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from collections import namedtuple
from typing import NamedTuple, Optional, Tuple
import numpy as np
import tensorflow as tf
from gpflow.kernels import Kernel
from .. import likelihoods, posteriors
from ..base import InputData, MeanAndVariance, RegressionData
from ..config import default_float, default_jitter
from ..covariances.dispatch import Kuf, Kuu
from ..inducing_variables import InducingPoints
from ..mean_functions import MeanFunction
from ..utilities import add_noise_cov, to_default_float
from .model import GPModel
from .training_mixins import InternalDataTrainingLossMixin
from .util import data_input_to_tensor, inducingpoint_wrapper
[docs]class SGPRBase_deprecated(GPModel, InternalDataTrainingLossMixin):
"""
Common base class for SGPR and GPRFITC that provides the common __init__
and upper_bound() methods.
"""
def __init__(
self,
data: RegressionData,
kernel: Kernel,
inducing_variable: InducingPoints,
*,
mean_function: Optional[MeanFunction] = None,
num_latent_gps: Optional[int] = None,
noise_variance: float = 1.0,
):
"""
`data`: a tuple of (X, Y), where the inputs X has shape [N, D]
and the outputs Y has shape [N, R].
`inducing_variable`: an InducingPoints instance or a matrix of
the pseudo inputs Z, of shape [M, D].
`kernel`, `mean_function` are appropriate GPflow objects
This method only works with a Gaussian likelihood, its variance is
initialized to `noise_variance`.
"""
likelihood = likelihoods.Gaussian(noise_variance)
X_data, Y_data = data_input_to_tensor(data)
num_latent_gps = Y_data.shape[-1] if num_latent_gps is None else num_latent_gps
super().__init__(kernel, likelihood, mean_function, num_latent_gps=num_latent_gps)
self.data = X_data, Y_data
self.num_data = X_data.shape[0]
self.inducing_variable = inducingpoint_wrapper(inducing_variable)
[docs] def upper_bound(self) -> tf.Tensor:
"""
Upper bound for the sparse GP regression marginal likelihood. Note that
the same inducing points are used for calculating the upper bound, as are
used for computing the likelihood approximation. This may not lead to the
best upper bound. The upper bound can be tightened by optimising Z, just
like the lower bound. This is especially important in FITC, as FITC is
known to produce poor inducing point locations. An optimisable upper bound
can be found in https://github.com/markvdw/gp_upper.
The key reference is
::
@misc{titsias_2014,
title={Variational Inference for Gaussian and Determinantal Point Processes},
url={http://www2.aueb.gr/users/mtitsias/papers/titsiasNipsVar14.pdf},
publisher={Workshop on Advances in Variational Inference (NIPS 2014)},
author={Titsias, Michalis K.},
year={2014},
month={Dec}
}
The key quantity, the trace term, can be computed via
>>> _, v = conditionals.conditional(X, model.inducing_variable.Z, model.kernel,
... np.zeros((model.inducing_variable.num_inducing, 1)))
which computes each individual element of the trace term.
"""
X_data, Y_data = self.data
num_data = to_default_float(tf.shape(Y_data)[0])
Kdiag = self.kernel(X_data, full_cov=False)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
I = tf.eye(tf.shape(kuu)[0], dtype=default_float())
L = tf.linalg.cholesky(kuu)
A = tf.linalg.triangular_solve(L, kuf, lower=True)
AAT = tf.linalg.matmul(A, A, transpose_b=True)
B = I + AAT / self.likelihood.variance
LB = tf.linalg.cholesky(B)
# Using the Trace bound, from Titsias' presentation
c = tf.reduce_sum(Kdiag) - tf.reduce_sum(tf.square(A))
# Alternative bound on max eigenval:
corrected_noise = self.likelihood.variance + c
const = -0.5 * num_data * tf.math.log(2 * np.pi * self.likelihood.variance)
logdet = -tf.reduce_sum(tf.math.log(tf.linalg.diag_part(LB)))
err = Y_data - self.mean_function(X_data)
LC = tf.linalg.cholesky(I + AAT / corrected_noise)
