# Variational Gaussian Processes¶

The Variational Free Energy (VFE) method [Tit09] applied to GPs, is an approach for approximating the posterior $$p(f | \mathbf{y}, \mathbf{X})$$ of a Gaussian Process. Instead of computing the exact GP posterior, VFE approximates it by another Gaussian distribution, which is cheaper to manipulate when making predictions or evaluating the marginal likelihood $$p(\mathbf{y} | \mathbf{X})$$. Similarly to other GP approximations, VFE achieves lower a computational cost by making sparsity assumptions. However, whereas other sparse methods like FITC and DTC make sparsity assumptions on the likelihood, VFE makes these assumptions on the approximate posterior and this has several important effects. $$\def\Kxx{\mathbf{K}_{\mathbf{X}\mathbf{X}}} \def\Kxb{\mathbf{K}_{\mathbf{X}\mathbf{\bar{X}}}} \def\Kbx{\mathbf{K}_{\mathbf{\bar{X}}\mathbf{X}}} \def\Kbb{\mathbf{K}_{\mathbf{\bar{X}}\mathbf{\bar{X}}}} \def\Ksb{\mathbf{K}_{\mathbf{X^*}\mathbf{\bar{X}}}} \def\Kbs{\mathbf{K}_{\mathbf{\bar{X}}\mathbf{X^*}}} \def\Ksx{\mathbf{K}_{\mathbf{X^*}\mathbf{\bar{X}}}} \def\Kxs{\mathbf{K}_{\mathbf{\bar{X}}\mathbf{X^*}}} \def\Kss{\mathbf{K}_{\mathbf{X^*}\mathbf{X^*}}} \def\fx{\mathbf{f}_{\mathbf{X}}} \def\fnb{f_{\neq \mathbf{\bar{X}}}} \def\fb{\mathbf{f}_{\mathbf{\bar{X}}}} \def\fstar{\mathbf{f}_{\mathbf{X^*}}} \def\X{\mathbf{x}} \def\xstar{\mathbf{x^*}} \def\X{\mathbf{X}} \def\Xb{\mathbf{\bar{X}}} \def\lrb[#1]{\left(#1\right)} \def\lrs[#1]{\left[#1\right]} \def\mb[#1]{\mathbf{#1}} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \newcommand{\bs}{\boldsymbol{#1}} \newcommand{\bm}{\mathbf{#1}}$$

## The Variational Free Energy¶

Often, the log marginal likelihood of the data can be costly or intractable to compute. To circumvent this difficulty, one solution is to optimise an alternative objective instead of the log-marginal. This objective should be chosen such that (1) it is cheap to evaluate; (2) optimising it cannot lead to worse overfitting than the original model. The Variational Free Energy (VFE), also known as the ELBO, can be used to meet both these criteria. The precise same bound is used for a variety of other models involving latent variables including Gassian Mixture Models, Variational Autoencoders or Bayesian Networks.

The VFE meets the two criteria above because (1) it can be cheap to evaluate if we make sensible approximation choices; (2) it lower bounds the marginal likelihood, so we can optimise it with respect to the hyperparameters without fear of overfitting any worse than the exact model would. The latter point is especially important because it decouples modelling assumptions (which we choose to make because they are appropriate for the data) from our approximations (which we make out of computational necessity). A user can therefore state their assumptions clearly upfornt and buy themselves as good an approximation to the exact model as their comptational budget can afford them. We’ll expand on this further, later.

Here is a short derivation for the VFE, which is applicable to any latent variable model. Suppose we want to evalate the marginal likelihood of some data $$\mathbf{y}$$ given some other data $$\mathbf{x}$$ under a model with latent variables $$\mathbf{f}$$ and parameters $$\bs{\theta}$$

\begin{align} p(\mathbf{y} | \mathbf{x}, \bs{\theta}) = \int p(\mathbf{y}, \mathbf{f} | \mathbf{x}, \bs{\theta}) d \mathbf{f} \end{align}

Now, defining a new probability distribution over $$\mathbf{f}$$, written $$q(\mathbf{f})$$. This is widely referred to as the variational posterior. If we subtract the KL divergence between the variational and the true posterior from the marginal likelihood, we get the inequality

\begin{split}\begin{align} \log p(\mathbf{y} | \mathbf{x}) \geq \mathcal{F}(q, \bs{\theta}) &= \int q(\mathbf{f}) \log p(\mathbf{y} | \mathbf{x}, \bs{\theta}) d \mathbf{f} + \int q(\mathbf{f}) \log \frac{p(\mathbf{f} | \mathbf{y}, \mathbf{x}, \bs{\theta})}{q(\mathbf{f})} d\bm{f} \\ &= \int q(\mathbf{f}) \log \frac{p(\mathbf{y}, \mathbf{f} | \mathbf{x}, \bs{\theta})}{q(\mathbf{f})} d\bm{f}, \end{align}\end{split}

where we have used the fact that the KL is non-negative. This shows that the VFE is a lower bound to the exact marginal likelihood. We made no assumptions about the distribution $$q$$ and the inequality always holds, irrespective of our choice of variational posterior. The inequality becomes an equality when the variational posterior is equal to the true posterior, where the KL divergence becomes zero, attaining its minimum value. Although we could pick the variational posterior to be any distribution we like, we should select a distribution which makes the VFE as large as possible while still remaining computationally tractable. It turns out that the VFE approximation for GPs achieves an excellent tradeoff between approximating the true posterior and keeping the computational cost low.

