tfd_variational_gaussian_process: Posterior predictive of a variational Gaussian process

Description

This distribution implements the variational Gaussian process (VGP), as described in Titsias (2009) and Hensman (2013). The VGP is an inducing point-based approximation of an exact GP posterior. Ultimately, this Distribution class represents a marginal distribution over function values at a collection of index_points. It is parameterized by

a kernel function,
a mean function,
the (scalar) observation noise variance of the normal likelihood,
a set of index points,
a set of inducing index points, and
the parameters of the (full-rank, Gaussian) variational posterior distribution over function values at the inducing points, conditional on some observations.

Usage

tfd_variational_gaussian_process(
  kernel,
  index_points,
  inducing_index_points,
  variational_inducing_observations_loc,
  variational_inducing_observations_scale,
  mean_fn = NULL,
  observation_noise_variance = 0,
  predictive_noise_variance = 0,
  jitter = 1e-06,
  validate_args = FALSE,
  allow_nan_stats = FALSE,
  name = "VariationalGaussianProcess"
)

Arguments

kernel

PositiveSemidefiniteKernel-like instance representing the GP's covariance function.

index_points

float Tensor representing finite (batch of) vector(s) of points in the index set over which the VGP is defined. Shape has the form [b1, ..., bB, e1, f1, ..., fF] where F is the number of feature dimensions and must equal kernel$feature_ndims and e1 is the number (size) of index points in each batch (we denote it e1 to distinguish it from the numer of inducing index points, denoted e2 below). Ultimately the VariationalGaussianProcess distribution corresponds to an e1-dimensional multivariate normal. The batch shape must be broadcastable with kernel$batch_shape, the batch shape of inducing_index_points, and any batch dims yielded by mean_fn.

inducing_index_points

float Tensor of locations of inducing points in the index set. Shape has the form [b1, ..., bB, e2, f1, ..., fF], just like index_points. The batch shape components needn't be identical to those of index_points, but must be broadcast compatible with them.

variational_inducing_observations_loc

float Tensor; the mean of the (full-rank Gaussian) variational posterior over function values at the inducing points, conditional on observed data. Shape has the form [b1, ..., bB, e2], where b1, ..., bB is broadcast compatible with other parameters' batch shapes, and e2 is the number of inducing points.

variational_inducing_observations_scale

float Tensor; the scale matrix of the (full-rank Gaussian) variational posterior over function values at the inducing points, conditional on observed data. Shape has the form [b1, ..., bB, e2, e2], where b1, ..., bB is broadcast compatible with other parameters and e2 is the number of inducing points.

mean_fn

function that acts on index points to produce a (batch of) vector(s) of mean values at those index points. Takes a Tensor of shape [b1, ..., bB, f1, ..., fF] and returns a Tensor whose shape is (broadcastable with) [b1, ..., bB]. Default value: NULL implies constant zero function.

observation_noise_variance

float Tensor representing the variance of the noise in the Normal likelihood distribution of the model. May be batched, in which case the batch shape must be broadcastable with the shapes of all other batched parameters (kernel$batch_shape, index_points, etc.). Default value: 0.

predictive_noise_variance

float Tensor representing additional variance in the posterior predictive model. If NULL, we simply re-use observation_noise_variance for the posterior predictive noise. If set explicitly, however, we use the given value. This allows us, for example, to omit predictive noise variance (by setting this to zero) to obtain noiseless posterior predictions of function values, conditioned on noisy observations.

jitter

float scalar Tensor added to the diagonal of the covariance matrix to ensure positive definiteness of the covariance matrix. Default value: 1e-6.

validate_args

Logical, default FALSE. When TRUE distribution parameters are checked for validity despite possibly degrading runtime performance. When FALSE invalid inputs may silently render incorrect outputs. Default value: FALSE.

allow_nan_stats

Logical, default TRUE. When TRUE, statistics (e.g., mean, mode, variance) use the value NaN to indicate the result is undefined. When FALSE, an exception is raised if one or more of the statistic's batch members are undefined.

name

name prefixed to Ops created by this class.

Value

a distribution instance.

Details

A VGP is "trained" by selecting any kernel parameters, the locations of the inducing index points, and the variational parameters. Titsias (2009) and Hensman (2013) describe a variational lower bound on the marginal log likelihood of observed data, which this class offers through the variational_loss method (this is the negative lower bound, for convenience when plugging into a TF Optimizer's minimize function). Training may be done in minibatches.

Titsias (2009) describes a closed form for the optimal variational parameters, in the case of sufficiently small observational data (ie, small enough to fit in memory but big enough to warrant approximating the GP posterior). A method to compute these optimal parameters in terms of the full observational data set is provided as a staticmethod, optimal_variational_posterior. It returns a MultivariateNormalLinearOperator instance with optimal location and scale parameters.