v = tf.linalg.triangular_solve(LC, tf.linalg.matmul(A, err) / corrected_noise, lower=True)
quad = -0.5 * tf.reduce_sum(tf.square(err)) / corrected_noise + 0.5 * tf.reduce_sum(
tf.square(v)
)
return const + logdet + quad
[docs]class SGPR_deprecated(SGPRBase_deprecated):
"""
Sparse Variational GP regression. The key reference is
::
@inproceedings{titsias2009variational,
title={Variational learning of inducing variables in
sparse Gaussian processes},
author={Titsias, Michalis K},
booktitle={International Conference on
Artificial Intelligence and Statistics},
pages={567--574},
year={2009}
}
"""
CommonTensors = namedtuple("CommonTensors", ["A", "B", "LB", "AAT", "L"])
[docs] def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.elbo()
def _common_calculation(self):
"""
Matrices used in log-det calculation
:return: A , B, LB, AAT with :math:`LLᵀ = Kᵤᵤ , A = L⁻¹K_{uf}/σ, AAT = AAᵀ, B = AAT+I, LBLBᵀ = B`
A is M x N, B is M x M, LB is M x M, AAT is M x M
"""
x, _ = self.data
iv = self.inducing_variable
sigma_sq = self.likelihood.variance
kuf = Kuf(iv, self.kernel, x)
kuu = Kuu(iv, self.kernel, jitter=default_jitter())
L = tf.linalg.cholesky(kuu)
sigma = tf.sqrt(sigma_sq)
# Compute intermediate matrices
A = tf.linalg.triangular_solve(L, kuf, lower=True) / sigma
AAT = tf.linalg.matmul(A, A, transpose_b=True)
B = add_noise_cov(AAT, tf.cast(1.0, AAT.dtype))
LB = tf.linalg.cholesky(B)
return self.CommonTensors(A, B, LB, AAT, L)
[docs] def logdet_term(self, common: NameError):
"""
Bound from Jensen's Inequality:
.. math::
log |K + σ²I| <= log |Q + σ²I| + N * log (1 + tr(K - Q)/(σ²N))
:param common: A named tuple containing matrices that will be used
:return: log_det, lower bound on -.5 * output_dim * log |K + σ²I|
"""
LB = common.LB
AAT = common.AAT
x, y = self.data
num_data = to_default_float(tf.shape(x)[0])
outdim = to_default_float(tf.shape(y)[1])
kdiag = self.kernel(x, full_cov=False)
sigma_sq = self.likelihood.variance
# tr(K) / σ²
trace_k = tf.reduce_sum(kdiag) / sigma_sq
# tr(Q) / σ²
trace_q = tf.reduce_sum(tf.linalg.diag_part(AAT))
# tr(K - Q) / σ²
trace = trace_k - trace_q
# 0.5 * log(det(B))
half_logdet_b = tf.reduce_sum(tf.math.log(tf.linalg.diag_part(LB)))
# N * log(σ²)
log_sigma_sq = num_data * tf.math.log(sigma_sq)
logdet_k = -outdim * (half_logdet_b + 0.5 * log_sigma_sq + 0.5 * trace)
return logdet_k
[docs] def quad_term(self, common: NamedTuple) -> tf.Tensor:
"""
:param common: A named tuple containing matrices that will be used
:return: Lower bound on -.5 yᵀ(K + σ²I)⁻¹y
"""
A = common.A
LB = common.LB
x, y = self.data
err = y - self.mean_function(x)
sigma_sq = self.likelihood.variance
sigma = tf.sqrt(sigma_sq)
Aerr = tf.linalg.matmul(A, err)
c = tf.linalg.triangular_solve(LB, Aerr, lower=True) / sigma
# σ⁻² yᵀy
err_inner_prod = tf.reduce_sum(tf.square(err)) / sigma_sq
c_inner_prod = tf.reduce_sum(tf.square(c))
quad = -0.5 * (err_inner_prod - c_inner_prod)
return quad
[docs] def elbo(self) -> tf.Tensor:
"""
Construct a tensorflow function to compute the bound on the marginal
likelihood. For a derivation of the terms in here, see the associated
SGPR notebook.
"""
common = self._common_calculation()
output_shape = tf.shape(self.data[-1])
num_data = to_default_float(output_shape[0])
output_dim = to_default_float(output_shape[1])
const = -0.5 * num_data * output_dim * np.log(2 * np.pi)
logdet = self.logdet_term(common)
quad = self.quad_term(common)
return const + logdet + quad
[docs] def predict_f(self, Xnew: InputData, full_cov=False, full_output_cov=False) -> MeanAndVariance:
# could copy into posterior into a fused version
"""
Compute the mean and variance of the latent function at some new points
Xnew. For a derivation of the terms in here, see the associated SGPR
notebook.
