Activation-level uncertainty in deep neural networks
Authors: Pablo Morales-Alvarez
Daniel Hernández-Lobato
Rafael Molina
José Miguel Hernández-Lobato
Venue: ICLR 2021 Poster (2021)
Keywords: Gaussian Processes
Uncertainty estimation
Deep Gaussian Processes
Bayesian Neural Networks

Abstract: Current approaches for uncertainty estimation in deep learning often produce too confident results. Bayesian Neural Networks (BNNs) model uncertainty in the space of weights, which is usually high-dimensional and limits the quality of variational approximations. The more recent functional BNNs (fBNNs) address this only partially because, although the prior is specified in the space of functions, the posterior approximation is still defined in terms of stochastic weights. In this work we propose to move uncertainty from the weights (which are deterministic) to the activation function. Specifically, the activations are modelled with simple 1D Gaussian Processes (GP), for which a triangular kernel inspired by the ReLu non-linearity is explored. Our experiments show that activation-level stochasticity provides more reliable uncertainty estimates than BNN and fBNN, whereas it performs competitively in standard prediction tasks. We also study the connection with deep GPs, both theoretically and empirically. More precisely, we show that activation-level uncertainty requires fewer inducing points and is better suited for deep architectures.

Published as a conference paper at ICLR 2021
ACTIVATION-LEVEL UNCERTAINTY IN DEEP NEURAL NETWORKS
Pablo Morales-Alvarezz
Daniel Hernández-Lobato
Department of Computer Science and Al University of Granada, Spain
Department of Computer Science
Universidad Autónoma de Madrid, Spain
pablomors r L decsai. egr.es
Rafael Molina
José Miguel Hernández-Lobato Department of Engineering University of Cambridge, UK Alan Turing Institute, London, UK
Department of Computer Science and Al University of Granada, Spain
ABSTRACT
Current approaches for uncertainty estimation in deep learning often produce too confident results. Bayesian Neural Networks (BNNs) model uncertainty in the space of weights, which is usually high-dimensional and limits the quality of variational approximations. The more recent functional BNNs (fBNNs) address this only partially because, although the prior specified in the space of functions, the posterior approximation is still defined in terms of stochastic weights. In this work we propose to move uncertainty from the weights (which are deterministic) to the activation function. Specifically, the activations are modelled with simple ID Gaussian Processes (GP), for which a triangular kernel inspired by the ReLu non-linearity is explored. Our experiments show that activation levell stochasticity provides more reliable uncertainty estimates than BNN and IBNN, whereas it performs competitively in standard prediction tasks. We also study the connection with deep GPs, both theoretically and empirically. More precisely, we show that activation- levell uncertainty requires fewer inducing points and is better suited for deep architectures.
INTRODUCTION
Deep Neural Networks (DNNs) have achieved state-of-the-art art performance many different tasks. such speech recognition (Hinton et al. 2012), natural language processing (Mikolov et al. 2013) or computer vision (Krihevsky et al.) 2012) ). spite of their predictive power, DNNs are limited in terms of uncertainty estimation. This has been classical concern in the field (MacKay 1992 Hinton & Van Camp 1993 Barber & Bishop 1998), which has attracted a lot of attention in the last years (Lakshminarayann)
al.) 2017|Guo Guo et al. 2017 Sun et al. 2019 Wenzel et al. 2020). Indeed, this ability to "know what is not known" is essential for critical applications such as medical diagnosis
[Estva et al. 2017 Mobiny et al. 2019) autonomous driving (Kendall & Gal 2017 Gal 2016).
Bayesian Neural Networks (BNNs) address this problem through a Bayesian treatment of the network weight: (MacKay 1992 Neal 1995). This will be refered to as weight-space stochasticity. However, dealing with uncertainty in weight space is challenging. since it contains many symmetries and is highly dimensional (Wenzel et al. 2020 [un et al. [019 Snoek al. 2019 Fort et al.) 2019). Here we focus on two specific limitations. First, it has been recently shown that BNNs with well-established inference methods such as Bayes by Backprop (BBP) (Blundell al. 2015) and MC-Dropout (Gal &
Ghahramani 2016) underestimate the predictive uncertainty for instances located nn-between two clusters of training points (Foong et al. 2020 2019 Yao et al. 2019). Second, the weight- prior does not allow BNNs to guide extrapolation to out-of -iito--iittibuttin (OOD) data (Sun et al. 2019
Nguyen et al. 2015 Ren et al. 2019). Both aspects are illustrated graphically Figure
more
details in Section 3.1
"Wor developed mostly while visiting Cambridge University, UK.
'The hias term will be absorbed within the weights throughout the work.

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Published as a conference paper at ICLR 2021
Déterninistic
Stochastic
w
wi
w'a
1-dim
D ddim
a) NN
bJ BNN
c) auNN
d) DGP
Figure I: Graphical representation of the artificial neurons for closely related methods. The subscript d and the superscript refer to the d-th unit in the (-th layer, respectively. (a) In standard Neural Networks (NN), both the weights and the activation function are deterministic. (b) In Bayesian (c) In auNN (this work), weights are deterministic and the activation is stochastic. (d) Deep GPs do not have a linear projection through weights, and the output is modelled directly with GP defined on the D dime-diinnso input space.
NNs, weights are stochastic and the activation is deterministic.
As an alternative to standard BNNs, Functional Bayesian Neural Nets (fBNN) specify the prior and perform inference directly in function space (Sun et al. 2019). This provides a mechanism to guide the extrapolation in OOD data, e-g. predictions no observed data. However, the posterior stochastic process is still defined by factorized Gaussian on the network weights (i.e. as in BBP), see (Sun al. 2019 Sect. 3.1). We will show that this makes fBNN inherit the problem of underestimating the predictive uncertainty for in-between data.
be encouraged to revert the prior in regions of
In this work, we adopt a different approach by moving stochasticity from the weights the activation function, see Figure [1] This will be referred to as auNN (activation-leww uncertainty for Neural Networks). The activation functions are modelled with (ine-dimensiona) GP priors, for which a triangular kernel inspired by the ReLu non-linearity (Nair & Hinton 2010: Glorot et al. (01T) is used. Since non- con-lincarities are typically simple functions (c.g. ReLu, sigmoid, tanb), our GPs are sparsified with few inducing points. The network weights are deterministic parameters which are estimated to maximize the marginal likelihood of the model. The motivation behind auNN is to avoid inference in the complex space of weights. We hypothesise that it could be enough introduce stochasticity in the activation functions that follow the linear projections to provide sensible uncertainty estimations.