## Sparse VFE approximation¶

Using the idea of introducing inducing points, much like those of the DTC[SWL03] or FITC[SG05] approximations, we will see how the VFE approximation will result in an approximate form which is significantly cheaper to work with than the exact GP. Unlike the DTC or FITC however, VFE approximates the posterior over the latent function $$f$$ rather than the model. Whereas models like DTC or FITC can overfit when optimising their inducing points and model parameters $$\theta$$, the VFE approximation does not run this risk at all because it uses the Free Energy to optimise inducing points and parameters. Since the Free Energy is a lower bound to the exact log-marginal likelihood, VFE is guaranteed not to overfit any more than the exact GP would.

### Model and approximate $$q$$¶

Let’s start by stating our assumptions and inference approximations. First, define the input variables $$\mathbf{X} = (\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_N)^\top$$ and the values of the function $$f$$ at these input locations as $$\mathbf{f}_\mathbf{X} = (f_{\mathbf{x}_1}, f_{\mathbf{x}_2}, ..., f_{\mathbf{x}_N})^\top$$, placing a zero-mean GP prior over them, so that

\begin{align} p(\mathbf{f}_\mathbf{X} |\mathbf{X}, \bs{\theta}) \sim \mathcal{N}\lrb[\mathbf{0}, \bm{K}_{\bm{X}\bm{X}}]. \end{align}

Now define the observed variables $$\mathbf{y} = (y_1, y_2, ..., y_N)^\top$$, obtained by adding noise to $$\mathbf{f}$$

\begin{align} p(\mathbf{y}| \mathbf{f}_\mathbf{X}) \sim \mathcal{N}\lrb[\mathbf{f}_\mathbf{X}, \sigma^2 \bm{I}], \end{align}

so that the marginal likelihood of $$\mathbf{y}$$ given $$\bm{X}$$ is

\begin{align} p(\mathbf{y}| \bs{\theta}) \sim \mathcal{N}\lrb[\mathbf{0}, \bm{K}_{\bm{X}\bm{X}} + \sigma^2 \bm{I}]. \end{align}

This is the exact GP model, describing precisely how the inputs $$\mathbf{X}$$ and outputs $$\mathbf{y}$$ are related. Exact inference and learning in this model is impossible because it involves the inversion of the full rank matrix $$\bm{K}_{\bm{X}\bm{X}} + \sigma^2 \bm{I}$$. To circumvent this, the VFE method places an approximate posterior based on inducing points. Let $$\mathbf{\bar{X}} = (\mathbf{\bar{x}}_1, \mathbf{\bar{x}}_2, ..., \mathbf{\bar{x}}_M)^\top$$ be a set of inducing inputs, different from $$\mathbf{X}$$, and let $$\fb = (f_{\mathbf{\bar{x}}_1}, f_{\mathbf{\bar{x}}_2}, ..., f_{\mathbf{\bar{x}}_N})^\top$$ be the corresponding function values at these points, called the inducing values. The VFE approximation chooses the approximate posterior to be

\begin{align} q(f) = p\left(\fnb | \fb\right) q\left(\fb\right), \end{align}

where $$f$$ is the full latent function and $$q\left(\fb\right)$$ is a free-form distribution to be optimised. Note that we have slightly abused notation, using $$q$$ to denote both the variational distribution over the full latent function $$f$$ as well as the variational distribution over the inducing point values $$\fb$$. This $$q$$ separates $$\fnb$$ from the data $$\mathbf{y}$$ and instead uses the inducing values $$\fb$$ to summarise the dataset and bring the approximate posterior $$q(f)$$ close to the true posterior.

### Maximising the VFE¶

We now look to maximise $$\mathcal{F}$$ with respect to $$q$$

\begin{align} q^* = \argmax_{q} \mathcal{F}(q, \bs{\theta}). \end{align}

This maximisation can be performed in closed form. Perhaps unsuprisingly, the optimal $$q^*$$ is also a Gaussian distribution.

