Sparse Function-space Representation of Neural Networks
Deep neural networks (NNs) are known to lack uncertainty estimates and
struggle to incorporate new data. We present a method that mitigates these
issues by converting NNs from weight space to function space, via a dual
parameterization. Importantly, the dual parameterization enables us to
formulate a sparse representation that captures information from the entire
data set. This offers a compact and principled way of capturing uncertainty and
enables us to incorporate new data without retraining whilst retaining
predictive performance. We provide proof-of-concept demonstrations with the
proposed approach for quantifying uncertainty in supervised learning on UCI
benchmark tasks.
Aidan Scannell, Riccardo Mereu, Paul Chang, Ella Tamir, Joni Pajarinen, Arno Solin
# Sparse Function-space Representation of Neural Networks

###### Abstract

Deep neural networks (NNs) are known to lack uncertainty estimates and struggle to incorporate new data. We present a method that mitigates these issues by converting NNs from weight space to function space, via a dual parameterization. Importantly, the dual parameterization enables us to formulate a sparse representation that captures information from the entire data set. This offers a compact and principled way of capturing uncertainty and enables us to incorporate new data without retraining whilst maintaining predictive performance. We provide proof-of-concept demonstrations with the proposed approach for quantifying uncertainty in supervised learning on UCI benchmark tasks.

Machine Learning, ICML 2023 Workshop on Duality for Modern Machine Learning, Honolulu, Hawaii, USA. Copyright 2023 by the author(s).

Machine Learning, ICML 2023 Workshop on Duality for Modern Machine Learning, Honolulu, Hawaii, USA. Copyright 2023 by the author(s).

## 1 Introduction

Deep learning (Goodfellow et al., 2016) has become the cornerstone of contemporary artificial intelligence, proving remarkably effective in tackling supervised and unsupervised learning tasks in the _large data_, _offline_, and _gradient-based training_ regime. Despite its success, gradient-based learning techniques exhibit limitations. Firstly, how can we efficiently quantify uncertainty without resorting to expensive and hard-to-interpret sampling in the model's weight space? Secondly, how to update the weights of an already trained model with new batches of data without compromising the performance on past data? These questions become central when used for sequential learning, such as continual learning (CL, Parisi et al., 2019; De Lange et al., 2021), Bayesian optimization (BO, Garnett, 2023) and reinforcement learning (RL, Sutton and Barto, 2018).

Recent techniques (_e.g._, Ritter et al., 2018; Khan et al., 2019; Daxberger et al., 2021; Fortuin et al., 2021; Immer et al., 2021a) apply a Laplace generalized Gauss-Newton (GGN) approximation to convert trained NNs into Bayesian neural networks (BNNs), that can provide uncertainty without sacrificing additional resources to training (Foong et al., 2019). Furthermore, the resultant weight-space posterior can be converted to function space as shown in Khan et al. (2019); Immer et al. (2021b). The function-space representation allows for a principled mathematical approach for analyzing its behaviour (Cho and Saul, 2009; Meronon et al., 2020), performing probabilistic inference (Khan et al., 2019), and quantifying uncertainty (Foong et al., 2019). These methods rely on linearizing the NN and the resultant neural tangent kernel (NTK, Jacot et al., 2018). The NN is characterized in function space by its first two moment functions, a mean function and covariance function (or kernel)--defining a Gaussian process (GP, Rasmussen and Williams, 2006). GPs provide a widely used probabilistic toolkit with principled uncertainty estimates.

Given an approximate inference technique, we demonstrate that NNs emit 'dual' parameters which are the building blocks of a GP (Csato and Opper, 2002; Adam et al., 2021; Chang et al., 2023). In contrast to previous work that utilizes subsets of training data (Immer et al., 2021a), our parameterization captures information from _all_ data points in a sparse representation, essential for scaling to deep learning data sets. Importantly, the dual parameterization can be used to _(i)_ sparsify the GP without requiring further optimization (_e.g._, variational inference) whilst capturing information from all data points, and _(ii)_ incorporate new data without retraining by conditioning on new data, in a computationally efficient manner. The dual parameterization establishes a connection between

Figure 1: **Regression example on an MLP with two hidden layers.** Left: Predictions from the trained neural network. Right: Our approach (sfr) equips trained NNs with uncertainty estimates.
NNs, GPs, and sparse approximations similar to sparse variational GPs (Titsias, 2009). We refer to our method as Sparse Function-space Representation (sfr)--a sparse GP derived from a trained NN; see Fig. 1 for an example.

