test_method_equivalence.py

# Copyright 2016 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 tensorflow as tf



import numpy as np
from numpy.testing import assert_allclose

import gpflow
from gpflow.test_util import GPflowTestCase


class TestEquivalence(GPflowTestCase):
    """
    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).

    Here, we make 5 models that should be the same, and make sure some
    similarites hold. The models are:

    1) GP Regression
    2) Variational GP (with the likelihood set to Gaussian)
    3) Sparse variational GP (likelihood is Gaussian, inducing points
       at the data)
    4) Sparse variational GP (as above, but with the whitening rotation
       of the inducing variables)
    5) Sparse variational GP Regression (as above, but there the inducing
       variables are 'collapsed' out, as in Titsias 2009)
    """

    def prepare(self):
        rng = np.random.RandomState(0)
        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
        self.Xtest = rng.rand(10, 1) * 10

        m1 = gpflow.models.GPR(
            X, Y, kern=gpflow.kernels.RBF(1),
            mean_function=gpflow.mean_functions.Constant())
        m2 = gpflow.models.VGP(
            X, Y, gpflow.kernels.RBF(1), likelihood=gpflow.likelihoods.Gaussian(),
            mean_function=gpflow.mean_functions.Constant())
        m3 = gpflow.models.SVGP(
            X, Y, gpflow.kernels.RBF(1),
            likelihood=gpflow.likelihoods.Gaussian(),
            Z=X.copy(),
            q_diag=False,
            mean_function=gpflow.mean_functions.Constant())
        m3.feature.trainable = False
        m4 = gpflow.models.SVGP(
            X, Y, gpflow.kernels.RBF(1),
            likelihood=gpflow.likelihoods.Gaussian(),
            Z=X.copy(), q_diag=False, whiten=True,
            mean_function=gpflow.mean_functions.Constant())
        m4.feature.trainable = False
        m5 = gpflow.models.SGPR(
            X, Y, gpflow.kernels.RBF(1),
            Z=X.copy(),
            mean_function=gpflow.mean_functions.Constant())

        m5.feature.trainable = False
        m6 = gpflow.models.GPRFITC(
            X, Y, gpflow.kernels.RBF(1), Z=X.copy(),
            mean_function=gpflow.mean_functions.Constant())
        m6.feature.trainable = False
        return [m1, m2, m3, m4, m5, m6]

    def test_all(self):
        with self.test_context() as session:
            models = self.prepare()
            likelihoods = []
            for m in models:
                opt = gpflow.train.ScipyOptimizer()
                opt.minimize(m, maxiter=300)
                neg_obj = tf.negative(m.objective)
                likelihoods.append(session.run(neg_obj).squeeze())
            assert_allclose(likelihoods, likelihoods[0], rtol=1e-6)
            variances, lengthscales = [], []
            for m in models:
                if hasattr(m.kern, 'rbf'):
                    variances.append(m.kern.rbf.variance.read_value())
                    lengthscales.append(m.kern.rbf.lengthscales.read_value())
                else:
                    variances.append(m.kern.variance.read_value())
                    lengthscales.append(m.kern.lengthscales.read_value())
            variances, lengthscales = np.array(variances), np.array(lengthscales)
            assert_allclose(variances, variances[0], 1e-5)
            assert_allclose(lengthscales, lengthscales.mean(), 1e-4)
            mu0, var0 = models[0].predict_y(self.Xtest)
            for i, m in enumerate(models[1:]):
                mu, var = m.predict_y(self.Xtest)
                assert_allclose(mu, mu0, 1e-3)
                assert_allclose(var, var0, 1e-4)


class VGPTest(GPflowTestCase):
    def test_vgp_vs_svgp(self):
        with self.test_context():
            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)

            kern = gpflow.kernels.Matern52(DX)
            likelihood = gpflow.likelihoods.StudentT()

            m_svgp = gpflow.models.SVGP(
                X, Y, kern, likelihood, X.copy(), whiten=True, q_diag=False)
            m_vgp = gpflow.models.VGP(X, Y, kern, likelihood)

            m_svgp.compile()
            m_vgp.compile()

            q_mu = np.random.randn(N, DY)
            q_sqrt = np.random.randn(DY, N, N)

            m_svgp.q_mu = q_mu
            m_svgp.q_sqrt = q_sqrt

            m_vgp.q_mu = q_mu
            m_vgp.q_sqrt = q_sqrt

            L_svgp = m_svgp.compute_log_likelihood()
            L_vgp = m_vgp.compute_log_likelihood()
            assert_allclose(L_svgp, L_vgp, rtol=1e-2)

            pred_svgp = m_svgp.predict_f(Xs)
            pred_vgp = m_vgp.predict_f(Xs)
            assert_allclose(pred_svgp[0], pred_vgp[0])
            assert_allclose(pred_svgp[1], pred_vgp[1])