We show that auNN obtains well-calibrated estimations for in in-between data, and its prior allows to guide the extrapolation to OOD data by reverting to the empirical mean. This will be visualized in simple 1D example (Figure 3 and Table Tp. Moreover, auNN obtains competitive performance in standard benchmarks. is scalable (datasets of up to ten millions training points are used), and can be readily used for classification. The use of GPs for the activations establishes an interesting connection with deep GPs (DGPs) (Damianou & Lawrence] 2013 Salimbeni & Deisenroth 2017) The main difference is the linear projection before the GP, recall Figure This allows auNN units to model simpler mappings between layers, which are defined along one direction of the input space, similarly to neural networks. However, DGP units model more complex mappings defined on the whole input space, see also Figure 2a. We will show that auNN units require fewer inducing points and are better suited for deep architectures, achieving superior performance. Also, a thorough discussion on additional related work will be provided in Section
In summary, the main contributions of this paper are: (1) a new approach to model uncertainty in DNNs. based deterministic weights and simple stochastic non-linearisies (in principle, not necessarily modelled by GPs); (2) the specific use of non-maraett GPs as a prior, including the triangular kernel inspired by the ReLu; (3) auNN addresses well-known limitation of BNNs and IBNNs (uncertainty underestimation for in-between data), can guide the extrapolation to OOD data by reverting to the empirical mean, and competitive in standard prediction tasks; (4) auNN units require fewer inducing points and are better suited for deep architectures than DGP ones, achieving superior performance.
2 PROBABILISTIC MODEL AND INFERENCE
Model specification. We focus on a supervised task (e.g. regression or classification) with training data (x,,> y yr I The graphical model in Figure 2 will be useful throughout this section. We
"The output represented vector since all the derivations apply for the multi-output case.
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Published as a conference paper at ICLR 2021
Plosted value:
Plotzed value: f
Train
Test
Train
F
A
W
Z'
U
A1 DGP
f
BUNN
(a)
(C)
Figure (a) Type of mappings modelled by DGP and auNN units (colours represent different values). Whereas DGP units describe complex functions defined on the whole D' dimensional input space, the linear projection through y in auNN yields simpler functions defined on just one direction. This is closer in spirit to NNs, requires fewer inducing points, and is better suited for deep architectures. The inducing points are shown in black (for auNN, these correspond to (hyper)planes in the input space before the projection). (b) Probabilistic graphical model for auNN layer. Yellow variables are to be estimated (light ones through point estimates and the dark one through a posterior distribution). The box highlights the auxiliary variables (inducing points and their values). (c) Graphical representation of the UCI gap splits. red, segment that crosses the gap joining two training points from different components, which will be used in the experiments.
assume model of L layers, each one with D' units in Figure Dr. Each activation is modelled with a (1D) GP prior, i.e. fa((aa) ~ GP k.). with p : R + R and ky IR x R - R. The GP hyperparameters 0'2 will be omitted for clarity (for the kernels used here, 022 includes the amplitude and the lengthscale). Assuming independence between units, each layer depends on the previous one as:
D'F
I2 P(Egla))
where Fl is the N x D' matrix of outputs of the (-th layer for N inputs, w is the D' x D' matrix of weights in that layer, and A is the N x D' matrix of pre-activations, i.e. = W As usual, the columns and rows of F' are denoted as f and Kh respectively (and analogously for the
= other matrices). Since the activation is defined by GP, we have = (resp. KC) the result of evaluating n1 (resp. k2y on a' (that is, /1, is N-dimensional vector and K a N x N matrix). To fully specify the model, the output Y is defined from the last layer with - distribution that factorizes across data points, i.e. p( ( I p(yn.. (.A.). This formulation resembles that of DGPs (Damianou & Lawrence) 2013 Salimbeni Deisenroth (017). The main
K'y), with u
difference is that we model F' r' through D' ID GPs evaluated on the pre- are-activations A' (i.e. the
projections of F through W whereas DGPs use D' GPs of dimension D evaluated directly on Fi-1, recall Figure 1 c-dd..
Variational Inference. Inference in the proposed model is intractable. To address this, we follow standard sparse variational GP approaches (Titsiss 2009 Hensman et al.) [013]] 2013). similarly to the Doubly Stochastic Variational Inference (DSVI) for DGPs (Salimbeni Specifically, in each unit of each layer we introduce M inducing values u'z which are the result of evaluating the GP on the one-dimensional inducing points z/z We naturally write U² and Z for the corresponding M x D' matrices associated to the 1-th layer, respectively. Following eq. (T. the augmented model for one layer
Deisenroth] 2017).
p(F', U' W Z') = JU' A', z' (U''Z') Th' PCClll, a 2 2))p(1))))) (2)
Variational inference (VT) involves the approximation of the true posterior p( (F', U' Y). Following (Hensman et al.) 2013 Salimbeni & Deisenroth 2017). we propose posterior given by p(F and a parametrie Gaussian on U:
q(1F'. u' I' P(F' U' A z (')' IT' I. ((E) u/, a/,, ())))
(3)
where q(u()) = Vu' m/y, Sy),, with m M and S E Mr xM' variational parameters to be estimated. Minimizing the KL divergence between q((F', U'})) and the true posterior is equivalent maximizing the following evidence lower bound (ELBO):
N
L D
log pl Y (W', Z'))
ELBO
E Eairz logplyn.. 24.