## 2 Background

We consider supervised learning with inputs \(\mathbf{x}_{i}\in\mathbb{R}^{D}\) and outputs \(\mathbf{y}_{i}\in\mathbb{R}^{C}\) (_e.g._, regression) or \(\mathbf{y}_{i}\in\{0,1\}^{C}\) (_e.g._, classification), giving a data set \(\mathcal{D}=\{(\mathbf{x}_{i},\mathbf{y}_{i})\}_{i=1}^{N}\). We introduce a NN \(f_{\mathbf{w}}:\mathbb{R}^{D}\to\mathbb{R}^{C}\) with weights \(\mathbf{w}\in\mathbb{R}^{P}\) and use a likelihood function \(p(\mathbf{y}_{i}\,|\,f_{\mathbf{w}}(\mathbf{x}_{i}))\) to link the function values to the output \(\mathbf{y}_{i}\) (_e.g._, categorical for classification). For notational conciseness, we stick to scalar outputs \(y_{i}\). We denote the sets of all inputs as \(\mathbf{X}=\{\mathbf{x}_{i}\}_{i=1}^{N}\) and the set of all outputs \(\mathbf{y}=\{y_{i}\}_{i=1}^{N}\).

**BNNs** In Bayesian deep learning, we place a prior over the weights \(p(\mathbf{w})\) and aim to calculate the posterior over the weights given the data \(p(\mathbf{w}\,|\,\mathcal{D})\). Given the weight posterior \(p(\mathbf{w}\,|\,\mathcal{D})\), we can make probabilistic predictions with

\[p_{\text{BNN}}(y_{i}\,|\,\mathbf{x}_{i},\mathcal{D})=\mathbb{E}_{p(\mathbf{w} \,|\,\mathcal{D})}\left[p(y_{i}\,|\,f_{\mathbf{w}}(\mathbf{x}_{i}))\right]. \tag{1}\]

The posterior \(p(\mathbf{w}\,|\,\mathcal{D})\propto p(\mathbf{y}\,|\,f_{\mathbf{w}}(\mathbf{ X}))\,p(\mathbf{w})\) is generally not available in closed form so we resort to approximations.

**MAP** It is common to train NN weights \(\mathbf{w}\) to minimize the (regularized) empirical risk,

\[\mathbf{w}^{*} =\arg\min_{\mathbf{w}}\mathcal{L}(\mathcal{D},\mathbf{w})\] \[=\arg\min_{\mathbf{w}}\sum_{i=1}^{N}\frac{\ell(f_{\mathbf{w}}( \mathbf{x}_{i}),y_{i})}{-\log p(y_{i}\,|\,f_{\mathbf{w}}(\mathbf{x}_{i}))}+ \frac{\delta\mathcal{R}(\mathbf{w})}{-\log p(\mathbf{w})}. \tag{2}\]

This objective corresponds to the log-joint distribution \(\mathcal{L}(\mathcal{D},\mathbf{w})=p(\mathcal{D},\mathbf{w})\) as the loss can be interpreted as a negative log-likelihood \(\ell(f_{\mathbf{w}}(\mathbf{x}_{i}),y_{i})=-\log(p(y_{i}\,|\,f_{\mathbf{w}}( \mathbf{x}_{i}))\) and the regularizer corresponds to a log prior over the weights \(\delta\mathcal{R}(\mathbf{w})=-\log p(\mathbf{w})\). For example, a weight decay regularizer \(\mathcal{R}(\mathbf{w})=\frac{1}{2}\|\mathbf{w}\|_{2}^{2}\) corresponds to a Gaussian prior over the weights \(p(\mathbf{w})=\mathcal{N}(\mathbf{0},\delta^{-1}\mathbf{I})\), with prior precision \(\delta\). As such, we can view Eq. (2) as the maximum _a posteriori_ (MAP) solution.