Lemma (Optimal $$q^*$$) The variational posterior $$q^*$$ which maximises the variational free energy is

\begin{align} q^*\lrb[\fb] = \mathcal{N}\lrb[\fb;~\sigma^{-2} \Kbb \bs{\Sigma}^{-1} \Kbx \mathbf{y},~\Kbb\bs{\Sigma}^{-1}\Kbb] \end{align}

where $$\bs{\Sigma} = \lrb[\Kbb + \sigma^{-2} \Kbx \Kxb]$$. The corresponding free energy is

\begin{align} \mathcal{F}(q^*, \bs{\theta}) = \mathcal{N}\lrb[\bm{y}; \bm{0}, \sigma^2 \bm{I} + \Kxb \Kbb^{-1} \Kbx] - \frac{1}{2\sigma^2} \text{Tr}\lrb[\Kxx - \Kxb\Kbb^{-1}\Kbx]. \end{align}

Derivation: Optimal $$q^*$$

Substituting the approximate posterior into $$\mathcal{F}$$, we obtain

\begin{split}\begin{align} \require{cancel} \mathcal{F}(q, \bs{\theta}) &= \int p\lrb[f_{\neq \Xb} | \fb] q\lrb[\fb | \mathbf{y}] \log \frac{p(\mathbf{y}, f | \bs{\theta})}{p\lrb[f_{\neq \Xb} | \fb] q\lrb[\mathbf{f}_{\mathbf{\bar{X}}} | \mathbf{y}]} df \\ &= \int p\lrb[f_{\neq \Xb} | \fb] q\lrb[\fb | \mathbf{y}] \log \frac{p(\mathbf{y} | \fx, \bs{\theta}) \cancel{p\lrb[f_{\neq \Xb} | \fb]} p\lrb[\fb]}{\cancel{p\lrb[f_{\neq \Xb} | \fb]} q\left(\fb | \mathbf{y} \right)} df \\ &= \int p\lrb[\fx | \fb] q\lrb[\fb | \mathbf{y}] \log \frac{p(\mathbf{y} | \fx, \bs{\theta}) p(\fb)}{ q\lrb[\fb | \mathbf{y}]} d\fx d\fb. \end{align}\end{split}

Now we seek to minimise $$\mathcal{F}(q, \bs{\theta})$$ with respect to $$q$$. This requires a variational approach since $$q$$ is a function, and also requires a Lagrange multiplier because $$q$$ must be constrained to integrate to $$1$$ since it is a distribution. Optimising $$\mathcal{F}(q, \bs{\theta})$$ subject to this constraint is equivalent to optimising the Lagrangian $$\mathcal{L}$$ without constraints

\begin{split}\begin{align} \mathcal{L} &= \mathcal{F}(q, \bs{\theta}) - \lambda \left( \int q\lrb[\fb | \mathbf{y}] d\fb - 1 \right) \\ &= \int p\lrb[\fx | \fb] q\lrb[\fb | \mathbf{y}] \log \left[p(\mathbf{y} | \fx, \bs{\theta}) p(\fb)\right] d\fx d\fb - \int p\lrb[\fx | \fb] q\lrb[\fb | \mathbf{y}] \log q\lrb[\fb | \mathbf{y}] d\fx d\fb - \lambda \left( \int q\lrb[\fb | \mathbf{y}] d\fb - 1 \right). \end{align}\end{split}

Setting the (variational) derivative of $$\mathcal{F}$$ w.r.t. $$q$$ to 0, we obtain:

\begin{align} \frac{\delta \mathcal{L}}{\delta q} &= \int p\lrb[\fx | \fb] \log p(\mathbf{y} | \fx, \bs{\theta}) d\fx + \log p\lrb[\fb] - \lrb[ \log q^*\lrb[\fb | \mathbf{y}] + 1] - \lambda = 0 \end{align}
\begin{align} q^*\lrb[\fb | \mathbf{y}] = \frac{1}{Z} p\lrb[\fb]\exp \int p\lrb[\fx | \fb] \log p(\mathbf{y} | \fx, \bs{\theta}) d\fx \end{align}

Substituting $$p\lrb[\mathbf{y} | \fx] = \mathcal{N}\lrb[\mathbf{y}; \fx, \sigma^2 \mathbf{I}]$$ we obtain

\begin{align} q^*\lrb[\fb | \mathbf{y}] = \frac{1}{Z} p\lrb[\fb] \exp \lrb[-\frac{N}{2}\log\lrb[2\pi\sigma^2] - \frac{1}{2\sigma^2} \underbrace{\int \lrb[\mathbf{y}^\top \mathbf{y} - 2 \mathbf{y}^\top \fx + \fx^\top \fx ] p\lrb[\fx | \fb] d\fx}_{= M}]. \end{align}

Using the fact that $$\fx^\top \fx = \text{Tr}\lrb[\fx \fx^\top]$$ and substituting for $$p\left(\fx | \fb \right)$$ and $$p\lrb[\fb]$$ into $$\mathcal{F}$$, we evaluate the integral as:

\begin{align} M = \mathbf{y}^\top \mathbf{y} - 2 \mathbf{y}^\top \Kxb \Kbb^{-1} \fb + \fb^\top \Kbb^{-1} \Kbx \Kxb \Kbb^{-1} \fb + \text{Tr}\lrb[\Kxx - \Kxb \Kbb^{-1} \Kbx]. \end{align}