ΣΣ KL (q(u))ll(u))) (4)
==1
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In the ELBO, the KL term can be computed in closed-form, as both q(u/) and p(u)))
Gaussians. The log likelihood term can be approximated by sampling from the marginal pos- terior q(( ) which can be done efficiently through univariate Gaussians in (Salimbeni &
Deisenroth| 2017). Specifically, U can be analytically marginalized in eq. (3) which yields q((F'))) II q(' F'-1, W aI ((( Pt's En)), with:
L (la(ataa) + + (aa(a (m') pl(()))),
(5)
Edd -k dzu) ( sy)al(d(a)
(6)
where (2)(x) = (x, (x, lk) and an. = w Importantly, the marginal posterior q ((
is a Gaussian that depends only on a/".. which in turn only depends q(C,."). Therefore, sampling from f. is straightforward using the reparametrization trick (Kingma Welling 2013):
fus (i + e [ E2
1/2 with (O,,)), and C = Xn,:-
Training consists in maximizing the ELBO, eq. w.r.f. variational parameters {m/, S.), inducing points {z']- and model parameters (i.e. weights {w'}} and kernel parameters {0²}). This can be done in batches, allowing for scalability to very large datasets. The complexity evaluate the ELBO is ONNM? +... D))),, the same as DGPs with DSVI (Salimbeni & Deisenroth 2017
As in (Salimbeni & Deisenroth| 2017). ). this can be approximated by sampling S values up to the (L- - 1)
Predictions. Given a new X..:> we want to compute [(f X, S,Y)) = (u((U))) (p(ff,((''
layer with the same eq. (2). but starting with x.... Then, p(f!: [X, Y) is given by the mixture of the S Gaussians distributions obtained from eqs. 5
Triangular kernel. One of the most popular kemels in GPs is the RBF (Williams & Rasmussen
2006). which produces very smooth functions. However, the ReLu non-linearity led to a general boost in performance in DNNs (Nair & Hinton| 2010 Glorot et al., 2011), 1), and aim to model similar activations. Therefore, we introduce the use of the triangular (TRI) kemne. Just like RBF, TRI is an isotropic kernel, i.e. it depends on the distance between the inputs, k(r, y) - 9[[ [2:
yl/ with y and l the amplitude and lengthscale. For RBF, g() = e 1/2 For TRI, g() = max - t,, 0). This is valid kernel (Williams & Rasmussen 2006, Section 4.2.1). Similarly to the ReLu, the functions modelled by TRI are piecewise linear, see Figure O in the main text and Figure [ in Appendix [
Comparison with DGP. The difference between auNN and DGP units is graphically illustrated in Figure 2p. Whereas DGP mappings from one layer to the next are complex functions defined on DJ-1 dimensions (Di-1 = 2 in the figure), auNN mappings are defined just along one direction via the weight projection. This is closer in spirit to NNs, whose mappings are also simpler and better suited for feature extraction and learning more abstract concepts. Moreover, since the GP is defined on a ID space, auNN requires fewer inducing points than DGP (which, intuitively, can be interpreted as inducing (hyper) in the -d-mennnnso space before the projection).
EXPERIMENTS
In this section, auNN is compared to BNN, IBNN (Sun al. 2019) and DSVI DGP (Salimbeni & Deisenroth 2017). BNNs are trained with BBP Blundell et al. 2015), since auNN also leverages simple VI-based inference approach. In each section we will highlight the most relevant experimental aspects, and all the details can be found in Appendix B In the sequel, NLL stands for Negative Log
Likelihood. Anonymized code for auNN is provided in the supplementary material, along with a script to run it for the ID illustrative example of Section 3.1
3.1 AN ILLUSTRATIVE EXAMPLE
Here we illustrate the two aspects that were highlighted in the introduction: the underestimation of predictive uncertainty for instances located in-between two clusters of training points and the
As in Salimbeni & Deisenroth 2017). there exists also cubic term ((M +... D= that is dominated by the former (since the batch size N is typically larger than M). Moreover, in auNN we have the multiplication by weights, with complexity O(ND' D') for each layer. This also dominated by the former. "The distribution p(y_.IX Y) is obtained as the expectation of the likelihood over p(f!. [X. Y). A Gaussian likelihood used for regression, and the Robust-Max Soemandez--Lohate et al. 201T for classification.
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Table Visual overview of con- clusions from the ID experiment in Figure This shows that NN, BNN, IBNN and auNN increas- ingly expand their capabilities.
1-
0
M
BNN
aukN RBF
aUNN
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
0.0
2.5
5.0
-5.0
-2.5
0.0
2.5
5.0
Figure 3: Predictive distribution (mean and one standard devia- tion) after training on a ID dataset with two clusters of points. This simple example illustrates the main limitations of NN, BNN and IBNN, which are overcome by the novel auNN. See Table for summary and the text for details.
Epistemic uncertainty
Reverts to the mean
In betweween uncertainty
NN
x
X x
BNN
x
BNN
/
X
auNN
/
extrapolation to OOD data. Figure 3 shows the predictive distribution of NN, BNN, fBNN and auNN (with RBF and TRI kernels) after training on a simple 1D dataset with two clusters of points. All the methods have one hidden layer with 25 units, and 5 inducing points are used for auNN
In Figure 3 the deterministic nature of NNs prevents them from providing epistemic uncertainty (i.e. the one originating from the model (Kendall Gal 2017)). Moreover, there is no prior to guide the extrapolation to OOD data. BNNs provide epistemic uncertainty. However, the prior in the complex space of weights does not allow for guiding the extrapolation to OOD data (e.g. by reverting to the empirical mean). Moreover, note that BNNs underestimate the predictive uncertainty in the region between the two clusters, where there is no observed data (this region is usually called the gap). More specifically, as shown in (Foong et al., 2020), the predictive uncertainty for data points in the gap is limited by that on the extremes. By specifying the prior in function space, fBNN can induce properties in the output, such reverting the empirical mean for OOD data through a zero-mean GP prior. However, the underestimation of in-between uncertainty persists, since the posterior stochastic process for IBNN is based on a weight-space factorized Gaussian (as BNN with BBP), see (Sun et al. 2019 Section 3) for details. Finally, auNN (either with RBF or TRI kernel) addresses both aspects through the novel activation-level modelling of uncertainty, which utilizes zero-mean GP prior for the activations. Table [ summarizes the main characteristics of each method. Next, a more comprehensive experiment with deeper architectures and more complex multidimensional datasets is provided.
3.2 UCI REGRESSION DATASETS WITH GAP SPLITS
Standard splits are not appropriate to evaluate the quality of uncertainty estimates for in in-between data, since both train and test sets may cover the space equally. This motivated the introduction of gap splits (Foong et al. 2019). Namely, a set with D dimensions admits D such train-test partitions by considering each dimension, sorting the points according to its value, and selecting the middle 1/3 for test (and the outer 2/3 for training), see Figure P. With these partitions, overconfident predictions for data points in the gap manifest as very high values of test negative log likelihood.
Using the gap splits, it was recently shown that BNNs yield overconfident predictions for in-between data (Foong et al. 2019). The authors highlight the case of Energy and Naval datasets, where BNNs fail catastrophically. Figure 4 reproduces these results for BNNs and checks that IBNNs also obtain overcontident predictions, as theoretically expected. However, notice that activation-level stochasticity performs better, specially through the triangular kernel, which dramatically improves the results (see the plot scale). Figure44 confirms that the difference is due to the underestimation of uncertainty, since the predictive performance in terms of RMSE is on a similar scale for all the methods. In all cases, D 50 hidden units are used, and auNN uses M = 10 inducing points.
