# Global inducing points for BNNs¶

One central challenge in working with Bayesian Neural Networks (BNNs) is handling their posterior distributions. Since the posterior over the parameters of a BNN is analytically intractable, working with it typically requires sampling or making approximations. In Variational Inference (VI), we approximate the exact posterior using another tractable distibution, which can be used for making predictions. Perhaps unsurpsrisingly, VI is sensitive to the choice of approximate posterior, and naive choices can lead to undesirable modelling behaviours.[TS11] For example, a usual choice is to approximate the posteerior using a factored Gaussian,[BCKW15] which typically leads to underfitting and poor estimates of the predictive uncertainty. [FLHLT19] Recently, Ober and Aitchison [OA21a] introduced an approximate posterior which goes beyond the factored approximation and yields improved results. Their approximation is based on inducing points, an idea which is common in the Gaussian Process (GP) literature. In particular, this approximate posterior can be used for Deep GPs (DGPs) as well as BNNs, highlighting the similarities between these models.

## Posterior over the last layer¶

We first define the model and prior assumptions. Consider a regression task, where we want to learn a mapping from inputs $$\mathbf{X} \subseteq \mathbb{R}^{K \times D_x}$$ to the corresponding outputs $$\mathbf{Y} \subseteq \mathbb{R}^{K \times D_y}$$. Consider a fully connected neural network made up of $$L$$ hidden layers, each containing $$N_1, \dots, N_L$$ hidden units, followed by by a last linear layer which maps from the $$N_L$$ units to the dimension of the data $$D_y$$. Let the weights in each layer be

\begin{align} \mathbf{W} = \{\mathbf{W}_l \in \mathbb{R}^{N_l \times N_{l+1}} \}_{l = 1}^{L+1}, \end{align}

where $$N_0 = D_x$$ and $$N_{L+1} = D_y$$. Thus the network takes the form

\begin{split}\begin{align} \mathbf{F}_1 &= \mathbf{X} \mathbf{W}_1, \\ \mathbf{F}_{l+1} &= \phi(\mathbf{F}_{l}) \mathbf{W}_l, \text{ for } l = 1, \dots, L, \end{align}\end{split}

where $$\phi$$ is a nonlinearity. Note that under this notation, the weights post-multiply the activations rather than pre-multiplying them. Now suppose we place a diagonal Gaussian prior

\begin{align} p(\mathbf{W}) = \prod_{l = 1}^{L+1} \prod_{ij} \mathcal{N}(w_{l, i, j}; 0, \sigma_l^2), \end{align}

over the weights, where $$w_{l, i, j}$$ is the weight corresponding to the $$(i, j)$$ entry of the matrix $$\mathbf{W}_l$$, together with the Gaussian likelihood function

\begin{align} p(\mathbf{Y} | \mathbf{W}, \mathbf{X}) = \prod_{k = 1}^K \mathcal{N}(\mathbf{y}_{k, \cdot}; \mathbf{f}_{L, k, \cdot} \mathbf{W}_{L+1}, \sigma_n^2 I), \end{align}

where $$\mathbf{y}_{k, \cdot}$$ is the $$k^{th}$$ row of the $$\mathbf{Y}$$ matrix and $$\mathbf{f}_{L, k, \cdot}$$ is the $$k^{th}$$ row of the $$\mathbf{F}_{L+1}$$ matrix. The posterior over the weights of this network is not analytic and must therefore be approximated. However, conditioned on the weights of all previous layers, the posterior over the weights of the last layer is analytic. In particular, we have

\begin{align} p(\mathbf{w}_{L+1, \cdot, d} | \mathbf{W}_{1:L}, \mathbf{X}) \propto \underbrace{p(\mathbf{w}_{L+1, \cdot, d})}_{\text{Prior term}}~\underbrace{\mathcal{N}(y_{\cdot, d}; \mathbf{f}_{L, d, \cdot}^\top \mathbf{w}_{L+1, \cdot, d}, \sigma_n^2 I)}_{\text{Likelihood term}}, \end{align}

where $$\mathbf{w}_{L+1, \cdot, d}$$ is the $$d^{th}$$ column of $$\mathbf{W}_{L+1}$$. Note that the features of the last layer depend on the weights of all previous layers, which we are conditioning on. From here we can show that the conditional posterior over $$\mathbf{W}_{L+1, \cdot, d}$$ is also Gaussian, and takes the form

\begin{align} p(\mathbf{w}_{L+1, \cdot, d} | \mathbf{W}_{1:L}, \mathbf{X}) = \mathcal{N}\left(\mathbf{w}_{L+1, \cdot, d}; \mathbf{f}_{L, k, \cdot}^\top \mathbf{w}_{L+1, \cdot, d}, (\sigma_n^{-2} I)\right), \end{align}

Ober and Aitchison [OA21a] draw inspiration from this to propose an approximate posterior, in which the weights of a layer given all previous layers are conditionally Gaussian, but the full posterior is not.