We can now read off the $$q^*$$ distribution easily as follows. Since $$M$$ is a quadratic form in $$\fb$$, the whole exponential term above is an unnormalised Gaussian. The $$p\lrb[\fb]$$ term is also a Gaussian, so its product with the exponential term will also be an unnormalised Gaussian in $$\fb$$, thus arriving at the result that the optimal $$q^*$$ is also a Gaussian. We only have to determine the mean and covariance of the overall $$q^*$$ and need not bother with constants at this stage. Using the standard results for the mean and covariance of a product of Gaussians we obtain

\begin{align} q^*\lrb[\fb | \mathbf{y}] = \mathcal{N}\lrb[\fb;~\sigma^{-2} \Kbb \bs{\Sigma}^{-1} \Kbx \mathbf{y},~\Kbb\bs{\Sigma}^{-1}\Kbb] \end{align}

where $$\bs{\Sigma}^{-1}$$ is given by

\begin{align} \bs{\Sigma}^{-1} &= \lrb[\Kbb + \sigma^{-2} \Kbx \Kxb]^{-1}. \end{align}

Substituting this back into the free energy we obtain

\begin{align} \mathcal{F}(q^*, \bs{\theta}) = \mathcal{N}\lrb[\bm{y}; \bm{0}, \sigma^2 \bm{I} + \Kxb \Kbb^{-1} \Kbx] - \frac{1}{2\sigma^2} \text{Tr}\lrb[\Kxx - \Kxb\Kbb^{-1}\Kbx]. \end{align}

Looking at this expression for $$\mathcal{F}$$, we observe that the first term is a Gaussian involving a covariance with special structure. The $$\sigma^2 \bm{I}$$ term is diagonal, while the $$\Kxb \Kbb^{-1} \Kbx$$ has rank $$\min(M, N)$$. Such matrices can be inverted in $$\mathcal{O}\lrb[\min(NM^2, MN^2)]$$ time, using the Woodbury Identity. The second term involves the trace of a matrix which, when $$M < N$$, requires $$\mathcal{O}\lrb[NM^2]$$ to compute. Therefore, the overal complexity of computing the VFE is $$\mathcal{O}\lrb[NM^2]$$, which is a significant gain over $$\mathcal{O}\lrb[N^3]$$ if $$M << N$$.

### Posterior predictive¶

We are also interested in making predictions at new test points. In order to achieve this in a computationally efficient way, we can use $$q^*$$ and evaluate

\begin{align} p\lrb[\bm{y}^* | \bm{X}^*, \Xb] \approx \int p\lrb[\bm{y}^* | \fstar] p\lrb[\fstar | \fb] q\lrb[\fb] d\fb d\fstar. \end{align}

Again, we can do this calculation in closed form to obtain an (approximate) predictive posterior.

Lemma (Approximate predictive posterior) Under the optimal approximate posterior $$q^*$$, the predictive posterior of the VFE approximation is

\begin{align} p\lrb[\bm{y}^* | \X^*, \Xb] = \mathcal{N}\lrb[\bm{y}^*; \sigma^{-2} \Ksb \bs{\Sigma}^{-1} \Kbx \mathbf{y}, \Kss - \Ksb \Kbb^{-1}\Kbs + \Ksb \bs{\Sigma}^{-1} \Kbs + \sigma^2 \mathbf{I}]. \end{align}

Derivation: Approximate predictive posterior

Starting from the expression for the approximate predictive posterior

\begin{align} p\lrb[\bm{y}^* | \bm{X}^*, \Xb] \approx \int p\lrb[\bm{y}^* | \fstar] p\lrb[\fstar | \fb] q\lrb[\fb] d\fb d\fstar, \end{align}

and remembering that the exact $$p\lrb[\fstar | \fb]$$ is

\begin{align} p\lrb[\fstar | \fb] = \mathcal{N}\lrb[\fstar;~\Ksb\Kbb^{-1}\fb,~\Kss - \Ksb\Kbb^{-1}\Kbs], \end{align}

we obtain, using standard results for the product of Gaussians and marginals of Gaussians

\begin{align} p\lrb[\bm{y}^* | \X^*, \Xb] = \mathcal{N}\lrb[\bm{y}^*; \sigma^{-2} \Ksb \bs{\Sigma}^{-1} \Kbx \mathbf{y}, \Kss - \Ksb \Kbb^{-1}\Kbs + \Ksb \bs{\Sigma}^{-1}\Kbs + \sigma^2 \mathbf{I} ]. \end{align}

To evaluate this predictive posterior, we need to invert the $$M \times M$$ matrix $$\Kbb$$, costing us $$\mathcal{O}\lrb[M^3]$$, and also to compute certain matrix products, the worst of which will cost us $$\mathcal{O}\lrb[NM^2]$$, making the overall cost scale as $$\mathcal{O}\lrb[NM^2]$$ when $$M < N$$, just as with the free energy.