To further understand the intuition behind the different results, Figure [ shows the predictive distribu- tion over a segment that crosses the gap. recall Figure We observe that activation- approaches obtain more sensitive (less confident) uncertainties in the gap, where there is no observed data. For instance, BNN and IBNN predictions in Naval are unjustifiably overconfident, since the output in that dataset ranges from 0.95 to 1. Also, to illustrate the internal mechanism of auNN, Figure 6 shows one example of the activations learned when using each kernel. Although it is just one example, it allows for visualising the different nature: smoother for RBF and piccewise linear for TRI. All the activations for particular network and for both kernels are shown in Appendix C (Figure 8)
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Energy
Naval
Energy
Naval
BNN 3
BNN 3
1B884 2
1BN4-2
3UMN-88
BUNN-RRFF-
9UMN-RBF
TRI22-R1122
JUKI TRL-2
1800 2000
0.03
(a)
(b)
Figure 4: Test NLL (a) and RMSE (b) for the gap splits in Energy and Naval datasets (mean and one standard error; the lower the better). Activation-level uncertainty, specially through the triangular kernel, avoids the dramatic failure of BNN and IBNN in terms of NLL (see the scale). The similar values in RMSE reveal that this failure actually comes from an extremely overconfident estimation by BNN and IBNN, see also Figure
BMN-2
SNI-3
TBAN-2
ISNN-3
W.NN-ABBF2
RBF-
aUNI--TR-2
24 SUNN-TRR-3
350
520
2.05
3.01
2.02
2.02
550 dm
onp dm
250 ch
540 dm
Cip dm
240 dmm
Ch0 dm
CP0 dm
Figure 5: Predictive distribution (mean and one standard deviation) over segment that crosses the gap, joining two training points from different connected components. auNN avoids overconfident predictions by allocating more uncertainty in the gap, where there is no observed data.
In addition to the paradigmatic cases of Energy and Naval illustrated here, four more datasets are included Appendix [ Figure there analogous Figure 4 here, and Tables 4 and there the full numeric results and ranks. We observe that auNN, specially through the triangular kernel, obtains the best results and does not fail catastrophically in any dataset (unlike BNN and fBNN, which do in Energy and Naval). Finally, the performance on the gap splits is complemented by that on standard splits, see Tables 6 in Appendix C This shows that, in addition to the enhanced uncertainty estimation, auNN is a competitive alternative in general practice.
there show
3.3 COMPARISON WITH DGPS
As explained Section 2 the choice of a GP prior for activation stochasticity establishes a strong connection with DGPs. The main difference that auNN performs linear projection from D to D' dimensions before applying D ID GPs, whereas DGPs define D' GPs directly on the D dimensional space. This means that auNN units are simpler than those of DGP, recall Figure we show two practical implications of this.
Here
First, it reasonable to hypothesise that DGP units may require a higher number inducing points M than auNN, since they need to cover a multi-dimensional input space. By contrast, auNN may require a higher number of hidden units D, since these are simpler. Importantly, the computational cost is not symmetric in M and D. but significantly cheaper on D, recall Section 2 Figure 6p shows the performance
auNN and DGP for different values of M and D on the UCI Kin8 set (with one hidden layer; depth will be analyzed next). As expected, note the differentt influence by M and D: whereas auNN improves "by rows" (i.e. as D grows), DGP does it "by columns" (i.e. as M grows Next section (Section 3.4) will show that this makes auNN faster than DGP in practice. An analogous figure for RMSE and full numeric results are in Appendix (Figure and Tables p I0
Second, auNN simpler units might be better suited for deeper architectures. Figure 6C shows the performance on the UCI Power dataset when depth is additionally considered. It can be observed that auNN is able to take greater advantage of depth, which translates into better overall performance.
'Interestingly, the fact that DGP is not greatly influenced by D could be appreciated in its recommended value in the original work (Salimbeni & Deisenroth 2017). They set D min(30. D?), where 5 is the input dimension. This limits D maximum value of 30.

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aUNN-RBF
auNN-TRI
DGP
JUNN TRI
2,875
10
25
2.850
0
2.825
10
-5.0
-2.5
9.0
2.5
5.5
-5.0
-2.5
0.0
255
5.0
25
(a)
50-
2.775
-0.75
2.725
50
10.55 75100
1025500
1025 75100
-1.25
50
20
10255075100 10255505500
M
(b)
(c)
Figure 6: (a) One example of activation function (mean and standard deviation) learned by auNN with each kernel. RBF's one smoother, whereas TRT's piecewise linear, inspired by ReLu. Black dots represent (the mean of) the inducing point values. Green dots are the locations of input data when propagated to the corresponding unit. (b) Test NLL of auNN and DGP for different values of M (number of inducing points) and D (number of hidden units). The lower the better. The results are the average over five independent runs with different splits. Whereas DGP improves "by columns" (i.e. with M). auNN does it "by rows" (i.e. with D). This is as hypothesized, and is convenient from a scalability viewpoint. (c) Test NLL with increasing depth (7. 2, 3, 4). This supports that auNN might benefit more than DGP from deeper networks. Moreover, the aforementioned different influence of M and D on DGP and auNN is confirmed here.
Moreover, the aforementioned different influence of and on DGP and on auNN is also confirmed here. The results RMSE are similar, see Figure an and Tables IT in Appendix C
Finally, it may be argued that auNN closely resembles DGP with additive kernel (Duvenaud et al., 2011 Durrande et al.) (211) (DGP-add hereafter). Recall that additive kernel models functions fo (x Therefore, the model for a'+ I|a in auNN very similar that of f(+ If in DGP-add, see Figure 11 in Appendix C Specifically, in both cases, the input (a auNN, f in DGP-add) goes through TD GPs and then these are aggregated (linear combination through W auNN, summation in DGP-add) yield the output (a'+I in auNN, pl+1 in DGP-add). However, there exists key difference. In auNN, all the nodes in the (1 + 1)-th layer (i.e. (+1) aggregate a shared set of distinct functions (namely, f.), each node using its own weights aggregate them. While in DGP-add, there not such shared set of functions, and each node the + 1h layer (i.e. f!++) aggregates a different set of GP realizations (i.e. the unlabelled blue nodes in Figure I.p). This subtle theoretical difference has empirical implications, since many more functions need be learned for DGP-add. Indeed, Figures and i Appendix compare the performance of DGP-add and auNN-RBF (the experimental setting is analogous to that of Figure We observe that the results obtained by DGP-add are worse than those by auNN-RBE, probably due to the larger number of functions that need to be learned in DGP-add.
that are decomposed as f(x) = = fi(1) (z1)).... + +
3.4 CLASSIFICATION SCALABILITY, AND ADDITIONAL METRICS
So far, we have experimented with small to medium regression datasets, and uncertainty estimation has been measured through the (negative) log likelihood and the visual inspection of the predictive distribution (Figures 3 and 5). Here we focus on two large scale classification datasets (up to 10" instances), and additional metrics that account for uncertainty calibration are reported. We use the well-known particle physics binary classification sets HIGGS (N 11M, D 28) and SUSY (N 5M, D= 18) (Baldi et al. 2014). We consider DGP as a baseline, as obtained state-of-the-as results for these datasets (Salimbeni & Deisenroth] 2017). For all the methods, consider Robust-Max classification likelihood (Hermández--Lobato et al. 2011).