**Laplace approximation** The Laplace approximation MacKay (1992); Daxberger et al. (2021) builds upon this and approximates the weight posterior \(p(\mathbf{w}\,|\,\mathcal{D})\approx q(\mathbf{w})=\mathcal{N}(\mathbf{w}^{*},\mathbf{\Sigma})\) around the MAP weights (\(\mathbf{w}^{*}\)) by setting the covariance to the Hessian of the posterior,

\[\mathbf{\Sigma}=-\left[\nabla_{\mathbf{w}\mathbf{w}}^{2}\log p(\mathbf{w}\,| \,\mathcal{D})|_{\mathbf{w}=\mathbf{w}^{*}}\right]^{-1}. \tag{3}\]

Computing Eq. (3) requires calculating the Hessian of the log-likelihood \(\nabla_{\mathbf{w}\mathbf{w}}^{2}\log p(\mathbf{y}\,|\,f_{\mathbf{w}}( \mathbf{X}))\) from Eq. (2). As highlighted in Immer et al. (2021), computing this Hessian is often infeasible and in practice it is common to adopt the generalized Gauss-Newton (GGN) approximation,

\[\nabla_{\mathbf{w}\mathbf{w}}^{2}\log p(\mathbf{y}\,|\,f_{\mathbf{w}}(\mathbf{ X}))\approx\mathcal{J}_{\mathbf{w}}(\mathbf{X})^{\top}\nabla_{\mathbf{if}}^{2} \log(\mathbf{y}\,|\,\mathbf{f})\mathcal{J}_{\mathbf{w}}(\mathbf{X}).\]

where the Jacobian is given by \(\mathcal{J}_{\mathbf{w}}(\mathbf{x})\coloneqq\left[\nabla_{\mathbf{w}}f_{ \mathbf{w}}(\mathbf{x})\right]^{\top}\in\mathbb{R}^{C\times P}\). Immer et al. (2021) highlighted that the GGN approximation corresponds to a local linearization of the NN, \(f_{\mathbf{w}^{*}}^{\text{lin}}(\mathbf{x})=f_{\mathbf{w}}(\mathbf{x})+ \mathcal{J}_{\mathbf{w}^{*}}(\mathbf{x})(\mathbf{w}-\mathbf{w}^{*})\). This suggests that predictions should be made with a generalized linear model (GLM), given by

\[p_{\text{GLM}}(y_{i}\,|\,\mathbf{x}_{i},\mathcal{D})=\mathbb{E}_{q(\mathbf{w})} \left[p(y_{i}\,|\,f_{\mathbf{w}^{*}}^{\text{lin}}(\mathbf{x}_{i}))\right]. \tag{4}\]

**Gaussian processes** As Gaussian distributions remain tractable under linear transformations, we can convert the linear model from weight space to function space (see Ch. 2.1 in Rasmussen and Williams, 2006). As such (and shown in Immer et al. (2021)), the Bayesian GLM in Eq. (4) has an equivalent GP formulation,

\[p_{\text{GP}}(y_{i}\,|\,\mathbf{x}_{i},\mathcal{D})=\mathbb{E}_{q(f_{i})} \left[p(y_{i}\,|\,f_{i})\right], \tag{5}\]

where \(q(f_{i})=\mathcal{N}\left(f_{\mathbf{w}^{*}}(\mathbf{x}_{i}),\sigma_{i}^{2}\right)\) is the GP's predictive posterior with variance,

\[\sigma_{i}^{2}=k_{ii}-\mathbf{k}_{\mathbf{xi}}^{\top}(\mathbf{K}_{\mathbf{x} \mathbf{x}}+\mathbf{\Lambda}^{-1})^{-1}\mathbf{k}_{\mathbf{xi}}, \tag{6}\]

where the kernel \(\kappa(\mathbf{x},\mathbf{x}^{\prime})=\frac{1}{\delta}\mathcal{J}_{\mathbf{w} ^{*}}(\mathbf{x})\,\mathcal{J}_{\mathbf{w}^{*}}^{\top}(\mathbf{x}^{\prime})\) is the Neural Tangent Kernel (NTK, Jacot et al., 2018) and \(f_{i}=f_{\mathbf{w}}(\mathbf{x}_{i})\) is the function output at \(\mathbf{x}_{i}\). The \(ij^{\text{th}}\) entry of matrix \(\mathbf{K}_{\mathbf{x}\mathbf{x}}\in\mathbb{R}^{N\times N}\) is \(\kappa(\mathbf{x}_{i},\mathbf{x}_{j})\), \(\mathbf{k}_{\mathbf{xi}}\) is a vector where each \(j^{\text{th}}\) element is \(\kappa(\mathbf{x}_{i},\mathbf{x}_{j})\), \(k_{ii}=\kappa(\mathbf{x}_{i},\mathbf{x}_{i})\) and \(\mathbf{\Lambda}=-\nabla_{\mathbf{if}}^{2}\log p(\mathbf{y}\,|\,\mathbf{f})\) can be interpreted as per-input noise.