### Cost and quality¶

Let’s take a moment to review the VFE approximation. How did we achieve the improvement from $$\mathcal{O}\lrb[N^3]$$ to $$\mathcal{O}\lrb[NM^2]$$? In the original GP posterior predictive, we considered statistical relationships between all datapoints $$\X, \bm{y}$$ and the prediction points $$\X^*, \bm{y}^*$$

By approximating $$p\lrb[f | \bm{y}, \X]$$ with the distribution $$p\lrb[f_{\neq \Xb} | \fb] q\lrb[\fb]$$, we instead account for statistical relationships only between $$\fb$$ and $$f$$, thereby reducing the cost of evaluating the posterior predictive. The information of how $$\mathbf{y}$$ affects $$\mathbf{y}^*$$ is entirely contained in the distribution $$q\lrb[\fb]$$, the form of which is picked to approximate the the true posterior as accurately as possible. Although this summarised representation reduces the cost of making predictions, it also limits the set of posterior distributions which can be accurately represented by our model. A way in which this approximation might fail is if there are not enough inducing points to sufficiently constrain the approximate posterior - consider the extreme case of using a single (or zero) inducing points to model many data points. This could be either due to picking too small an $$M$$, or placing the inducing points’ inputs $$\Xb$$ at poor locations, leaving large areas of the input space uncovered.

As the dataset size $$N$$ increases, a greater number of inducing points may be needed. In particular, if we wish to approximate the exact posterior $$p\lrb[\fx | \fb]$$ sufficiently accurately throughout a larger input region, we may need more inducing points and thus a larger $$M$$. The scaling of $$M$$ will therefore depend on (1) the distribution of input data $$\bm{X}$$, (2) the type of kernel used and (3) the specified quality of approxmation, for example in KL distance.

Clearly, the positions $$\Xb$$ of the inducing points are important and we haven’t talked about how to optimise those. One of the main contributions of the VFE approximation is to provide a principled way of selecting the inducing point locations, which is to optimise $$\mathcal{F}$$ with respect to $$\Xb$$. This can be done without fear of overfitting, because we are optimising $$\mathcal{F}$$ which is a lower bound to the exact log-marginal likelihood. Adding more inducing points or optimising their locations for longer will only bring the approximate posterior closer to the true posterior. This is not so for model-approximating methods like DTC[SWL03] or FITC[SG05] which approximate the model rather than the posterior. In this case, there is no guarantee that adding more inducing points or optimising their locations won’t overfit. This is one of the strengths of the VFE approximation.

## Implementation¶

We are now at a position to implement the sparse VFE approximation for GPs. We will fit the VFE approximation to data sampled from an exact GP for which we know the ground truth.

### Sampling the exact GP¶

First, we implement a constant GP mean class and an EQ covariance class, both as tf.keras.Models, so that we can later re-use them to make a trainable VFEGP.

import tensorflow as tf
import tensorflow_probability as tfp

class ConstantMean(tf.keras.Model):

def __init__(self,
dtype,
name='constant_mean'):

super().__init__(name=name, dtype=dtype)

self.constant = tf.Variable(tf.constant(0., dtype=dtype))

def __call__(self, x):
return self.constant * tf.ones(x.shape, dtype=self.dtype)

class EQcovariance(tf.keras.Model):

def __init__(self,
log_coeff,
log_scales,
dim,
dtype,
trainable=True,
name='eq_covariance',
**kwargs):

super().__init__(name=name, dtype=dtype, **kwargs)

# Convert parameters to tensors
log_coeff = tf.convert_to_tensor(log_coeff, dtype=dtype)
log_scales = tf.convert_to_tensor(log_scales, dtype=dtype)

# Reshape parameter tensors
log_coeff = tf.squeeze(log_coeff)
log_scales = tf.reshape(log_scales, (-1,))

# Set input dimensionality
self.dim = dim

# Set EQ parameters
self.log_scales = tf.Variable(log_scales, trainable=trainable)
self.log_coeff = tf.Variable(log_coeff, trainable=trainable)

def __call__(self,
x1,
x2,
diag=False,
epsilon=None):

# Convert to tensors
x1 = tf.convert_to_tensor(x1, dtype=self.dtype)
x2 = tf.convert_to_tensor(x2, dtype=self.dtype)

# Get vector of lengthscales
scales = self.scales

if not diag:

x1 = x1[:, None, :]
x2 = x2[None, :, :]

scales = self.scales[None, None, :] ** 2

# Compute quadratic, exponentiate and multiply by coefficient
quad = - 0.5 * (x1 - x2) ** 2 / scales
eq_cov = self.coeff ** 2 * tf.exp(quad)

if epsilon is not None:
eq_cov = eq_cov + epsilon * tf.eye(eq_cov.shape,
dtype=self.dtype)

return eq_cov

@property
def scales(self):
return tf.math.exp(self.log_scales)

@property
def coeff(self):
return tf.math.exp(self.log_coeff)