The metrics to be used the Brier score (Gneiting & Raftery 2007) and the Expected Calibration Error (ECE) (Guo et al. (2077). The former is a proper score function that measures the accuracy of probabilistic predictions for categorical variables. In practice, it is computed as the mean squared
For fair comparison, here we auNN-RBF (and not TRT), hecause DGP-add leverages RBF kemel.
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Table 2: Brier score and expected calibration error (ECE) for auNN and DGP in the large scale classfication datasets HIGGS and SUSY (the lower the better in both metrics). The standard error (on three splits) close to zero in all cases, see Table 13in Appendix
auNN
DGP
RBF-2
RBF-3
RBF-4
TRI-2
TRI33
TRI
DGP-2
DGP-3
DGP-4
HIGGS
0.3363
0.3159
0.3098
0.3369
0.3172
0.3118
0.4527
0.4399
Brier
SUSY
5M
18
0.2746
0.2739
0.2737
0,2749
0.2742
0.2738
0,3815
0,3816
0,3804
HIGGS
0.2196
0.2427
0.2198
0.2390
0.2397
0.4352
0.4303
0.4251
ECE
SUSY
SM
18
0.3453
0.3496
0.3504
0,3462
0.3485
0.3465
0.5304
0.5291
0,5273
Table 3: Average training time per batch over 50 independent runs (in seconds). The standard error is low in all cases, see Table 14 in Appendix C
auNN
DGP
RBF-2
RBF-3
RBF-4
TRI-2
TRI.3
TRI4
DGP-2
DGP-3
DGP-4
HIGGS SUSY
0,0962
0.1607
0.2259
0,0922
0.1647
0.2308
0.1918
0.3102
0.3930
0.0926
0,1564
0.2245
0,0923
0.1563
0,2265
0.1430
0.2129
0.2771
difference between a one dimensional vector with the probability for each class label and the one-hot encoding of the actual class. The latter measures miscalibration as the difference in expectation between confidence and accuracy. This is done by partitioning the predictions in M equally spaced bins and taking weighted average of the bins' accuracy/confidence difference, see (Guo et al. 2017 (3)) for details.
Table 2 shows the Brier score and ECE for auNN and DGP for different values of (depth). We observe that auNN outperforms DGP in both metrics, achieving superior uncertainty estimation. Both TRI and RBF kernels obtain similar results for auNN. Notice that the Brier score generally improves with the network depth, whereas the performance in ECE decreases with depth. Interestingly, this behavior was also observed for standard NNs (Guo et al. 2017 Figure 2a).
Finally, as was theoretically justified in Section auNN can scale up to very large datasets (HIGGS has more than 10" training instances). Regarding the practical computational cost, Table shows the average training time per batch for both auNN and DGP in the previous datasets. Although the theoretical complexity is analogous for both methods (recall Section (2,, the experiments in Figures op-c showed that DGP requires larger values of M, whereas auNN needs larger 1) [ Since the computational cost is not symmetric on M and D, but significantly cheaper in the Tatter (recall Section 2). auNN is faster than DGP in practice.
RELATED WORK
Activation-level uncertainty is introduced here as an alternative to weight-space stochasticity. The expressiveness of the latter has been recently analyzed in the recent work (Wenzel et al. (020)) where the authors advocate a modified BNN objective. Alternatively, different prior specifications are studied in (Hafner al. 2020 Pearce al. 2019 Flaam-Shepherr et al.) (017)) in addition to the (BNN discussed here (Sun et al. 2019). However, none of these works consider stochasticity on the activations.
Since we present a straightforward use of VI for auNN, in this work we have compared empirically with the well-known VI-based BBP for BNNs. Yet, we expect auNN to benefit from independent inference refinements like those proposed over the last years for BNNs. For instance, natural-gradient
VI allows for leveraging techniques such as BatchNorm or data augmentation (Osawa al. 2019)
and the information contained in the SGD trajectory can be exploited as well (Maddox et al. 2019).
Also, getting rid of the gradient variance through deterministic approximate moments has provided enhanced results in BNNs (Wu et al. 2019)
"In this section, both DGP and auNN are trained with one hidden layer and their optimal configuration according to the previous experiment: large M for DGP (M 100, 1 is set as recommended by the authors, i.c. D min(30, 17)),, and large D) for auNN (D 50, M is set the intermediate value of M 25).
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A key aspect of auNN is the modelling of the activation function. This element of neural nets has been analyzed before. For instance, nerf-nooma neural nets (Klambauer et al.
2017) induce
the normalization that is explicitly performed in related approaches such as Batch orm (loffe &
Szegedy 2015) and weight and layer normalization (Salimans & Kingma 2016 Ba et al. (016))
Learnable deterministic activations have been explored too, e.g. (He al.] 2015 Agostinelli lli et al.. 2014). However, as opposed to auNN, in all these cases the activations are deterministic,
Probabilistic neural networks such as Vatural-Paramette Networks (NPN) (Wang et al. 2016) prop- agate probability distributions through layers of transformations. Therefore, the values of the activations are also described by probability distributions (specifically, the exponential family is used in NPN). Fast dropout training (Wang & Manning 2013) and certain variants of NPNs can be also viewed in this way (Shekhovtsov & Flach, [018 Postels et al.) 2019). However, in auNN the activations are modelled themselves as stochastic learnable components that follow GP prior. Along with the deterministic weights, this provides a conceptually different approach to model uncertainty.
A very preliminary study on GP-based activation functions is proposed in (Urban & van der Smagt 2018). However, the method is not empirically evaluated, no connection with deep GPs is provided, and the inference approach is limited. Namely, the output of each unit is approximated with
Gaussian whose mean and covariance are computed in closed-form, as was done in (Bui et al. 2016) for DGPs. However, this is only tractable for the RBF kernel (in particular, it cannot leverage the more convenient TRI kernel studied here), and the Gaussian approximation typically yields worse results than Monte Carlo approximations to the ELBO as used here (indeed, DSVI (Salimbeni & Deisenroth 2017) substantially improved the results for DGPs compared to (Bui et al. 2016)).