**Sparse GPs** The GP formulation from Immer et al. (2021) (Eq. (5)) requires inverting an \(N\times N\) matrix which has complexity \(O(N^{3})\). This limits its applicability to large data sets, which are common in deep learning. Sparse GPs reduce the computational complexity by representing the GP as a low-rank approximation at a set of inducing inputs \(\mathbf{Z}=\{\mathbf{z}_{j}\}_{j=1}^{M}\in\mathbb{R}^{M\times D}\) with corresponding inducing variables \(\mathbf{u}=f(\mathbf{Z})\) (see Quinonero-Candela and Rasmussen, 2005, for an early overview). The approach by Titsias (2009) (also used in the DTC approximation), defines the marginal predictive distribution as \(q_{\mathbf{u}}(f_{i})=\int p(f_{i}\,|\,\mathbf{u})\,q(\mathbf{u})\,\mathrm{d}\mathbf{u}\) where the variational distribution is parameterized as \(q(\mathbf{u})=\mathcal{N}\left(\mathbf{u}\,|\,\mathbf{m},\mathbf{S}\right)\)(as in Titsias (2009); Hensman et al. (2013)). Assuming a zero mean function, the sparse GP predictive posterior is

\[\mathbb{E}_{q_{\mathbf{u}}(f_{i})}[f_{i}] =\mathbf{k}_{\mathbf{xi}}^{\top}\mathbf{K}_{\mathbf{z}\mathbf{z}}^ {-1}\mathbf{m}, \tag{7a}\] \[\mathrm{Var}_{q_{\mathbf{u}}(f_{i})}[f_{i}] =k_{ii}-\mathbf{k}_{\mathbf{xi}}^{\top}(\mathbf{K}_{\mathbf{z} \mathbf{z}}^{-1}-\mathbf{K}_{\mathbf{z}\mathbf{z}}^{-1}\mathbf{S}\mathbf{K}_{ \mathbf{z}\mathbf{z}}^{-1})\mathbf{k}_{\mathbf{xi}},\]

where \(\mathbf{K}_{\mathbf{z}\mathbf{z}}\) and \(\mathbf{k}_{\mathbf{xi}i}\) are defined similarly to
the variational posterior (\(\mathbf{m}\) and \(\mathbf{S}\)) are usually obtained via variational inference, which requires further optimization.

Not only does the GP formulation from Immer et al. (2021) (shown in Eq. (5)) struggle to scale to large data sets but it also cannot incorporate new data by conditioning on it. This is because its posterior mean is the NN \(f_{\mathbf{w}^{*}}(\mathbf{x})\). Our method overcomes both of these limitations via a dual parameterization. Importantly, our dual parameterization enables us to _(i)_ sparsify the GP without further optimization, and _(ii)_ incorporate new data without retraining.

## 3 Sfr: Sparse function-space representation of NNs

In this section, we present our method, named sfr, which converts a trained NN into a GP. sfr is built upon a dual parameterization of the GP posterior. Early work by Csato & Opper (2002) showed that Expectation Propagation gives rise to a dual parameterization. More recent work by Adam et al. (2021); Chang et al. (2023) showed a different dual parameterization arising from the evidence lower bound for sparse variational GPs (Hensman et al., 2013; Titsias, 2009). The dual parameterization from Adam et al. (2021); Chang et al. (2023), consists of parameters \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\), which gives rise to the predictive posterior,

\[\mathbb{E}_{p(f_{i}\mid\mathbf{y})}[f_{i}] =\mathbf{k}_{\mathbf{xi}}^{\top}\boldsymbol{\alpha}, \tag{8a}\] \[\mathrm{Var}_{p(f_{i}\mid\mathbf{y})}[f_{i}] =k_{ii}-\mathbf{k}_{\mathbf{xi}i}^{\top}(\mathbf{K}_{\mathbf{xx} }+\text{diag}(\boldsymbol{\beta})^{-1})^{-1}\mathbf{k}_{\mathbf{x}i}.\]