## Approximate posterior¶

While the posterior over the weights of the last layer, conditioned on the weights of all previous layers. This is because the posterior over weights of a linear-in-the parameters model with a Gaussian prior, is also Gaussian. Inspired by this observation, Ober and Aitchison propose using an approximate posterior, where, conditioned on the weights of all previous layers, the weights of a given layer are Gaussian distributed, as in

\begin{align} q(\mathbf{W}) = q(\mathbf{W}_{L+1} | \mathbf{W}_L, \dots \mathbf{W}_1) \dots q(\mathbf{W}_2 | \mathbf{W}_1) q(\mathbf{W}_1). \end{align}

However, the intermediate layers do not have observations associated with them, and so we have the question of how to parameterise the posterior. Ober and Aitchison introduce a set of variational parameters, $$\mathbf{v}_l \in \mathbb{R}^{M \times N_l}$$ and $$\boldsymbol{\Lambda}_l = \text{diag}(\boldsymbol{\lambda}_l) \in \mathbb{R}^{N_{l+1} \times N_{l+1}}$$ for each layer, which play the role of pseudo-observations and pseudo-noise precision for the intermediate layers. They also introduce a set of corresponding pseudo-inputs $$\mathbb{U}_0 \in \mathbb{R}^{M \times N_1}$$, which they propagate through the network according to

\begin{split}\begin{align} \mathbf{U}_1 &= \mathbf{U}_0 \mathbf{W}_1, \\ \mathbf{U}_{l+1} &= \phi(\mathbf{U}_{l}) \mathbf{W}_l, \text{ for } l = 1, \dots, L, \end{align}\end{split}

to obtain inducing inputs at every layer of the network. This is in contrast to approaches such as Doubly Stochastic VI [SD17] which use different $$\mathbb{U}_l$$ for each layer. They then pick an approximate posterior that is analogous to the product of a pseudo-likelihood term and the prior term

\begin{align} q\left(\mathbf{W}_l | \{\mathbf{W}_{l'}\}_{l' = 1}^{l-1}\right) &\propto \underbrace{p(\mathbf{w}_{l, d})}_{\text{Prior term}} \prod_{d = 1}^{D_l} \underbrace{\mathcal{N}\left(\mathbf{v}_{l, \cdot, d}; \phi(\mathbf{U}_{l-1}) \mathbf{w}_{l, \cdot, d}, \boldsymbol{\Lambda}^{-1}_l\right)}_{\text{Pseudo-likelihood terms}}. \end{align}

Note the similarity of this term to the exact posterior. Note also that under the conditional $$q$$ is a Gaussian which factorises over weight columns. However, the full posterior, that is the product of the $$q$$ terms, is not factorised, neither is it Gaussian. Therefore, this approximation may be able to capture a wide family of distributions, and can be more expressive than a mean-field posterior. Now if we rearrange this expression, to get the normalised distribution over $$\mathbf{w}_d^l$$, we obtain

\begin{split}\begin{align} q\left(\mathbf{W}_l | \{\mathbf{W}_{l'}\}_{l' = 1}^{l-1}\right) &= \prod_{d = 1}^{D_l} \mathcal{N}\left(\mathbf{w}_{l, \cdot, d}; \boldsymbol{\mu}_{l, d}, \boldsymbol{\Sigma}_l\right), \\ \boldsymbol{\mu}_{l, d} &= \boldsymbol{\Sigma}_l \phi\left(\mathbf{U}_{l-1}\right)^\top \boldsymbol{\Lambda}_l \mathbf{v}_{l, \cdot, d}, \\ \boldsymbol{\Sigma}_l &= \left( D_l \mathbf{I} + \phi\left(\mathbf{U}_{l-1}\right)^\top \boldsymbol{\Lambda}_l \phi\left(\mathbf{U}_{l-1}\right) \right)^{-1}. \end{align}\end{split}

We can train this model by maximising the Evidence Lower Bound (ELBO). We can estimate the ELBO in the following recursive manner. First, we evaluate $$q$$ at the $$l^{th}$$ layer given all previous inducing inputs and weights, we sample $$\mathbf{W}_l$$ and use these weights to propagate the network activations and inducing inputs to obtain $$\mathbf{F}_l$$ and $$\mathbf{U}_l$$. As we propagate this information through the network, we also need to keep track of the empirical KL divergence between the approximate posterior $$q$$ and the prior $$p$$ over the weights of the layer. At the last layer, we can use sampled weights to evaluate the likelihood of the data, conditioned on the sampled weights. Summing the KL contribution from all layers together with the likelihood of the data, we obtain an unbiased estimate of the ELBO.