"""
X_data, Y_data = self.data
num_inducing = self.inducing_variable.num_inducing
err = Y_data - self.mean_function(X_data)
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
Kus = Kuf(self.inducing_variable, self.kernel, Xnew)
sigma = tf.sqrt(self.likelihood.variance)
L = tf.linalg.cholesky(kuu)
A = tf.linalg.triangular_solve(L, kuf, lower=True) / sigma
B = tf.linalg.matmul(A, A, transpose_b=True) + tf.eye(
num_inducing, dtype=default_float()
) # cache qinv
LB = tf.linalg.cholesky(B)
Aerr = tf.linalg.matmul(A, err)
c = tf.linalg.triangular_solve(LB, Aerr, lower=True) / sigma
tmp1 = tf.linalg.triangular_solve(L, Kus, lower=True)
tmp2 = tf.linalg.triangular_solve(LB, tmp1, lower=True)
mean = tf.linalg.matmul(tmp2, c, transpose_a=True)
if full_cov:
var = (
self.kernel(Xnew)
+ tf.linalg.matmul(tmp2, tmp2, transpose_a=True)
- tf.linalg.matmul(tmp1, tmp1, transpose_a=True)
)
var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1]) # [P, N, N]
else:
var = (
self.kernel(Xnew, full_cov=False)
+ tf.reduce_sum(tf.square(tmp2), 0)
- tf.reduce_sum(tf.square(tmp1), 0)
)
var = tf.tile(var[:, None], [1, self.num_latent_gps])
return mean + self.mean_function(Xnew), var
[docs] def compute_qu(self) -> Tuple[tf.Tensor, tf.Tensor]:
"""
Computes the mean and variance of q(u) = N(mu, cov), the variational distribution on
inducing outputs. SVGP with this q(u) should predict identically to
SGPR.
:return: mu, cov
"""
X_data, Y_data = self.data
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
sig = kuu + (self.likelihood.variance ** -1) * tf.matmul(kuf, kuf, transpose_b=True)
sig_sqrt = tf.linalg.cholesky(sig)
sig_sqrt_kuu = tf.linalg.triangular_solve(sig_sqrt, kuu)
cov = tf.linalg.matmul(sig_sqrt_kuu, sig_sqrt_kuu, transpose_a=True)
err = Y_data - self.mean_function(X_data)
mu = (
tf.linalg.matmul(
sig_sqrt_kuu,
tf.linalg.triangular_solve(sig_sqrt, tf.linalg.matmul(kuf, err)),
transpose_a=True,
)
/ self.likelihood.variance
)
return mu, cov
[docs]class GPRFITC(SGPRBase_deprecated):
"""
This implements GP regression with the FITC approximation.
The key reference is
::
@inproceedings{Snelson06sparsegaussian,
author = {Edward Snelson and Zoubin Ghahramani},
title = {Sparse Gaussian Processes using Pseudo-inputs},
booktitle = {Advances In Neural Information Processing Systems},
year = {2006},
pages = {1257--1264},
publisher = {MIT press}
}
Implementation loosely based on code from GPML matlab library although
obviously gradients are automatic in GPflow.
"""
def common_terms(self):
X_data, Y_data = self.data
num_inducing = self.inducing_variable.num_inducing
err = Y_data - self.mean_function(X_data) # size [N, R]
Kdiag = self.kernel(X_data, full_cov=False)
kuf = Kuf(self.inducing_variable, self.kernel, X_data)
kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter())
Luu = tf.linalg.cholesky(kuu) # => Luu Luu^T = kuu
V = tf.linalg.triangular_solve(Luu, kuf) # => V^T V = Qff = kuf^T kuu^-1 kuf
diagQff = tf.reduce_sum(tf.square(V), 0)
nu = Kdiag - diagQff + self.likelihood.variance
B = tf.eye(num_inducing, dtype=default_float()) + tf.linalg.matmul(
V / nu, V, transpose_b=True
)
L = tf.linalg.cholesky(B)
beta = err / tf.expand_dims(nu, 1) # size [N, R]
alpha = tf.linalg.matmul(V, beta) # size [N, R]
gamma = tf.linalg.triangular_solve(L, alpha, lower=True) # size [N, R]
return err, nu, Luu, L, alpha, beta, gamma
[docs] def maximum_log_likelihood_objective(self) -> tf.Tensor:
return self.fitc_log_marginal_likelihood()
[docs] def fitc_log_marginal_likelihood(self) -> tf.Tensor:
"""
Construct a tensorflow function to compute the bound on the marginal
likelihood.
"""