# Set random seed and tf.dtype
np.random.seed(0)
dtype = tf.float64

# Num. observations (N)
N = 100

# EQ covariance hyperparameters
log_coeff = 0.
log_scale = 0.
noise = 1e-2
dim = 1

# Initialise covariance
ground_truth_cov = EQcovariance(log_coeff=log_coeff,
log_scales=log_scale,
dim=dim,
dtype=dtype)

# Pick inputs at random
x_train = np.random.uniform(low=-4., high=4., size=(N, 1))

# Compute covariance matrix terms
K_train_train = ground_truth_cov(x_train, x_train, epsilon=1e-12)
I_noise = noise * np.eye(N)

# Sample f_ind | x_ind
y_train = np.dot(np.linalg.cholesky(K_train_train + I_noise),
np.random.normal(loc=0., scale=1., size=(N, 1)))

# Locations to plot mean and variance of generative model, y_plot | f_ind, x_plot
x_plot = np.linspace(-8., 8., 100)[:, None]

# Covariances between inducing points and input locations
K_train_plot = ground_truth_cov(x_train, x_plot)
K_plot_train = ground_truth_cov(x_plot, x_train)
K_plot_diag = ground_truth_cov(x_plot, x_plot, diag=True)

# Mean and standard deviation of y_plot | f_ind, x_plot
y_plot_mean = np.dot(K_plot_train, np.linalg.solve(K_train_train + I_noise, y_train))[:, 0]
f_plot_var = K_plot_diag - np.diag(np.dot(K_plot_train,
np.linalg.solve(K_train_train + I_noise, K_train_plot)))
y_plot_var = f_plot_var + noise
y_plot_std = y_plot_var ** 0.5

# Plot inducing points and observed data
plt.figure(figsize=(10, 3))

# Plot exact posterior predictive
plt.plot(x_plot, y_plot_mean - 2*y_plot_std, '--', color='purple',  zorder=2)
plt.plot(x_plot, y_plot_mean, color='purple',  zorder=2, label='Exact post.')
plt.plot(x_plot, y_plot_mean + 2*y_plot_std, '--', color='purple',  zorder=2)

# Plot sampled data
plt.scatter(x_train,
y_train,
color='red',
marker='+',
zorder=3,
label=r'Observed $\mathbf{y}$')

# Plot formatting
plt.title('Synthetic data and ground truth', fontsize=22)
plt.xticks(np.arange(-8, 9, 4), fontsize=14)
plt.yticks(np.arange(-8, 9, 4), fontsize=14)
plt.legend(loc='lower right', fontsize=14)
plt.xlim([-8., 8.])
plt.show() ### Implementing the VFE GP¶

We can now put things together and implement the VFEGP. The post_pred and free_energy methods usee the Woodbury identity to speed up the inversion of the matrices required in this computation. It turns out however that when using the Woodbury identity, some of the matrices involved are poorly condtitioned. To alleviate this issue, I’ve found it necessary to make use of a conditioning trick also used in the GPFlow library.

class VFEGP(tf.keras.Model):

def __init__(self,
x_train,
y_train,
x_ind_init,
mean,
cov,
log_noise,
trainable_noise,
dtype,
name='vfe_gp',
**kwargs):

super().__init__(name=name, dtype=dtype, **kwargs)

# Set training data and inducing point initialisation
self.x_train = tf.convert_to_tensor(x_train, dtype=dtype)
self.y_train = tf.convert_to_tensor(y_train, dtype=dtype)

# Set inducing points
self.x_ind = tf.convert_to_tensor(x_ind_init, dtype=dtype)
self.x_ind = tf.Variable(self.x_ind)

# Set mean and covariance functions
self.mean = mean
self.cov = cov

# Set log of noise parameter
self.log_noise = tf.convert_to_tensor(log_noise, dtype=dtype)
self.log_noise = tf.Variable(self.log_noise, trainable=trainable_noise)

@property
def noise(self):
return tf.math.exp(self.log_noise)

def post_pred(self, x_pred):

# Number of training points
N = self.y_train.shape
M = self.x_ind.shape

# Compute covariance terms
K_ind_ind = self.cov(self.x_ind, self.x_ind, epsilon=1e-9)
K_train_ind = self.cov(self.x_train, self.x_ind)
K_ind_train = self.cov(self.x_ind, self.x_train)
K_pred_ind = self.cov(x_pred, self.x_ind)
K_ind_pred = self.cov(self.x_ind, x_pred)
K_pred_pred_diag = self.cov(x_pred, x_pred, diag=True)

# Compute shared matrix and its cholesky:
# L = chol(K_ind_ind)
# U = iL K_ind_train
# A = U / noise
# B = I + A A.T
L = tf.linalg.cholesky(K_ind_ind)
LT = tf.transpose(L, (1, 0))
U = tf.linalg.triangular_solve(L, K_ind_train, lower=True)
A = U / self.noise
B = tf.eye(M, dtype=self.dtype) + tf.matmul(A, A, transpose_b=True)
B_chol = tf.linalg.cholesky(B)