Eq. (8) states that the first two moments of the resultant posterior process (which may not be a GP), can be parameterized via the dual parameters \(\boldsymbol{\alpha},\boldsymbol{\beta}\in\mathbb{R}^{N}\), defined as

\[\alpha_{i} \coloneqq\mathbb{E}_{p(\mathbf{w}\mid\mathbf{y})}[\nabla_{f} \log p(y_{i}\,|\,f)|_{f=f_{i}}], \tag{9a}\] \[\beta_{i} \coloneqq-\mathbb{E}_{p(\mathbf{w}\mid\mathbf{y})}[\nabla_{ff} ^{2}\log p(y_{i}\,|\,f_{i})|_{f=f_{i}}]. \tag{9b}\]

Eq. (9) holds for generic likelihoods and involves no approximations since the expectation is under the exact posterior, given that the model can be expressed in a kernel formulation. Eq. (8) and Eq. (9) highlight that the approximate inference technique, usually viewed as a posterior approximation, can alternatively be interpreted as an approximation of the expectation of loss (likelihood) gradients.

Dual parameters from NNGiven that we use a Laplace approximation of the NN, we remove the expectation over the posterior (see Ch. 3.4.1 in Rasmussen & Williams, 2006, for derivation) and our dual parameters are given by

\[\hat{\alpha}_{i} \coloneqq\nabla_{f}\log p(y_{i}\,|\,f)|_{f=f_{i}}, \tag{10a}\] \[\hat{\beta}_{i} \coloneqq-\nabla_{ff}^{2}\log p(y_{i}\,|\,f)|_{f=f_{i}}. \tag{10b}\]

Substituting Eq. (10) into Eq. (8), we obtain our GP based on the trained NN. Making predictions with Eq. (8) costs \(O(N^{3})\), which limits its applicability on large data sets.

Sparsification via dual parametersGiven that we have computed the dual parameters derived from our NN predictions and a kernel function, we could essentially employ any sparsification method (Quinonero-Candela & Rasmussen, 2005). In this work, we opt for the approach suggested by Titsias (2009); Hensman et al. (2013), which is shown in Eq. (7). However, instead of parameterizing the variational distribution as \(q(\mathbf{u})=\mathcal{N}\left(\mathbf{u}\,|\,\mathbf{m},\mathbf{S}\right)\), we follow insights from Adam et al. (2021) that the posterior under this model bears a structure akin to Eq. (8). As such, we project the dual parameters onto the inducing points giving sparse dual parameters. Using this sparse definition of the dual parameters, our sparse GP posterior is given by

\[\mathbb{E}_{q_{\mathbf{u}}(\mathsf{f})}[f_{i}] =\mathbf{k}_{\mathbf{zi}}^{\top}\mathbf{K}_{\mathbf{z}\mathbf{ z}}^{-1}\boldsymbol{\alpha}_{\mathbf{u}}, \tag{11a}\] \[\mathrm{Var}_{q_{\mathbf{u}}(\mathsf{f})}[f_{i}] =k_{ii}-\mathbf{k}_{\mathbf{zi}}^{\top}[\mathbf{K}_{\mathbf{z} \mathbf{z}}^{-1}-(\mathbf{K}_{\mathbf{z}\mathbf{z}}+\boldsymbol{B}_{\mathbf{ u}})^{-1}]\mathbf{k}_{\mathbf{zi}},\]

with sparse dual parameters,

\[\boldsymbol{\alpha}_{\mathbf{u}}=\sum_{i=1}^{N}\mathbf{k}_{\mathbf{zi}}\,\hat {\alpha}_{i}\quad\text{and}\quad\boldsymbol{B}_{\mathbf{u}}=\sum_{i=1}^{N} \mathbf{k}_{\mathbf{zi}}\,\hat{\beta}_{i}\,\mathbf{k}_{\mathbf{zi}}^{\top}. \tag{12}\]

Note that the sparse dual parameters are now a sum over _all data points_, with \(\boldsymbol{\alpha}_{\mathbf{u}}\in\mathbb{R}^{M}\) and \(\boldsymbol{B}_{\mathbf{u}}\in\mathbb{R}^{M\times M}\). Contrasting Eq. (11) and Eq. (8), we can see that the computational complexity went from \(\mathcal{O}(N^{3})\) to \(\mathcal{O}(M^{3})\), with \(M\ll N\). Crucially, given the structure of our probabilistic model, our sparse dual parameters Eq. (12) are a compact representation of the full model projected using the kernel.