    def test_vgp_vs_opper_archambeau(self):
        with self.test_context():
            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)

            kern = gpflow.kernels.Matern52(DX)
            likelihood = gpflow.likelihoods.StudentT()

            m_vgp = gpflow.models.VGP(X, Y, kern, likelihood)
            m_vgp_oa = gpflow.models.VGP_opper_archambeau(X, Y, kern, likelihood)
            m_vgp.compile()
            m_vgp_oa.compile()

            q_alpha = np.random.randn(N, DX)
            q_lambda = np.random.randn(N, DX) ** 2

            m_vgp_oa.q_alpha = q_alpha
            m_vgp_oa.q_lambda = q_lambda

            K = kern.compute_K_symm(X) + np.eye(N) * gpflow.settings.jitter
            L = np.linalg.cholesky(K)
            L_inv = np.linalg.inv(L)
            K_inv = np.linalg.inv(K)

            mean = K.dot(q_alpha)
            prec_dnn = K_inv[None, :, :] + np.array([np.diag(l ** 2) for l in q_lambda.T])
            var_dnn = np.linalg.inv(prec_dnn)

            m_svgp_unwhitened = gpflow.models.SVGP(
                X, Y, kern, likelihood, X.copy(),
                whiten=False, q_diag=False)

            m_svgp_unwhitened.q_mu = mean
            m_svgp_unwhitened.q_sqrt = np.linalg.cholesky(var_dnn)

            m_svgp_unwhitened.compile()

            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)

            m_vgp.q_mu = mean_white_nd
            m_vgp.q_sqrt = q_sqrt_nnd

            L_vgp = m_vgp.compute_log_likelihood()
            L_svgp_unwhitened = m_svgp_unwhitened.compute_log_likelihood()
            L_vgp_oa = m_vgp_oa.compute_log_likelihood()
            assert_allclose(L_vgp, L_vgp_oa, rtol=1e-2)
            assert_allclose(L_vgp, L_svgp_unwhitened, rtol=1e-2)

            pred_vgp = m_vgp.predict_f(Xs)
            pred_svgp_unwhitened = m_svgp_unwhitened.predict_f(Xs)
            pred_vgp_oa = m_vgp_oa.predict_f(Xs)

            assert_allclose(pred_vgp[0], pred_vgp_oa[0])
            assert_allclose(pred_vgp[0], pred_svgp_unwhitened[0])
            assert_allclose(pred_vgp[1], pred_vgp_oa[1], rtol=1e-4)  # jitter?
            assert_allclose(pred_vgp[1], pred_svgp_unwhitened[1], rtol=1e-4)

    #def test_recompile(self):
    #    with self.test_context():
    #        N, DX, DY = 100, 2, 2
    #        np.random.seed(1)
    #        X = np.random.randn(N, DX)
    #        Y = np.random.randn(N, DY)
    #        kern = gpflow.kernels.Matern52(DX)
    #        likelihood = gpflow.likelihoods.StudentT()
    #        m_vgp = gpflow.models.VGP(X, Y, kern, likelihood)
    #        m_vgp_oa = gpflow.models.VGP_opper_archambeau(X, Y, kern, likelihood)
    #        for m in [m_vgp, m_vgp_oa]:
    #            m.compile()
    #            opt = gpflow.train.ScipyOptimizer()
    #            opt.minimize(m, maxiter=1)
    #            m.X = X[:-1, :]
    #            m.Y = Y[:-1, :]
    #            opt.minimize(m, maxiter=1)


class TestUpperBound(GPflowTestCase):
    """
    Test for upper bound for regression marginal likelihood
    """

    def setUp(self):
        self.X = np.random.rand(100, 1)
        self.Y = np.sin(1.5 * 2 * np.pi * self.X) + np.random.randn(*self.X.shape) * 0.1

    def test_few_inducing_points(self):
        with self.test_context() as session:
            vfe = gpflow.models.SGPR(self.X, self.Y, gpflow.kernels.RBF(1), self.X[:10, :].copy())
            opt = gpflow.train.ScipyOptimizer()
            opt.minimize(vfe)

            full = gpflow.models.GPR(self.X, self.Y, gpflow.kernels.RBF(1))
            full.kern.lengthscales = vfe.kern.lengthscales.read_value()
            full.kern.variance = vfe.kern.variance.read_value()
            full.likelihood.variance = vfe.likelihood.variance.read_value()

            lml_upper = vfe.compute_upper_bound()
            lml_vfe = - session.run(vfe.objective)
            lml_full = - session.run(full.objective)

            self.assertTrue(lml_upper > lml_full > lml_vfe)


if __name__ == '__main__':
    tf.test.main()