## Implementation¶

We will define a GlobalInducingDenseLayer, which handdles propagating the data activations $$\mathbf{F}_l$$, the inducing activations $$\mathbf{U}_l$$ and computes the contribution of the layer to the total KL divergence.

class GlobalInducingDenseLayer(tfk.layers.Layer):

def __init__(self,
num_input,
num_output,
num_inducing,
nonlinearity,
prior_scale_factor,
dtype,
name="global_inducing_fully_connected_layer",
**kwargs):

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

self.num_input = num_input + 1
self.num_output = num_output
self.num_inducing = num_inducing
self.prior_scale_factor = prior_scale_factor

# Set nonlinearity for the layer
self.nonlinearity = (lambda x: x) if nonlinearity is None else \
getattr(tf.nn, nonlinearity)

def build(self, input_shape):

# Set up prior mean, scale and distribution
self.prior_mean = tf.zeros(
shape=(self.num_output, self.num_input),
dtype=self.dtype
)

self.prior_scale = tf.ones(
shape=(self.num_output, self.num_input),
dtype=self.dtype
)
self.prior_scale = self.prior_scale / self.num_input**0.5
self.prior_scale = self.prior_scale * self.prior_scale_factor

self.prior = tfd.MultivariateNormalDiag(
loc=self.prior_mean,
scale_diag=self.prior_scale
)

# Set up pseudo observation means and variances
self.pseudo_means = tf.zeros(
shape=(self.num_inducing, self.num_output),
dtype=self.dtype
)
self.pseudo_mean = tf.Variable(self.pseudo_means)

self.pseudo_log_prec = tf.zeros(
shape=(self.num_inducing,),
dtype=self.dtype
)
self.pseudo_log_prec = tf.Variable(self.pseudo_log_prec)

@property
def pseudo_precision(self):
return tf.math.exp(self.pseudo_log_precision)

def q_prec_cov_chols(self, Uin):

phiU = self.nonlinearity(Uin)
pseudo_prec = tf.math.exp(self.pseudo_log_prec)

# Compute precision matrix of multivariate normal
phiT_lambda_phi = tf.einsum("mi, m, mj -> ij", phiU, pseudo_prec, phiU)

q_prec = tf.linalg.diag(self.prior_scale[0, :]**-2.) + phiT_lambda_phi

# Compute cholesky of approximate posterior precision
q_prec_chol = tf.linalg.cholesky(q_prec)

# Compute cholesky of approximate posterior covariance
iq_prec_chol = tf.linalg.triangular_solve(
q_prec_chol,
tf.eye(q_prec_chol.shape[0]),
lower=True
)

q_cov = tf.matmul(iq_prec_chol, iq_prec_chol, transpose_a=True)
q_cov = q_cov + 1e-5 * tf.eye(q_cov.shape[0])
q_cov_chol = tf.linalg.cholesky(q_cov)

return q_prec_chol, q_cov_chol

def q_mean(self, Uin, prec_chol):

phiU = self.nonlinearity(Uin)
pseudo_prec = tf.math.exp(self.pseudo_log_prec)

mean = tf.matmul(
phiU,
pseudo_prec[:, None] * self.pseudo_mean,
transpose_a=True
)

mean = tf.linalg.cholesky_solve(prec_chol, mean)
mean = tf.transpose(mean, [1, 0])

return mean

def call(self, Fin, Uin):

# Augment input features with ones to absorb bias
Fones = tf.ones(shape=(Fin.shape[0], 1), dtype=self.dtype)
Fin = tf.concat([Fin, Fones], axis=-1)

Uones = tf.ones(shape=(Uin.shape[0], 1), dtype=self.dtype)
Uin = tf.concat([Uin, Uones], axis=-1)

Din = self.num_input
Dout = self.num_output
M = self.num_inducing

# Check shape of input features Fin and pseudo-means
check_shape(
[Fin, Uin, self.pseudo_means],
[(-1, Din), (M, Din), (M, Dout)]
)

# Compute cholesky factors of q precision and covariance.
# These are common between all weight columns, i.e. the covariance
# between weights leading to a neuron in the next layer is shared
# between all next neurons.
q_prec_chol, q_cov_chol = self.q_prec_cov_chols(Uin)

check_shape(
[q_prec_chol, q_cov_chol],
[(Din, Din), (Din, Din)]
)

# Compute means of q. There is a different mean vector for
# each column of weights.
q_mean = self.q_mean(Uin, q_prec_chol)

check_shape(q_mean, (Dout, Din))

