)
svgp_model_unwhitened = _create_svgp_model(kernel, likelihood, mean, np.linalg.cholesky(var_dnn), whiten=False)
mean_white_nd = L_inv.dot(mean)
var_white_dnn = np.einsum('nN,dNM,mM->dnm', L_inv, var_dnn, L_inv)
q_sqrt_nnd = np.linalg.cholesky(var_white_dnn)
vgp_model = _create_vgp_model(kernel, likelihood, mean_white_nd, q_sqrt_nnd)
likelihood_vgp = vgp_model.log_likelihood()
likelihood_vgp_oa = vgp_oa_model.log_likelihood()
likelihood_svgp_unwhitened = svgp_model_unwhitened.log_likelihood(DatumVGP.data)
assert_allclose(likelihood_vgp, likelihood_vgp_oa, rtol=1e-2)
assert_allclose(likelihood_vgp, likelihood_svgp_unwhitened, rtol=1e-2)
vgp_oa_mu, vgp_oa_var = vgp_oa_model.predict_f(DatumVGP.Xs)
svgp_unwhitened_mu, svgp_unwhitened_var = svgp_model_unwhitened.predict_f(DatumVGP.Xs)
vgp_mu, vgp_var = vgp_model.predict_f(DatumVGP.Xs)
assert_allclose(vgp_oa_mu, vgp_mu)
assert_allclose(vgp_oa_var, vgp_var, rtol=1e-4) # jitter?
assert_allclose(svgp_unwhitened_mu, vgp_mu)
assert_allclose(svgp_unwhitened_var, vgp_var, rtol=1e-4)
def test_upper_bound_few_inducing_points():
"""
Test for upper bound for regression marginal likelihood
"""
model_vfe = gpflow.models.SGPR((DatumUpper.X, DatumUpper.Y), gpflow.kernels.SquaredExponential(),
inducing_variable=DatumUpper.X[:10, :].copy(),
mean_function=Constant())
opt = gpflow.optimizers.Scipy()
@tf.function(autograph=False)
def model_vfe_closure():
return - model_vfe.log_marginal_likelihood()
opt.minimize(model_vfe_closure, variables=model_vfe.trainable_variables, options=dict(maxiter=500))
full_gp = gpflow.models.GPR((DatumUpper.X, DatumUpper.Y),
kernel=gpflow.kernels.SquaredExponential(),
mean_function=Constant())
full_gp.kernel.lengthscale.assign(model_vfe.kernel.lengthscale)
full_gp.kernel.variance.assign(model_vfe.kernel.variance)
full_gp.likelihood.variance.assign(model_vfe.likelihood.variance)
full_gp.mean_function.c.assign(model_vfe.mean_function.c)
lml_upper = model_vfe.upper_bound()
lml_vfe = model_vfe.log_marginal_likelihood()
lml_full_gp = full_gp.log_marginal_likelihood()
assert lml_vfe < lml_full_gp
assert lml_full_gp < lml_upper