We highlight that sfr differs from the GP subset from Immer et al. (2021), which results in several advantages. First, sfr leverages a dual parameterization to construct a sparse GP, which captures information from the entire data set. In contrast, Immer et al. (2021) utilize a data subset which ignores information from the rest of the data set. Second, sfr makes predictions with the GP mean, whereas Immer et al. (2021) center predictions around NN predictions \(f_{\mathbf{w}^{*}}(\cdot)\). As such, sfr can incorporate new data without retraining whilst the GP subset cannot.

To the best of our knowledge, we are the first to formulate a dual GP from a NN. Our dual parameterization has two main benefits: _(i)_ it enables us to construct a sparse GP without requiring us to perform further optimization, and _(ii)_ it enables us to incorporate new data without retraining by conditioning on new data (using Eq. (12)).

## 4 Experiments

We evaluate sfr on eight UCI (Dua & Graff, 2017) binary and multi-class classification tasks. Following Immer et al. (2021), we used a two-layer MLP with width 50, tanh activation functions and a \(70\%\) (train) \(:15\%\) (validation) \(:15\%\) (test) data split. We trained the NN using Adam (Kingma & Ba, 2015) with a
learning rate of \(10^{-4}\) and a batch size of \(128\). Training was stopped when the validation negative log predictive density (NLPD) stopped decreasing after 1000 steps. The checkpoint with the lowest NLPD was the NN MAP. Each experiment was run for \(5\) seeds.

We first compare sfr to the Laplace approximation when making predictions with _(i)_ the nonlinear NN (BNN) and _(ii)_ the generalised linear model (GLM) in Eq. (4). We use the Laplace PyTorch library (Daxberger et al., 2021) with the full Hessian. It is common practice to tune the prior precision \(\delta\) after training the NN. That is, find a prior precision \(\delta_{\text{tune}}\) which has a better NLPD on the validation set than the \(\delta\) used to train the NN. Note that this leads to a different prior precision being used for training and inference. We highlighted that technically this invalidates the Laplace/SFR methods as the NN weights are no longer the MAP weights. Nevertheless, we report results when the prior precision \(\delta\) is tuned (Table 1 right) and is not tuned (Table 1 left). When tuning the prior precision \(\delta\), sfr matches the performance of the Laplace methods (BNN/GLM). Interestingly, sfr also performs well without tuning the prior precision \(\delta\), unlike the Laplace methods.

As sfr captures information from all of the training data in the inducing points, we compare sfr to making GP predictions with a subset of the training data (GP subset). To make the comparison fair, we use sfr's inducing inputs \(\mathbf{z}\) as the subset. Table 1 shows that sfr is able to summarize the full data set more effectively than the GP subset method as it maintains predictive performance whilst using fewer inducing points. This is further illustrated in Fig. 2, which shows that as the number of inducing points is lowered from \(M=100\%\) of \(N\) to \(M=1\%\) of \(N\), sfr is able to maintain a much better NLPD. This demonstrates that sfr captures information from the entire data set and as a result can use fewer inducing points than the GP subset.

## 5 Discussion and conclusion

We introduced sfr, a novel approach for representing NNs in function space. It leverages a dual parameterization to obtain a low-rank (sparse) approximation that captures information from the entire data set. We showcased sfr's uncertainty quantification in UCI benchmark classification tasks (Table 1). Unlike the Laplace approximation, sfr achieves good predictive performance without tuning the prior precision \(\delta\). This is interesting and we believe it stays more true to Bayesian inference.

In practical terms, sfr serves a role similar to a sparse GP. However, unlike a conventional GP, it does not provide a straightforward method for specifying the prior.This limitation can be addressed indirectly: the architecture of the NN and the choice of activation functions can be used to implicitly specify the prior assumptions. It is important to note that sfr linearizes the NN around the MAP weights \(\mathbf{w}^{*}\), resulting in the function-space prior (and posterior) being a locally linear approximation of the NN.