# Sample approximate posterior for the weights
q_cov_chol = tf.stack([q_cov_chol]*Dout, axis=0)
q = tfd.MultivariateNormalTriL(loc=q_mean, scale_tril=q_cov_chol)
wT = q.sample()
w = tf.transpose(wT, [1, 0])

check_shape(w, (Din, Dout))

# Compute contibution to ELBO
kl_term = q.kl_divergence(self.prior)
kl_term = tf.reduce_sum(kl_term)

# Compute log-probability of weights under prior
log_p = self.prior.log_prob(wT)
log_p = tf.reduce_sum(log_p)

# Compute log-probability of weights under approximate posterior
log_q = q.log_prob(wT)
log_q = tf.reduce_sum(log_q)

# Compute Fout and Uout and return
Fout = tf.matmul(self.nonlinearity(Fin), w)
Uout = tf.matmul(self.nonlinearity(Uin), w)

return Fout, Uout, kl_term, log_p, log_q


We can then stack a few GlobalInducingDenseLayers to form a GlobalInducingFullyConnectedNetwork. We use an architecture using two hidden layers, each using $$50$$ units, as done by Ober and Aitchinson.

class GlobalInducingFullyConnectedNetwork(tfk.Model):

def __init__(self,
num_input,
num_output,
inducing_points,
nonlinearity,
prior_scale_factor,
dtype,
name="global_inducing_fully_connected",
**kwargs):

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

self.num_input = num_input
self.num_output = num_output
self.inducing_points = inducing_points
self.num_inducing = inducing_points.shape[0]
self.nonlinearity = nonlinearity
self.prior_scale_factor = prior_scale_factor
self.num_hidden = [50, 50]

def build(self, input_shape):

self.inducing_points = tf.Variable(self.inducing_points)

self.l1 = GlobalInducingDenseLayer(
num_input=self.num_input,
num_output=self.num_hidden[0],
num_inducing=self.num_inducing,
nonlinearity=None,
prior_scale_factor=self.prior_scale_factor,
dtype=self.dtype
)

self.l2 = GlobalInducingDenseLayer(
num_input=self.num_hidden[0],
num_output=self.num_hidden[1],
num_inducing=self.num_inducing,
nonlinearity=self.nonlinearity,
prior_scale_factor=self.prior_scale_factor,
dtype=self.dtype
)

self.l3 = GlobalInducingDenseLayer(
num_input=self.num_hidden[1],
num_output=self.num_output,
num_inducing=self.num_inducing,
nonlinearity=self.nonlinearity,
prior_scale_factor=self.prior_scale_factor,
dtype=self.dtype
)

self.log_noise = tf.Variable(
tf.convert_to_tensor(-2., dtype=self.dtype)
)

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

@tf.function
def call(self, x):

F1, U1, kl1, log_p1, log_q1 = self.l1(x, self.inducing_points)
F2, U2, kl2, log_p2, log_q2 = self.l2(F1, U1)
F3, U3, kl3, log_p3, log_q3 = self.l3(F2, U2)

means = F3
scales = self.noise * tf.ones_like(F3)

kl = tf.reduce_sum([kl1, kl2, kl3])

log_p = tf.reduce_sum([log_p1, log_p2, log_p3])
log_q = tf.reduce_sum([log_q1, log_q2, log_q3])

return means, scales, kl, log_p, log_q

def elbo(self, x, y):

means, scales, kl, _, _ = self(x)

cond_lik = tfd.Normal(loc=means, scale=scales)
cond_lik = tf.reduce_sum(cond_lik.log_prob(y))

elbo = cond_lik - kl

return elbo, cond_lik, kl

def iwbo(self, x, y, num_samples):

iwbo = []

for i in range(num_samples):

means, scales, kl, log_p, log_q = self(x)

cond_lik = tfd.Normal(loc=means, scale=scales)
cond_lik = tf.reduce_sum(cond_lik.log_prob(y))

iwbo.append(cond_lik + log_p - log_q)

iwbo = tf.stack(iwbo, axis=0)
iwbo = tf.math.reduce_logsumexp(iwbo) - np.log(num_samples)

return iwbo


We now want a dataset to fit tthis model to. We’ll use the same one-dimensional toy example that Ober and Aitchinson used, as shown below. This is an interesting example, because of the gap in the middle of the dataset. Recently Foong et al.[FLHLT19] have proved that applying mean-field VI to a, somewhat limited, class of neural networks, fails to capture predictive uncertainty in between different clusters of datapoints. The GIP approximate posterior is not a mean-field posterior, and may represent in-between uncertainty.