CONCLUSIONS AND FUTURE WORK
We proposed a novel approach for uncertainty estimation in neural network architectures. Whereas previous methods are mostly based on a Bayesian treatment of the weights, here we move the stochasticity to the activation functions, which are modelled with simple ID GP and triangular kernel inspired by the ReLu. Our experiments show that the proposed method obtains better calibrated uncertainty estimates and is competitive in standard prediction tasks. Moreover, the connection with deep GPs is analyzed. Namely, our approach requires fewer inducing points and is better suited for deep architectures, achieving superior performance.
We hope this work raises interest on alternative approaches to model uncertainty in neural networks. One of the main directions of future research is to deeply understand the properties induced by each one of the kernels considered here (i.c. the triangular one and RBF). In particular, it would be interesting to automatically learn the optimal kernel for each unit in a probabilistic way. Also, the use of GP prior for the activation function may hamper the scalability of auNN to wider and/or deeper networks. In these cases, the GP-based activation model could be substituted by a simpler Bayesian parametric one. This would allow for cheaper modelling of uncertainty within the activations. Finally, since only the activation function is modified, important deep learning elements such as convolutional layers can be still incorporated.
ACKNOWLEDGEMENTS
This work was supported by the "Agencia Estatal de Investigación" of the Spanish "Ministerio de Ciencia e Innovación" under contract the Spanish "Ministerio de Economía, Industria y Competitividad" under contract DP12016-77869. C2-2-R. DHL acknowlegess support from the Spanish "Ministerio de Ciencia Innovación" (projects TIN2016-76406- and HID2019-106827GB BAAGGB--OO////////000/00 13039/501 (00011033) PMA was funded by La Caixa Banking Foundation (ID 100010434, Barcelona, Spain) through La Caixa Fellowship for Doctoral Studies LCF/BQ/ES/7711100000 I, and the University of Granada through the program "Proyectos de Investigación Precompetitivos para Jóvenes Investigadores del Plan Propio 2019" (ref. PPJIB2019-03).
10001 1033, and
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A PRACTICAL SPECIFICATIONS FOR AUNN
Whitening transformation for q(u)). The proposed parametric posterior for each unit is given by the Gaussian q(u') = (N'u m' Syy) The GP prior on u is p(u)) = (u'llu K2). with u4a (2((2)) and K (z w) For numerical stability and to reduce the amount operations, we use a white representation for q as is common practice in (D)GPs (De G. Matthews et al. 2017 Salimbeni & Deisenroth 2017). That is, we consider the variable v22 ~ Mi S'2,, with 1 Pla + 1/2' Specifically, in the code the variable m' is denoted as mum and S is represented through its Cholesky factorization 1/2, which is named g_grr.
Initialization of the variational parameters (m'a). These are the mean of the posterior distribution on the inducing points. Therefore, their value determines the initialization of the activation function. If the RBF kernel is used, {m') are initialized to the prior ful = 122(2²) (since we are using the aforementioned white representation, q_ m_mu is initialized to zero). This is the most standard initialization in GP literature. For the TRI kernel, {m'} are initialized according to the ReLu which TRI is inspired by, i.e. m ReLu(z)) (22)-
Initialization of the variational parameters (S)) The posterior distribution covariance matrices are initialized to the prior KG (2 2 (that is, e_sgrt is initialized to the identity matrix). Following common practise for DGPs (Salimbeni & Deisenroth 2017). the covariance matrices of inner layers are scaled by 10-5,
Initialization of the weights. The Glorot uniform initializer (Glorot & Bengio 2010), also called Xavier uniform initializer, is used for the weights. The biases are initialized to zero.
Initialization of the kernel hyperparameters. The kernels used (RBF and TRI) have two hyper- parameters: the variance y and the lengthscale f. Both are always initialized to 1 (except for the lengthscale in the ID example in Section 3.1 where E is initialized to 0.1).
Initialization of the inducing points. In order to initialize z'z. the N input data points are propagated through the network with the aforementioned initial weights, biases, and activation function. Then, in each layer and unit, z initialized with linspace between the minimum and maximum of the N values there (the minimum (resp. the maximum) is decreased (resp. increased) by 0.1 to strictly contain the interval of interest).
Initialization of the regression likelihood noise. In the regression problems, we use Gaussian likelihood p(y f) N(y)f.o²²) The standard deviation of the noise is initialized to a 0.1.
Mean function. (and standard deviation equal one), a zero mean function allows for reverting the empirical mean for OOD data, as explained in the main text.
We always use a zero mean function. Since data is normalized to have zero mean
Optimizer and learning rate. Throughout the work, we use the Adam Optimizer (Kingma & Ba 2014) with default parameters and learning rate of 0.001.
B EXPERIMENTAL DETAILS FOR THE EXPERIMENTS
All the experiments were run on a NVIDIA Tesla P100. In order to predict, all the methods utilize 100 test samples in all the experiments. Details for each section are provided below.
An illustrative example (Section 3.1 in the main text). All the methods use two layers (i.e. one hidden layer). The hidden layer has D = 25 units in all cases. BNN and IBNN use ReLu activations. The auNN methods use M 10 inducing points in each unit (the rest of methods do not have such inducing points). The methods are trained during 5000 epochs with the whole dataset (no mini-batches). The dataset is synthetically generated to have two clusters of points around æ = +1. More specifically, 30 points are sampled uniformly in each interval (z 0.3, x + 0.3) for = +1, and the output is given by the sin function plus a Gaussian noise of standard deviation 0.1. We have also trained DGP and GP on this dataset, see Figure 14 Both methods M = 10 inducing points, and are trained during 5000 epochs with the whole dataset (no mini-batchess DGP has one one hidden layer with D 25 units.
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Boston
Concrete
Energy
Naval
Wine
Yacht
BNN-2
Tolt rag log
(BNN-2
(BNN--
JUNN RBF 2-
auN-t-1157-3
auMi TRI-2
auNNN-TUU-3
1000
2000
Boston
Concrete
Energy
Naval
Wine
Yacht
BINN
B1NN-3
Lae RIMSE
1BNN-2
FBNN-3
auNN--TU-3
100
15
0.02
0.03
0.7
0.8
0.9
10
15
Figure 7: Performance of the compared methods in the gap splits for six UCI datasets, Mean and one standard error of NLL (upper row) and RMSE (lower row) are shown, the lower the better.