# Compute mean
diff = self.y_train # - self.mean(self.x_train)[:, None]
beta = tf.linalg.cholesky_solve(B_chol, tf.matmul(U, diff))
beta = tf.linalg.triangular_solve(LT, beta, lower=False)
mean = tf.matmul(K_pred_ind / self.noise ** 2, beta)[:, 0]

C = tf.linalg.triangular_solve(L, K_ind_pred)
D = tf.linalg.triangular_solve(B_chol, C)

# Compute variance
var = K_pred_pred_diag + self.noise ** 2
var = var - tf.linalg.diag_part(tf.matmul(C, C, transpose_a=True))
var = var + tf.linalg.diag_part(tf.matmul(D, D, transpose_a=True))

return mean, var

def free_energy(self):

# Number of training points
N = self.y_train.shape
M = self.x_ind.shape

# Compute covariance terms
K_ind_ind = self.cov(self.x_ind, self.x_ind, epsilon=1e-9)
K_train_ind = self.cov(self.x_train, self.x_ind)
K_ind_train = self.cov(self.x_ind, self.x_train)
K_train_train_diag = self.cov(self.x_train, self.x_train, diag=True)

# Compute shared matrix and its cholesky:
# L = chol(K_ind_ind)
# U = iL K_ind_train
# A = U / noise
# B = I + A A.T
L = tf.linalg.cholesky(K_ind_ind)
U = tf.linalg.triangular_solve(L, K_ind_train, lower=True)
A = U / self.noise
B = tf.eye(M, dtype=self.dtype) + tf.matmul(A, A, transpose_b=True)
B_chol = tf.linalg.cholesky(B)

# Compute log-normalising constant of the matrix
log_pi = - N / 2 * tf.math.log(tf.constant(2 * np.pi, dtype=self.dtype))
log_det_B = - tf.reduce_sum(tf.math.log(tf.linalg.diag_part(B_chol)))
log_det_noise = - N / 2 * tf.math.log(self.noise ** 2)

# Log of determinant of normalising term
log_det = log_pi + log_det_B + log_det_noise

diff = self.y_train - self.mean(self.x_train)[:, None]
c = tf.linalg.triangular_solve(B_chol, tf.matmul(A, diff), lower=True) / self.noise
quad = - 0.5 * tf.reduce_sum((diff / self.noise) ** 2)

# Compute trace term
trace = - 0.5 * tf.reduce_sum(K_train_train_diag) / self.noise ** 2
trace = trace + 0.5 * tf.linalg.trace(tf.matmul(A, A, transpose_b=True))

free_energy = (log_det + quad + trace) / N

return free_energy

def plot(model,
ground_truth_cov,
x_pred,
x_train,
y_train,
x_ind_init,
step):

# Get exact and approximate posterior predictive
vfe_mean, vfe_var = model.post_pred(x_pred)

# Covariances between inducing points and input locations
K_train_plot = ground_truth_cov(x_train, x_plot)
K_plot_train = ground_truth_cov(x_plot, x_train)
K_plot_diag = ground_truth_cov(x_plot, x_plot, diag=True)

# Mean and standard deviation of y_plot | f_ind, x_plot
exact_mean = np.dot(K_plot_train, np.linalg.solve(K_train_train + I_noise, y_train))[:, 0]
exact_var = K_plot_diag - np.diag(np.dot(K_plot_train,
np.linalg.solve(K_train_train + I_noise, K_train_plot)))
exact_std = (exact_var + noise) ** 0.5

x_pred = x_pred[:, 0].numpy()
x_ind = model.x_ind[:, 0].numpy()

vfe_mean = vfe_mean.numpy()
vfe_var = vfe_var.numpy()

plt.figure(figsize=(10, 3))

# Plot posterior predictive
plt.plot(x_pred,
vfe_mean,
color='black',
zorder=1,
label='Approx. Post.')

plt.fill_between(x_pred,
vfe_mean - 2 * vfe_var ** 0.5,
vfe_mean + 2 * vfe_var ** 0.5,
color='gray',
alpha=0.3)

# Plot exact posterior
plt.plot(x_plot,
exact_mean - 2*exact_std,
'--',
color='purple',
zorder=1)

plt.plot(x_plot,
exact_mean,
color='purple',
zorder=1)

plt.plot(x_plot,
exact_mean + 2*exact_std,
'--',
color='purple',
zorder=1,
label='Exact Post.')