num_data = 100
num_input = 1
std_noise = 3.

x1 = tf.random.uniform(minval=-4., maxval=-2., shape=(num_data // 2, 1))
x2 = tf.random.uniform(minval=2., maxval=4., shape=(num_data // 2, 1))

x = tf.concat([x1, x2], axis=0)
y = tf.concat([x1, x2], axis=0) ** 3. + std_noise * tf.random.normal(shape=(num_data, 1))

x = (x - tf.reduce_mean(x)) / tf.math.reduce_std(x)
y = (y - tf.reduce_mean(y)) / tf.math.reduce_std(y)

# Figure to plot on
plt.figure(figsize=(11, 2))

# Plot data
plt.scatter(
x[:, 0],
y[:, 0],
marker="+",
c="black"
)

# Format plot
plt.xlim([-2.5, 2.5])
plt.ylim([-3.5, 3.5])

plt.xticks(np.linspace(-2., 2., 3), fontsize=20)
plt.yticks(np.linspace(-3., 3., 3), fontsize=20)

plt.xlabel("$x$", fontsize=20)
plt.ylabel("$y$", fontsize=20)

plt.show()


Now we can train the model. We’ll use a standard training procedure with Adam, and no special initialisation or scheduling tricks. We’ll also train the model for quite a while to ensure it has converged.

# We decorate a single gradient descent step with tf.function. On the first
# call of single_step, tensorflow will compile the computational graph first.
# After that, all calls to single_step will use the compiled graph which is
# much faster than the default eager mode execution. In this case, the gain
# is roughly a x20 speedup (with a CPU), which can be checked by commenting
# out the decorator and rerunning the training script.

@tf.function
def single_step(model, optimiser, x, y):

elbo, cond_lik, kl = model.elbo(x, y)
loss = - elbo / x.shape[0]

return elbo, cond_lik, kl

# Set model constants
num_input = 1
num_output = 1
num_inducing = 100
dtype = tf.float32
nonlinearity = "relu"
prior_scale_factor = 2.
num_steps = int(1e4)

# Initialise inducing points at subset of training points
inducing_idx = tf.random.shuffle(tf.range(x.shape[0]))[:num_inducing]
inducing_points = tf.gather(x, inducing_idx)

# Initialise model
model = GlobalInducingFullyConnectedNetwork(
num_input=num_input,
num_output=num_output,
inducing_points=inducing_points,
nonlinearity=nonlinearity,
prior_scale_factor=prior_scale_factor,
dtype=dtype
)

# Initialise optimiser

# Set progress bar and suppress warnings
progress_bar = tqdm(range(1, num_steps+1))
tf.get_logger().setLevel('ERROR')

# Set tensors for keeping track of quantities of interest
train_elbo = []
train_cond_lik = []
train_kl = []

# Train model
for i in progress_bar:

elbo, cond_lik, kl = single_step(
model=model,
optimiser=optimiser,
x=x,
y=y
)

if i % 1000 == 0:

progress_bar.set_description(
f"ELBO {elbo:.1f}, "
f"Cond-lik. {cond_lik:.1f}, "
f"KL {kl:.1f}"
)

train_elbo.append(elbo)
train_cond_lik.append(cond_lik)
train_kl.append(kl)


We can verify that training has converged by looking at the ELBO. Training further will likely not improve the model parameters or inducing points.

# Helper for computing moving average
def moving_average(array, n):

cumsum = np.cumsum(array)
cumsum[n:] = cumsum[n:] - cumsum[:-n]

moving_average = cumsum[n - 1:] / n

return moving_average

# Plot ELBO during optimisation
plt.figure(figsize=(11, 2))

plt.plot(
moving_average(tf.stack(train_elbo).numpy(), n=100),
color="black",
alpha=0.5
)

# Format plot
plt.xlabel("# training steps", fontsize=20)
plt.ylabel("ELBO", fontsize=20)

plt.xticks(np.linspace(0, num_steps, 6), fontsize=16)

plt.yticks(np.linspace(20, 80, 4), fontsize=16)

plt.xlim([0, 1.05*num_steps])
plt.ylim([15., 85.])
plt.show()


Checking the plot below, we see that the model has fit the data well and produced reasonable uncertainty estimates. This is quite pleasing since we haven’t had to use any tricks in the training procedure.

## How tight is the GIP ELBO?¶

We can also check how tight the ELBO of the GIP approximate posterior is, using importance sampling. We can estimate the same importance weighed lower bound of the marginal likelihood that importance weighed autoencoders {cite} burda2015importance use. This importance weighted lower bound (IWBO) requires multiple samples from the approximate posterior. When using a single sample, the IWBO is identical to the ELBO, and as we increase the number of samples, the IWBO approaches the marginal likelihood. To get reliable estimates, we will evaluate the ELBO and the IWBO num_repetitions = 10 times each. For the IWBO, we will use num_iwbo_samples = 1000 samples for each estimate.