Mathematical Details

Notation We will in general be concerned about three collections of index points, and it'll be good to give them names:

x[1], ..., x[N]: observation index points -- locations of our observed data.
z[1], ..., z[M]: inducing index points -- locations of the "summarizing" inducing points
t[1], ..., t[P]: predictive index points -- locations where we are making posterior predictions based on observations and the variational parameters.

To lighten notation, we'll use X, Z, T to denote the above collections. Similarly, we'll denote by f(X) the collection of function values at each of the x[i], and by Y, the collection of (noisy) observed data at each x[i]. We'll denote kernel matrices generated from pairs of index points as K_tt, K_xt, K_tz, etc, e.g.,

K_tz =
| k(t[1], z[1])    k(t[1], z[2])  ...  k(t[1], z[M]) |
| k(t[2], z[1])    k(t[2], z[2])  ...  k(t[2], z[M]) |
|      ...              ...                 ...      |
| k(t[P], z[1])    k(t[P], z[2])  ...  k(t[P], z[M]) |

Preliminaries A Gaussian process is an indexed collection of random variables, any finite collection of which are jointly Gaussian. Typically, the index set is some finite-dimensional, real vector space, and indeed we make this assumption in what follows. The GP may then be thought of as a distribution over functions on the index set. Samples from the GP are functions on the whole index set; these can't be represented in finite compute memory, so one typically works with the marginals at a finite collection of index points. The properties of the GP are entirely determined by its mean function m and covariance function k. The generative process, assuming a mean-zero normal likelihood with stddev sigma, is

f ~ GP(m, k)
Y | f(X) ~ Normal(f(X), sigma),   i = 1, ... , N

In finite terms (ie, marginalizing out all but a finite number of f(X), sigma), we can write

f(X) ~ MVN(loc=m(X), cov=K_xx)
Y | f(X) ~ Normal(f(X), sigma),   i = 1, ... , N

Posterior inference is possible in analytical closed form but becomes intractible as data sizes get large. See Rasmussen (2006) for details.

The VGP

The VGP is an inducing point-based approximation of an exact GP posterior, where two approximating assumptions have been made:

function values at non-inducing points are mutually independent conditioned on function values at the inducing points,
the (expensive) posterior over function values at inducing points conditional on obseravtions is replaced with an arbitrary (learnable) full-rank Gaussian distribution,

q(f(Z)) = MVN(loc=m, scale=S),

where m and S are parameters to be chosen by optimizing an evidence lower bound (ELBO). The posterior predictive distribution becomes

q(f(T)) = integral df(Z) p(f(T) | f(Z)) q(f(Z)) = MVN(loc = A @ m, scale = B^(1/2))

where

A = K_tz @ K_zz^-1
B = K_tt - A @ (K_zz - S S^T) A^T

The approximate posterior predictive distribution q(f(T)) is what the VariationalGaussianProcess class represents.

Model selection in this framework entails choosing the kernel parameters, inducing point locations, and variational parameters. We do this by optimizing a variational lower bound on the marginal log likelihood of observed data. The lower bound takes the following form (see Titsias (2009) and Hensman (2013) for details on the derivation):

L(Z, m, S, Y) = MVN(loc=
(K_zx @ K_zz^-1) @ m, scale_diag=sigma).log_prob(Y) -
(Tr(K_xx - K_zx @ K_zz^-1 @ K_xz) +
Tr(S @ S^T @ K_zz^1 @ K_zx @ K_xz @ K_zz^-1)) / (2 * sigma^2) -
KL(q(f(Z)) || p(f(Z))))

where in the final KL term, p(f(Z)) is the GP prior on inducing point function values. This variational lower bound can be computed on minibatches of the full data set (X, Y). A method to compute the negative variational lower bound is implemented as VariationalGaussianProcess$variational_loss.

Optimal variational parameters

As described in Titsias (2009), a closed form optimum for the variational location and scale parameters, m and S, can be computed when the observational data are not prohibitively voluminous. The optimal_variational_posterior function to computes the optimal variational posterior distribution over inducing point function values in terms of the GP parameters (mean and kernel functions), inducing point locations, observation index points, and observations. Note that the inducing index point locations must still be optimized even when these parameters are known functions of the inducing index points. The optimal parameters are computed as follows:

C = sigma^-2 (K_zz + K_zx @ K_xz)^-1
optimal Gaussian covariance: K_zz @ C @ K_zz
optimal Gaussian location: sigma^-2 K_zz @ C @ K_zx @ Y