UCT regression datasets with gap (and standard) splits (Section 3.2 in the main text). The methods use T, 2,3 layers. In all cases, the hidden layers have D = 50 units. BNN and fBNN use ReLu activations. The methods are trained during 10000 epochs, with mini-batch size that depends on the size of the dataset. For those with fewer than 5000 instances (i.e. Boston, Concrete, Energy, Wine and Yacht), the mini-batch size is 500. For those with more than 5000 (i.e. Naval), the mini-batch size 5000. Recall from the main text that each dataset has as many gap splits as dimensionality, with 2/3 for train and 1/3 for test. In the case of standard splits, each dataset uses 10 random 90%-10% train-test splits. Regarding the segment used Figure each extreme of the segment is a point from different connected component of the training set. These are chosen so that the function is well-known in the extremes (but not along the segment, which crosses the gap). Namely, the extremes chosen the training points who have minimum average distance to the closest five points in its connected component.
Comparison with DGPs (Section 3.3 in the main text). Here, different values of depth L, number of inducing points M and number of hidden layers D are studied (see the main text). auNN is trained during 5000 epochs, with mini-batch size of 5000 (20000 epochs are used for DGP, as proposed by the authors (Salimbeni & Deisenroth, 2017)). Each experiment repeated on five random 90%- -00% train-test splits. DGP uses RBF kernel. The experimental details for DGP-add are the same for DGP, with the only difference of the kernel. Namely, an additive kernel using RBF components is used for DGP-add.
Large scale experiments (Section 3.4 in the main text). Since we are dealing with classification datasets, a cobust-Max likelihood used in all cases (Hernández- Lobato al. 2011). The values of D and M are chosen following the conclusions from Section 3.3] That is, DGP needs large M (the largest = 100 is used), but less influenced by D (this is chosen as recommended by the authors (Salimbeni & Deisenroth 2017): : D min(30, D)), with D the dimensionality of the input data). auNN needs large D (the largest = 50 is used), but is less influenced by M (the intermediate value M = 25 is chosen). All the methods are trained during 100 epochs, with a mini-bated size of 5000. Three random train-test splits are used. In both datasets, 500000 instances are used for test (which leaves 10.5M and 4.5M training instances for HIGGS and SUSY, respectively).
C ADDITIONAL FIGURES AND TABLES
Finally, additional material is provided here. Every figure and table is referred from the main text.
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Table 4: Test NLL for the gap splits of the six UCI datasets (mean and one standard error, the lower the better). Last column is the per-group (weight-space stochasticity vs activation -evell stochasticity) average rank.
Boston
Conerete
Energy
Naval
Wine
Yacht
Rank
Rank (group)
BNN-2
3.29::0.10
0.58::0.09
114:44170669
2186, 1186.002-464.33
0.96:10.01
0.59
3.92:20.79
INNN 2
BNN-3
3,56-20.00
3.671.025
4.601:0.99
4,25=00.04
1116651.6.66
30.9++19997
1050.65::: 192.6)
618,44 117::117.50 447.99
2.80:10.31
0.98:+-002
4.10=-0:03
1.77:0.122
5.06::0.36
4.98==0.70
4.832
3.69:0.24
4,49::0.44
93324:56455
1000::44:247.111
198.76:330.24
147::0.15
5.36:00.00
.NNX-RF-2
5.1920.47
4.2720.26
39.91:1:00
279.552677.7 74
14:10
16822035
4.69:20611
JuNN R
mNN-TRI-2
2.77:20.06
5.681:0.75
0.05
0.50
50.48:1.8
3.99:1.144
352,9411.2.13
30.47:55.54
16.05:1133
1:062000
2.34:20.03
0.282:0233
5.29:20.89
3.25
4.17:10.40
JuNN 3
2.701.0.04
0.00
5.501:2.45
2.28
1.23:00044
2.68:00.30
3,47:80.80
Table 5: Test RMSE for the gap splits of the six UCI datasets (mean and standard error, the lower the better). Last column is the per-group (weight-space stochasticity vs activation-leve stochasticity)
average rank.
Bustun
Comcrete
Energy
Naval
Wine
Yachi
Rank
Rank (group)
BNN-2
6.54100.56
7.62:20.35
4.23211.91
0.03=0.000
0.63::0.01
1.18:0.11
4.091:066
[BNN-2
BNN-3
375:10.21
3.37:10.40
16.33:10.67
7.58:20:41
5.27:51.41
3.95
0:03=00:0
0.02:0.00
0.64:20.00
0.78::0.02
14.311:0.66
1.25:0008
6.151:0991
4,70110.55
4.1=00.77
IRNN-1
381:0020
1.5220.36
4.482
0.03=0.00
0.87
1.1320.12
4.71:::555
mNN--RBF-2
490:20047
188220477
3.41:1.46
0.0==0000
0.72
0.99::0118
4.37:::447
=NN-RBF-3
mNN--TRE-2
4.7:1:0299
401:20300
1.4::0:21
144420 18
2.72:20.70
2.72=11.03
0:03:0:00
0.02-0.00
0.67
0.82:00001
1.030001
1.112:000
4.27:1:.55
3.002::33
4.09
MINN-TRL-
3,78:0.19
7.03:20.23
3.362 1.25
1M02=0.00
0.68
3.80
3.85:::466
Table 6: Test NLL for the standard splits of the six UCI datasets (mean and one standard error, the lower the better). Last column is the per-group (weight-space stochasticity vs activation-level stochasticity) average rank.
test NIJ.
Boston
Concrete
Energy
Naval
Wine
Yacht
Rank
Rank (group)
BNN-2
2.71
3.12=0.02
0.65=0.04
-5.38=0.599
0.99=#0.02
1.01
3.78=0.41
BNN-3
3.62:20.05
4.24:0001
0.80=0.03
-5.02-20.33
1.01-10002
4.0620005
6.25=0070
4.5:10.39
I
2.3100.20
3.201:0.4
0.6-00004
6.17:10.02
1.55
0.77100.2
4.13:10.57
IBNN-3
2.75:10.14
3.1320.05
0.65
6.26:10.00
2M.43:1.112
0.7910002
3.83.0.85
mINN-BBF-2
3,38
3.1420.05
0.63-0.03
-5.0020.88
1.16+0.06
0.52
3.9720.00
auNN-RBF-3
3.89:20.47
3.25=0.13
0.53=0.07
-5.6920.033
8.98::1.51
0.54:0.03
4.42-0.85
4.5:10.43
auNN- TRI-2
2.56:10.05
3.08:10.02
1.47=0.04
4.81:00077
0.96:1..03
2.25
4.78:10.92
auNN TRI-3
2.50
2.98
1.02 0.02
3.43.1..22
1.10:10.07
2.26:10.01
4.83 1.01
Table 7: Test RMSE for the standard splits of the six UCI datasets (mean and one standard error, the lower the better). Last column the per-group (weight-space stochasticity vs activation-level stochasticity) average rank.