# Helper for computing the ELBO and IWBO
def evaluate_elbo_and_iwbo(model, num_repetitions, num_iwbo_samples):

# Compute ELBO and IWBO num_repetitions times
elbos = [
model.elbo(x=x, y=y)[0] for i in range(num_repetitions)
]

iwbos = [
model.iwbo(x=x, y=y, num_samples=num_iwbo_samples) for i in range(num_repetitions)
]

# Compute mean and error estimate for the mean, of the ELBO and IWBO
elbo_mean = tf.reduce_mean(elbos)
elbo_std = tf.math.reduce_std(elbos)/num_repetitions**0.5

iwbo_mean = tf.reduce_mean(iwbos)
iwbo_std = tf.math.reduce_std(iwbos)/num_repetitions**0.5

return elbo_mean, elbo_std, iwbo_mean, iwbo_std

# Number of times to estimate the ELBO/IWBO
# and number of samples to draw for IWBO
num_repetitions = 10
num_iwbo_samples = 1000

# Compute mean and standard deviation for each
elbo_mean, elbo_std, iwbo_mean, iwbo_std = evaluate_elbo_and_iwbo(
model=model,
num_repetitions=num_repetitions,
num_iwbo_samples=num_iwbo_samples
)

# Print results
print(
f"ELBO: {elbo_mean: 7.3f} +/- {2.*elbo_std:.3f} "
f"(estimated with {num_repetitions} ELBO samples)"
)

print(
f"IWBO: {iwbo_mean: 7.3f} +/- {2.*iwbo_std:.3f} "
f"(estimated with {num_repetitions} IWBO samples, "
f"each using {num_iwbo_samples} weight samples)"
)

ELBO:  61.173 +/- 3.931 (estimated with 10 ELBO samples)
IWBO:  66.156 +/- 0.166 (estimated with 10 IWBO samples, each using 1000 weight samples)


We see that there is a statistically significant gap between the IWBO ($$\approx$$ marginal likelihood). However, considering there are $$100$$ datapoints in the dataset, the per-datapoint gap is relatively small. This suggests that the variational posterior is a good approximation to the true posterior. For a full comparison with existing approaches, it would be good to compare this with baselines such as mean-field VI[BCKW15] or other structured approximations. Recently, Thang Bui performed a more extensive comparison[Bui21] of various VI approximations for neural networks, and found that GIP produced ELBOs close to the marginal log-likelihood, as estimated by Annealed Importance Sampling.[Nea01]

## More experiments¶

We next look at two factors which could affect the performance of GIP, the scale of the prior and the number of inducing points in the variational posterior. For convenience, we’ll define a helper function, which will train a GIP BNN some parameters, which we’ll then systematically sweep over.

def train_model(prior_scale_factor,
num_inducing,
learning_rate=1e-2,
num_steps=int(1e5),
num_eval_reps=10,
num_iwbo_samples=10):

@tf.function
def single_step(model, optimiser, x, y):

elbo, cond_lik, kl = model.elbo(x, y)
loss = - elbo / x.shape[0]

return elbo, cond_lik, kl

# Dictionary for holding experiment results
results = {
"train_elbo"     : [],
"train_cond_lik" : [],
"train_kl"       : [],
"elbo_mean"      : None,
"elbo_std"       : None,
"iwbo_mean"      : None,
"iwbo_std"       : None,
"model"          : None
}

# Set model constants
num_input = 1
num_output = 1
dtype = tf.float32
nonlinearity = "relu"

# Initialise inducing points at subset of training points
inducing_idx = tf.random.shuffle(tf.range(x.shape[0]))[:num_inducing]
inducing_points = tf.gather(x, inducing_idx)

# Initialise model
model = GlobalInducingFullyConnectedNetwork(
num_input=num_input,
num_output=num_output,
inducing_points=inducing_points,
nonlinearity=nonlinearity,
prior_scale_factor=prior_scale_factor,
dtype=dtype
)