# FITC approximation to the log marginal likelihood is
# log ( normal( y | mean, K_fitc ) )
# where K_fitc = Qff + diag( \nu )
# where Qff = Kfu kuu^{-1} kuf
# with \nu_i = Kff_{i,i} - Qff_{i,i} + \sigma^2
# We need to compute the Mahalanobis term -0.5* err^T K_fitc^{-1} err
# (summed over functions).
# We need to deal with the matrix inverse term.
# K_fitc^{-1} = ( Qff + \diag( \nu ) )^{-1}
# = ( V^T V + \diag( \nu ) )^{-1}
# Applying the Woodbury identity we obtain
# = \diag( \nu^{-1} ) - \diag( \nu^{-1} ) V^T ( I + V \diag( \nu^{-1} ) V^T )^{-1) V \diag(\nu^{-1} )
# Let \beta = \diag( \nu^{-1} ) err
# and let \alpha = V \beta
# then Mahalanobis term = -0.5* ( \beta^T err - \alpha^T Solve( I + V \diag( \nu^{-1} ) V^T, alpha ) )
err, nu, Luu, L, alpha, beta, gamma = self.common_terms()
mahalanobisTerm = -0.5 * tf.reduce_sum(
tf.square(err) / tf.expand_dims(nu, 1)
) + 0.5 * tf.reduce_sum(tf.square(gamma))
# We need to compute the log normalizing term -N/2 \log 2 pi - 0.5 \log \det( K_fitc )
# We need to deal with the log determinant term.
# \log \det( K_fitc ) = \log \det( Qff + \diag( \nu ) )
# = \log \det( V^T V + \diag( \nu ) )
# Applying the determinant lemma we obtain
# = \log [ \det \diag( \nu ) \det( I + V \diag( \nu^{-1} ) V^T ) ]
# = \log [ \det \diag( \nu ) ] + \log [ \det( I + V \diag( \nu^{-1} ) V^T ) ]
constantTerm = -0.5 * self.num_data * tf.math.log(tf.constant(2.0 * np.pi, default_float()))
logDeterminantTerm = -0.5 * tf.reduce_sum(tf.math.log(nu)) - tf.reduce_sum(
tf.math.log(tf.linalg.diag_part(L))
)
logNormalizingTerm = constantTerm + logDeterminantTerm
return mahalanobisTerm + logNormalizingTerm * self.num_latent_gps
[docs] def predict_f(self, Xnew: InputData, full_cov=False, full_output_cov=False) -> MeanAndVariance:
"""
Compute the mean and variance of the latent function at some new points
Xnew.
"""
_, _, Luu, L, _, _, gamma = self.common_terms()
Kus = Kuf(self.inducing_variable, self.kernel, Xnew) # [M, N]
w = tf.linalg.triangular_solve(Luu, Kus, lower=True) # [M, N]
tmp = tf.linalg.triangular_solve(tf.transpose(L), gamma, lower=False)
mean = tf.linalg.matmul(w, tmp, transpose_a=True) + self.mean_function(Xnew)
intermediateA = tf.linalg.triangular_solve(L, w, lower=True)
if full_cov:
var = (
self.kernel(Xnew)
- tf.linalg.matmul(w, w, transpose_a=True)
+ tf.linalg.matmul(intermediateA, intermediateA, transpose_a=True)
)
var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1]) # [P, N, N]
else:
var = (
self.kernel(Xnew, full_cov=False)
- tf.reduce_sum(tf.square(w), 0)
+ tf.reduce_sum(tf.square(intermediateA), 0)
) # [N, P]
var = tf.tile(var[:, None], [1, self.num_latent_gps])
return mean, var
[docs]class SGPR_with_posterior(SGPR_deprecated):
"""
This is an implementation of GPR that provides a posterior() method that
enables caching for faster subsequent predictions.
"""
[docs] def posterior(self, precompute_cache=posteriors.PrecomputeCacheType.TENSOR):
"""
Create the Posterior object which contains precomputed matrices for
faster prediction.
precompute_cache has three settings:
- `PrecomputeCacheType.TENSOR` (or `"tensor"`): Precomputes the cached
quantities and stores them as tensors (which allows differentiating
through the prediction). This is the default.
- `PrecomputeCacheType.VARIABLE` (or `"variable"`): Precomputes the cached
quantities and stores them as variables, which allows for updating
their values without changing the compute graph (relevant for AOT
compilation).
- `PrecomputeCacheType.NOCACHE` (or `"nocache"` or `None`): Avoids
immediate cache computation. This is useful for avoiding extraneous
computations when you only want to call the posterior's
`fused_predict_f` method.
"""
return posteriors.SGPRPosterior(
kernel=self.kernel,
data=self.data,
inducing_variable=self.inducing_variable,
likelihood_variance=self.likelihood.variance,
num_latent_gps=self.num_latent_gps,
mean_function=self.mean_function,
precompute_cache=precompute_cache,
)
[docs] def predict_f(self, Xnew: InputData, full_cov=False, full_output_cov=False) -> MeanAndVariance:
"""
For backwards compatibility, GPR's predict_f uses the fused (no-cache)
computation, which is more efficient during training.
For faster (cached) prediction, predict directly from the posterior object, i.e.,:
model.posterior().predict_f(Xnew, ...)
"""
return self.posterior(posteriors.PrecomputeCacheType.NOCACHE).fused_predict_f(
Xnew, full_cov=full_cov, full_output_cov=full_output_cov
)
[docs]class SGPR(SGPR_with_posterior):
# subclassed to ensure __class__ == "SGPR"
pass