# Plot training data
plt.scatter(x_train, y_train, color='red', marker='+', zorder=2)

# Plot initial inducing points
plt.scatter(x_ind_init,
-5.5 * tf.ones_like(x_ind_init),
color='green',
marker='+',
label=r'Init. $\bar{\mathbf{X}}$',
zorder=2)

# Plot current inducing points
plt.scatter(x_ind,
-5. * tf.ones_like(x_ind),
color='blue',
marker='+',
label=r'Curr. $\bar{\mathbf{X}}$',
zorder=2)

# Format plot
plt.title(f'VFE after {step} optimisation steps', fontsize=18)

plt.xticks(np.arange(-10, 11, 4), fontsize=14)
plt.yticks(np.arange(-6, 7, 3), fontsize=14)

plt.xlim([-8., 8.])
plt.ylim([-8., 4.])

plt.legend(loc='lower right', fontsize=10)

plt.show()

def print_info(model, step):

free_energy = model.free_energy()

print(
f'Step: {step:5>} '
f'Free energy: {free_energy.numpy():8.3f} '
f'Coeff: {model.cov.coeff.numpy():5.2f} '
f'Scales: {[round(num, 3) for num in model.cov.scales.numpy()]} '
f'Noise: {model.noise.numpy():5.2f}'
)


### Learning inducing points¶

As a sanity check, we first try learning the inducing point locations only, setting the covariance parameters to their ground truth values.

# Set random seed and tensor dtype
tf.random.set_seed(1)

# Number GP constants
M = 10
inducing_range = (-4., -2.)
log_noise = np.log(1e-1)
log_coeff = np.log(1e0)
log_scales = [np.log(1e0)]
trainable = False

# Define mean and covariance
mean = ConstantMean(dtype=dtype)

cov = EQcovariance(log_coeff=log_coeff,
log_scales=log_scales,
dim=1,
dtype=dtype,
trainable=trainable)

# Initial locations of inducing points
x_ind_dist = tfp.distributions.Uniform(*inducing_range)
x_ind_init = x_ind_dist.sample(sample_shape=(M, 1))
x_ind_init = tf.cast(x_ind_init, dtype=dtype)

# Define sparse VFEGP
vfe_gp = VFEGP(mean=mean,
cov=cov,
log_noise=log_noise,
x_train=x_train,
y_train=y_train,
x_ind_init=x_ind_init,
dtype=dtype,
trainable_noise=trainable)

num_steps = 1000

x_pred = tf.linspace(-8., 8., 100)[:, None]
x_pred = tf.cast(x_pred, dtype=dtype)

for step in range(num_steps + 1):

free_energy = vfe_gp.free_energy()
loss = - free_energy

# Print information and plot at start and end
if step % num_steps == 0:

print_info(vfe_gp, step)

plot(vfe_gp,
ground_truth_cov,
x_pred,
x_train,
y_train,
x_ind_init,
step)


Step: 0 Free energy:  -35.235 Coeff:  1.00 Scales: [1.0] Noise:  0.10 Step: 1000 Free energy:    0.532 Coeff:  1.00 Scales: [1.0] Noise:  0.10 Initially the positions of the inducing points (green) do not allow for an expressive enough posterior to match the exact GP posterior. By spreading out the inducing points, the model manages to recover an approximate posterior that is very close to the ground truth. We were expecting this to be possible since there are lots of observed data which are redundant, and should be summarisable by a smaller set of inducing points.

### Complete learning¶

We now turn to joint learning of learning the inducing point locations as well as the covariance parameters. This time around we initialise the hyperparameters to values far from the ground truth, to see whether the model is able to recover something close to the true posterior.

# Set random seed and tensor dtype
tf.random.set_seed(1)

# Number GP constants
M = 10
inducing_range = (-4., -2.)
log_noise = np.log(1e0)
log_coeff = np.log(1e1)
log_scales = [np.log(1e1)]
trainable = True

# Define mean and covariance
mean = ConstantMean(dtype=dtype)

cov = EQcovariance(log_coeff=log_coeff,
log_scales=log_scales,
dim=1,
dtype=dtype,
trainable=trainable)

# Initial locations of inducing points
x_ind_dist = tfp.distributions.Uniform(*inducing_range)
x_ind_init = x_ind_dist.sample(sample_shape=(M, 1))
x_ind_init = tf.cast(x_ind_init, dtype=dtype)

# Re-define sparse VFEGP with trainable noise
vfe_gp = VFEGP(mean=mean,
cov=cov,
log_noise=log_noise,
x_train=x_train,
y_train=y_train,
x_ind_init=x_ind_init,
dtype=dtype,
trainable_noise=trainable)

num_steps = 1000

x_pred = tf.linspace(-8., 8., 100)[:, None]
x_pred = tf.cast(x_pred, dtype=dtype)

for step in range(num_steps + 1):

free_energy = vfe_gp.free_energy()
loss = - free_energy

# Print information and plot at start and end
if step % num_steps == 0:

print_info(vfe_gp, step)

plot(vfe_gp,
ground_truth_cov,
x_pred,
x_train,
y_train,
x_ind_init,
step)


Step: 0 Free energy:   -1.232 Coeff: 10.00 Scales: [10.0] Noise:  1.00 Step: 1000 Free energy:    0.547 Coeff:  1.16 Scales: [1.115] Noise:  0.10 We observe that the model has been able to recover both an approximate posterior which is very close to the true posterior, as well as parameter values close to the ground truth. It achieves this with a complexity of just $$\mathcal{O}(NM^2)$$ as opposed to exact GP inference which requires $$\mathcal{O}(N^3)$$ operations.