# Initialise optimiser

# Set progress bar and suppress warnings
tf.get_logger().setLevel('ERROR')

# Set tensors for keeping track of quantities of interest
train_elbo = []
train_cond_lik = []
train_kl = []

# Train model
for i in range(num_steps):

elbo, cond_lik, kl = single_step(
model=model,
optimiser=optimiser,
x=x,
y=y
)

results["train_elbo"].append(elbo)
results["train_cond_lik"].append(cond_lik)
results["train_kl"].append(kl)

evaluation_results = evaluate_elbo_and_iwbo(
model=model,
num_repetitions=num_eval_reps,
num_iwbo_samples=num_iwbo_samples
)

results["elbo_mean"] = evaluation_results[0]
results["elbo_std"] = evaluation_results[1]
results["iwbo_mean"] = evaluation_results[2]
results["iwbo_std"] = evaluation_results[3]
results["model"] = model

return results


### Effect of prior scale¶

The choice of prior heavily affects any Bayesian model. In fact, and perhaps unsurprisingly, in preliminary tests the prior scale seemed to have a substantial effect on the quality of the fit.

num_prior_scales = 10
min_log10_prior_scale = -0.5
max_log10_prior_scale = 0.5
num_inducing = 100
learning_rate = 1e-2
num_steps = int(1e4)
num_eval_reps = 10
num_iwbo_samples = 1000

prior_scales = 10 ** np.linspace(
min_log10_prior_scale,
max_log10_prior_scale,
num_prior_scales+1
)

# Set progress bar and suppress warnings
progress_bar = tqdm(list(enumerate(prior_scales)))

# List for holding all experimental results
prior_results = []

for i, prior_scale_factor in progress_bar:

experiment_results = train_model(
prior_scale_factor=prior_scale_factor,
num_inducing=num_inducing,
learning_rate=learning_rate,
num_steps=num_steps,
num_eval_reps=num_eval_reps,
num_iwbo_samples=num_iwbo_samples
)

progress_bar.set_description(
f"Experiment {i+1:02} Prior scale {prior_scale_factor:.3f}"
)

prior_results.append(experiment_results)

elbo_means = np.array([result["elbo_mean"] for result in prior_results])
elbo_stds = np.array([result["elbo_std"] for result in prior_results])

iwbo_means = np.array([result["iwbo_mean"] for result in prior_results])
iwbo_stds = np.array([result["iwbo_std"] for result in prior_results])

# Figure to plot on
plt.figure(figsize=(11, 3))

# Plot ELBO and error bars
plt.plot(
prior_scales,
elbo_means,
color="grey",
marker="^",
label="ELBO"
)
plt.fill_between(
prior_scales,
elbo_means - 2.*elbo_stds,
elbo_means + 2.*elbo_stds,
color="grey",
alpha=0.3
)

# Plot IWBO and error bars
plt.plot(
prior_scales,
iwbo_means,
color="pink",
marker="^",
label="IWBO"
)

plt.fill_between(
prior_scales,
iwbo_means - 2.*iwbo_stds,
iwbo_means + 2.*iwbo_stds,
color="pink",
alpha=0.3
)

# Format figure
plt.xscale("log")

plt.tick_params(axis="both", which="major", labelsize=16)
plt.tick_params(axis="both", which="minor", labelsize=16)

plt.xlim([prior_scales[0], prior_scales[-1]])
plt.ylim([-160, 110])

plt.xlabel("Prior scale $\sigma_p$", fontsize=20)
plt.ylabel("ELBO and IWBO", fontsize=20)

plt.legend(loc="lower right", fontsize=16)

plt.show()


We see that the scale of the prior has a dramatic effect on the model performance. For small scales, the model performs very poorly. Plotting the fit for small prior scales shows that the model collapses to a constant or linear function and explains most, if not all, the variation in the data as noise. As the prior scale increases, the model performs much better, presumably because the prior no longer forces the weights close to zero. We also note that the gap between the ELBO and the IWBO remains relatively small throughout, reinforcing the conclusion that this approximate posterior is a good approximation of the true posterior.

### Effect of inducing points¶

Another interesting question is how the model performance depends on the number of inducing points. In variational approximations to GPs, the number of inducing points can heavily influence the quality of the approximation, and it would be no surprise if it did so here. However, the dependence of the approximate posterior and the predictive on the inducing points is less clear for GIP BNNs than it is for GPs. Here we vary the number of inducing points from a handful of points up to one inducing point per datapoint.

num_inducing_increment = 10
min_inducing_points = 10
max_inducing_points = 100
prior_scale_factor = 2.
learning_rate = 1e-2
num_steps = int(1e4)
num_eval_reps = 10
num_iwbo_samples = 1000

nums_inducing_points = np.arange(
min_inducing_points,
max_inducing_points+1,
num_inducing_increment
)

nums_inducing_points = np.concatenate(
[[1, 2, 5], nums_inducing_points]
)

