# Copyright 2016 James Hensman, Valentine Svensson, alexggmatthews, Mark van der Wilk
#
# 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.
from typing import Tuple
import numpy as np
import tensorflow as tf
from .. import kullback_leiblers
from ..base import Parameter
from ..conditionals import conditional
from ..config import default_float
from ..models.model import GPModel
from ..utilities import positive, triangular
from .util import inducingpoint_wrapper
class SVGP(GPModel):
"""
This is the Sparse Variational GP (SVGP). The key reference is
::
@inproceedings{hensman2014scalable,
title={Scalable Variational Gaussian Process Classification},
author={Hensman, James and Matthews, Alexander G. de G. and Ghahramani, Zoubin},
booktitle={Proceedings of AISTATS},
year={2015}
}
"""
def __init__(self,
kernel,
likelihood,
inducing_variable,
*,
mean_function=None,
num_latent: int = 1,
q_diag: bool = False,
q_mu=None,
q_sqrt=None,
whiten: bool = True,
num_data=None):
"""
- kernel, likelihood, inducing_variables, mean_function are appropriate
GPflow objects
- num_latent is the number of latent processes to use, defaults to 1
- q_diag is a boolean. If True, the covariance is approximated by a
diagonal matrix.
- whiten is a boolean. If True, we use the whitened representation of
the inducing points.
- num_data is the total number of observations, defaults to X.shape[0]
(relevant when feeding in external minibatches)
"""
# init the super class, accept args
super().__init__(kernel, likelihood, mean_function, num_latent)
self.num_data = num_data
self.q_diag = q_diag
self.whiten = whiten
self.inducing_variable = inducingpoint_wrapper(inducing_variable)
# init variational parameters
num_inducing = len(self.inducing_variable)
self._init_variational_parameters(num_inducing, q_mu, q_sqrt, q_diag)
def _init_variational_parameters(self, num_inducing, q_mu, q_sqrt, q_diag):
"""
Constructs the mean and cholesky of the covariance of the variational Gaussian posterior.
If a user passes values for `q_mu` and `q_sqrt` the routine checks if they have consistent
and correct shapes. If a user does not specify any values for `q_mu` and `q_sqrt`, the routine
initializes them, their shape depends on `num_inducing` and `q_diag`.
Note: most often the comments refer to the number of observations (=output dimensions) with P,
number of latent GPs with L, and number of inducing points M. Typically P equals L,
but when certain multioutput kernels are used, this can change.
Parameters
----------
:param num_inducing: int
Number of inducing variables, typically refered to as M.
:param q_mu: np.array or None
Mean of the variational Gaussian posterior. If None the function will initialise
the mean with zeros. If not None, the shape of `q_mu` is checked.
:param q_sqrt: np.array or None
Cholesky of the covariance of the variational Gaussian posterior.
If None the function will initialise `q_sqrt` with identity matrix.
If not None, the shape of `q_sqrt` is checked, depending on `q_diag`.
:param q_diag: bool
Used to check if `q_mu` and `q_sqrt` have the correct shape or to
construct them with the correct shape. If `q_diag` is true,
`q_sqrt` is two dimensional and only holds the square root of the
covariance diagonal elements. If False, `q_sqrt` is three dimensional.
"""
q_mu = np.zeros((num_inducing, self.num_latent)) if q_mu is None else q_mu
self.q_mu = Parameter(q_mu, dtype=default_float()) # [M, P]
if q_sqrt is None:
if self.q_diag:
ones = np.ones((num_inducing, self.num_latent), dtype=default_float())
self.q_sqrt = Parameter(ones, transform=positive()) # [M, P]
else:
q_sqrt = [np.eye(num_inducing, dtype=default_float()) for _ in range(self.num_latent)]
q_sqrt = np.array(q_sqrt)
self.q_sqrt = Parameter(q_sqrt, transform=triangular()) # [P, M, M]
else:
if q_diag:
assert q_sqrt.ndim == 2
self.num_latent = q_sqrt.shape[1]
self.q_sqrt = Parameter(q_sqrt, transform=positive()) # [M, L|P]
else:
assert q_sqrt.ndim == 3
self.num_latent = q_sqrt.shape[0]
num_inducing = q_sqrt.shape[1]
self.q_sqrt = Parameter(q_sqrt, transform=triangular()) # [L|P, M, M]
def prior_kl(self):
return kullback_leiblers.prior_kl(self.inducing_variable,
self.kernel,
self.q_mu,
self.q_sqrt,
whiten=self.whiten)
def log_likelihood(self, data: Tuple[tf.Tensor, tf.Tensor]) -> tf.Tensor:
"""
This gives a variational bound on the model likelihood.
"""
X, Y = data
kl = self.prior_kl()
f_mean, f_var = self.predict_f(X, full_cov=False, full_output_cov=False)
var_exp = self.likelihood.variational_expectations(f_mean, f_var, Y)
if self.num_data is not None:
num_data = tf.cast(self.num_data, kl.dtype)
minibatch_size = tf.cast(tf.shape(X)[0], kl.dtype)
scale = num_data / minibatch_size
else:
scale = tf.cast(1.0, kl.dtype)
return tf.reduce_sum(var_exp) * scale - kl
def elbo(self, data: Tuple[tf.Tensor, tf.Tensor]) -> tf.Tensor:
"""
This returns the evidence lower bound (ELBO) of the log marginal likelihood.
"""
return self.log_marginal_likelihood(data)
def predict_f(self, Xnew: tf.Tensor, full_cov=False, full_output_cov=False) -> tf.Tensor:
q_mu = self.q_mu
q_sqrt = self.q_sqrt
mu, var = conditional(Xnew,
self.inducing_variable,
self.kernel,
q_mu,
q_sqrt=q_sqrt,
full_cov=full_cov,
white=self.whiten,
full_output_cov=full_output_cov)
# tf.debugging.assert_positive(var) # We really should make the tests pass with this here
return mu + self.mean_function(Xnew), var