\begin{table}
\begin{tabular}{l c c|c c|c c c|c c c c} \hline  & & & & \multicolumn{4}{c}{No \(\delta\) tuning} & \multicolumn{4}{c}{\(\delta\) tuning} \\  & \(N\) & \(D\) & \(C\) & NN MAP & bnn & GLM & GP subset & sfr & bnn & GLM & GP subset & sfr \\ \hline Australian & 690 & 14 & 2 & **0.35**\(\pm\)0.06 & 0.71\(\pm\)0.03 & 0.43\(\pm\)0.04 & **0.39**\(\pm\)0.03 & **0.35**\(\pm\)0.04 & **0.34**\(\pm\)0.05 & **0.35**\(\pm\)0.05 & 0.41\(\pm\)0.04 & **0.35**\(\pm\)0.04 \\ Breast cancer & 683 & 10 & 2 & **0.09**\(\pm\)0.05 & 0.72\(\pm\)0.06 & 0.47\(\pm\)0.09 & 0.23\(\pm\)0.02 & 0.18\(\pm\)0.02 & **0.09**\(\pm\)0.05 & **0.09**\(\pm\)0.05 & **0.13**\(\pm\)0.03 & **0.08**\(\pm\)0.04 \\ Dorts & 351 & 34 & 2 & **0.07**\(\pm\)0.04 & 2.35\(\pm\)0.01 & 3.11\(\pm\)0.15 & **1.10**\(\pm\)0.02 & **1.07**\(\pm\)0.03 & **0.07**\(\pm\)0.03 & **0.07**\(\pm\)0.04 & 0.16\(\pm\)0.04 & **0.08**\(\pm\)0.03 \\ Glass & 214 & 9 & **1.02**\(\pm\)0.41 & 1.82\(\pm\)0.06 & 1.77\(\pm\)0.07 & 1.14\(\pm\)0.07 & **0.93**\(\pm\)0.08 & **0.87**\(\pm\)0.28 & **0.82**\(\pm\)0.27 & 1.19\(\pm\)0.08 & **0.92**\(\pm\)0.11 \\ Ionospheric & 846 & 18 & 4 & **0.38**\(\pm\)0.05 & 0.70\(\pm\)0.03 & **0.37**\(\pm\)0.04 & **0.48**\(\pm\)0.03 & **0.39**\(\pm\)0.03 & **0.38**\(\pm\)0.05 & **0.37**\(\pm\)0.05 & 0.44\(\pm\)0.03 & **0.39**\(\pm\)0.04 \\ Satellite & 1000 & 21 & 3 & **0.24**\(\pm\)0.02 & 1.83\(\pm\)0.02 & 0.78\(\pm\)0.04 & **0.32**\(\pm\)0.01 & **0.26**\(\pm\)0.02 & **0.24**\(\pm\)0.02 & **0.24**\(\pm\)0.02 & **0.43**\(\pm\)0.05 & 0.31\(\pm\)0.03 \\ Vehicle & 1797 & 64 & 10 & **0.40**\(\pm\)0.06 & 1.40\(\pm\)0.02 & 1.55\(\pm\)0.01 & **0.88**\(\pm\)0.02 & **0.85**\(\pm\)0.04 & **0.38**\(\pm\)0.06 & **0.37**\(\pm\)0.04 & 0.61\(\pm\)0.06 & **0.43**\(\pm\)0.02 \\ Waveform & 6435 & 35 & 6 & 0.40\(\pm\)0.05 & 1.10\(\pm\)0.01 & 1.00\(\pm\)0.02 & 0.44\(\pm\)0.03 & 0.38\(\pm\)0.02 & **0.35**\(\pm\)0.04 & **0.36**\(\pm\)0.03 & **0.36**\(\pm\)0.03 & **0.32**\(\pm\)0.03 \\ \hline \end{tabular}
\end{table}
Table 1: Comparisons on UCI data with negative log predictive density (NLPD\(\pm\)std, lower better). sfr (with \(M=20\%\) of \(N\)) is on par with the Laplace approximation (BNN/GLM) and outperforms the GP subset when the prior precision (\(\delta\)) is tuned (right). Interestingly, when the prior precision (\(\delta\)) is not tuned (left), sfr outperforms all other methods. The GP subset uses the same inducing points as sfr.

Figure 2: Comparison of convergence in terms of number of inducing points \(M\) in NLPD (mean\(\pm\)std over 5 seeds) on UCI classification tasks: sfr () vs. GP subset (). Our sfr converges fast for all cases showing clear benefits of its ability to summarize all the data onto a sparse set of inducing points. The number of inducing points \(M\) are specified as a percentage of the number of training data \(N\).