# Set progress bar and suppress warnings
progress_bar = tqdm(list(enumerate(nums_inducing_points)))

# List for holding all experimental results
inducing_results = []

for i, num_inducing in progress_bar:

experiment_results = train_model(
prior_scale_factor=prior_scale_factor,
num_inducing=num_inducing,
learning_rate=learning_rate,
num_steps=num_steps,
num_eval_reps=num_eval_reps,
num_iwbo_samples=num_iwbo_samples
)

progress_bar.set_description(
f"Experiment {i+1:02} # inducing {num_inducing:.3f}"
)

inducing_results.append(experiment_results)

elbo_means = np.array([result["elbo_mean"] for result in inducing_results])
elbo_stds = np.array([result["elbo_std"] for result in inducing_results])

iwbo_means = np.array([result["iwbo_mean"] for result in inducing_results])
iwbo_stds = np.array([result["iwbo_std"] for result in inducing_results])

# Figure to plot on
plt.figure(figsize=(11, 3))

# Plot ELBO and error bars
plt.plot(
nums_inducing_points,
elbo_means,
color="grey",
marker="^",
label="ELBO"
)

plt.fill_between(
nums_inducing_points,
elbo_means - 2.*elbo_stds,
elbo_means + 2.*elbo_stds,
color="grey",
alpha=0.3
)

# Plot IWBO and error bars
plt.plot(
nums_inducing_points, iwbo_means,
color="pink",
marker="^",
label="IWBO"
)

plt.fill_between(
nums_inducing_points,
iwbo_means - 2.*iwbo_stds,
iwbo_means + 2.*iwbo_stds,
color="pink",
alpha=0.3
)

# Format figure
plt.xlim([nums_inducing_points[0], nums_inducing_points[-1]])
plt.ylim([-110, 110])

plt.xticks(nums_inducing_points[3:], fontsize=18)
plt.yticks(np.linspace(-100, 100, 5), fontsize=18)

plt.xlabel("# inducing points", fontsize=20)
plt.ylabel("ELBO and IWBO", fontsize=20)

plt.legend(loc="lower right", fontsize=16)

plt.show()


With very few inducing points, the model does not perform well at all. As we increase the number of inducing points, the model performance increases accordingly, until it reaches a plateau, after which it does not improve further.

## Conclusion¶

The GIP method is a variational method for approximating the posterior over weights of BNNs. Ober and Aitchison also show that GIP posteriors are also applicable to deep Gaussian processes (DGPs), though we have not looked at this here. From the experiments done here, we have a few takeaways. GIP posteriors can represent in-between uncertainty between different clusters of data, overcoming a common pathology of mean-field VI. They are also relatively easy to train, requiring no particular initialisation or training tricks, which are used in other methods. Further, GIP posteriors seem to be fairly good approximations of the ground truth predictive posteriors which they are approximating. This was evidenced by the small gap between the ELBO and the IWBO in the experiments above. We also saw that the prior scale has a significant effect on the model fit, as is reasonable to expect from any Bayesian model. Last, we saw that GIP posteriors are fairly robust to the number of inducing points used, performing fairly well with even a relatively small number on inducing inputs. Ober and Aitchison include more extensive evaluations of GIP and competing methods on UCI datasets, for both BNNs and DGPs. They have also followed up GIP with subsequent work on deep kernel processes [AYO21] and deep Wishart processes.[OA21b] Overall, this reproduction shows that GIP is a fairly easy method to work with, and gives fairly good results in the context of BNNs.

## References¶

AYO21

Laurence Aitchison, Adam X. Yang, and Sebastian W. Ober. Deep kernel processes. 2021. arXiv:2010.01590.

BCKW15(1,2)

Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International Conference on Machine Learning. PMLR, 2015.

Bui21

Thang D Bui. Biases in variational bayesian neural networks. Bayesian Deep Learning workshop, 2021.

FLHLT19(1,2)

Andrew YK Foong, Yingzhen Li, Jose Miguel Hernandez-Lobato, and Richard E Turner. In-between uncertainty in bayesian neural networks. arXiv preprint arXiv:1906.11537, 2019.

Nea01

Radford M Neal. Annealed importance sampling. Statistics and computing, 2001.

OA21a(1,2)

Sebastian W Ober and Laurence Aitchison. Global inducing point variational posteriors for bayesian neural networks and deep gaussian processes. In International Conference on Machine Learning, 8248–8259. PMLR, 2021.

OA21b

Sebastian W. Ober and Laurence Aitchison. A variational approximate posterior for the deep wishart process. 2021. arXiv:2107.10125.

SD17

Hugh Salimbeni and Marc Deisenroth. Doubly stochastic variational inference for deep gaussian processes. 2017. arXiv:1705.08933.

TS11

R. E. Turner and M. Sahani. Two problems with variational expectation maximisation for time-series models. In Bayesian Time series models. Cambridge University Press, 2011.