test RMSE
Boston
Concrete
Energy
Naval
Wine
Yacht
Rank
Rank (group)
BNN-2
3.4
5.49:2..13
0.45=0.02
0.00=0.00
0.65=0.01
0.68:10.08
4.70::0.48
BNN-3
8.89:0.45
16.7120.20
0.00=0.00
0.67:00.2
13.49:::.94
6.50:20.64
4.59:20.41
1BNN-2
2.80::0.21
5.3420.13
0.47=0.02
0.00=0.00
0.70=20.22
0.33-10.04
3,70=:0.61
IBNN-3
2.74::0.16
5.07:0011
0.46=0.02
0.00=0.00
0.83-0.02
0.36:1:004
3.45=00888
auNN-BBF-2
3.16:::0.33
5.13::0116
0.45=0.02
0.00=0.00
0.67+0.02
0.11:0004
4.25::0.35
SUNN-RBF-3
3.01::025
4.51::018
0.41=0.03
0.00=0.00
0.76200.2
0.38:10.03
3.35
4.41:0.41
mNN--RR122
3.00-1::26
5.21+00.10
0.72-0.02
0.00-0.00
0.62:10.02
1.154:0.14
5.40-1-0.80
WINN-TRI-3
2811:0117
4.6740015
0.65=0.03
0.01=0.00
0.62-0.02
1.16::0.15
4.65+t1.00
Table 8: Standard error obtained by auNN and DGP in three splits of the large scale classification datasets HIGGS and SUSY.
auNN
DGP
N
RBF-2
RBF-3
RBF-4
TRI-2
TRI-3
TRI-4
DGP-2
DGP-3
DGP-4
HIGUS SUSY
SM
18
0.0004
0.0005
0.000S
0.000S
0.0005
0.0004
0.0005
0.0027
0.0035
15

---

Published as a conference paper at ICLR 2021
auNN-RBF
auNN-TRI
Layer
Layer 2
Layer
Layer I
Layer
Layer 3
0.0
-2
-0.5
-0.5
-0.5
0.5
-1
6.5
0.0
0.55
-3
-0.5
W
m
255
-25
0.0
0.00
0.0
-0.25
-1
-0.5
-1
Figure A complete example of the activation functions learned by auNN with RBF and TRI kernels. These were obtained for the Energy dataset with the first gap split, using three layers, 10 hidden units per (hidden) layer, and 5 inducing points in each unit. Whereas auNN-RBE learns smoother activations, auNN-TRI ones are piece-wing wise linear, inspired by the ReLu. Notice that auNN allows units to switch off they are not required. Black dots represent the five inducing points in each unit. Green points the locations of the input data when propagated to the corresponding unit.
16

---

Published as a conference paper at ICLR 2021
BUNN-RBF
aUNN-TRI
DGP
00.00
50
0.075
1025 75100
2025507310
502550 75100
N
Figure 9: Test RMSE auNN and DGP different values of M (number of inducing points) and D (number of hidden units). Results are the average over 5 independent runs on the UCI Kin8 dataset. The lower the better. Whereas DGP improves "by columns" (i.c. with M), auNN does it "by rows" (i.e. with D). This is theoretically expected, and it is convenient from scalability viewpoint.
auNN-RBF
auNN-TR TRI
DGP
4.3
25
4.2
10
4.0
3.9
5
50
3.6
10253075100
Figure 10: Test RMSE with increasing depth (L = 2, 3).. This supports that auNN might benefit more than DGP from deeper networks. Moreover, the aforementioned different influence of M and D on DGP and auNN is confirmed here.
17

---

Published as a conference paper at ICLR 1
Table 9: Test NLL of N and DGP for different values of M (number of inducing points) and D (number of hidden units). Mean and one standard error over 5 independent runs on the UCI Kin8 dataset are shown. The lower the better.
8INN-RBF
auNK-TRI
DGP
D
M
5
10
25
30
75
100
5
10
25
50
75
100
5
10
25
50
75
100
5
0..8500.01
-0.89110.01
-0.89:0000
0.89:0011
-0.89-0.01
-0.9020.011
-0.78::0000
-0.9:0:03
-0.78:00.05
-0.7720.04
-0.6710006
-0.1::004
-067::0011
-0.96:0..00
-1.19100001
-1.30:2:001
-1,33:0011
-1.34-0.01
10
1.06-0.01
11.0910.01
-109:0001
-109::0011
-.10=0002
-.10400.01
096110011
400220011
-1.03-00001
00996.0.03
0.94-10000
0,89:0003
0.69:10.01
-0.9600.00
-1.19:0.00
-13000001
-,3344001
-1.35-0.01
0.22700.2
-1.301:0.02
-1.30:0002
-1.30:1002
0.31
-1311000
-1091:001
-1.99000
-1.22:1001
-1.1520002
-1.11:00011
-1066:000
0.68:10.1
-0.9:00.00
-1.17::001
-1261001
-1229:000
-1.30-001
50
-1.33=0001
-1.34100000
-1.440000
-1.332001
-1.3400.2
-1.2200.0
-1.154001
-1.241001
-1.29:20011
-1.2620001
-1.34:0000
-1.19:002
-06910001
-0.96-0.01
-1.162000
-1314001
-12224001
-1.24=0.01
Table 10: Test RMSE of auNN and DGP for different values of M (number of inducing points) and D (number of hidden units). Mean and one standard error over 5 independent runs on the UCI Kin8 dataset are shown. The lower the better.
suNN-RBF
suNN-TRI
DGP
10
50
75
75
100
0,10+:000
0.10::0:00
0,1000.00
0.10=0.00
0100:0000
0.10:0000
0.114:000
0:11:0000
0.11000111
0.11::000
0.14000
0.12::000
0.12=0.00
0.9:00000
0.0700000
007:00000
0.0:::000
0.06::0000
0:0+00300
0.08-0.00
000000:00
0:0-:00000
0,92:0000
008-0030
0.08+0.00
00940000
009+0000
0.10+0000
0.12+0.00
007+0000
0.6400000
0.64:0000
0.96+-0.00
18
0.0700000
00700.000
0.0020000
0.0720:00
0072000
000:00000
0.07:0000
0.0200000
0.07:0.00
00800000
00e:0000
0.8000000
0.1220.00
009:0000
007:00000
0001:0000
0.073:000
0.06:20.00
0.06+:0000
0000:0000
0.06=0000
0:06:00:0
0.06+00000
0.00:0000
0,0740000
04700000
0.07=00000
0007
00770000
0.070000
0.12:0.00
0.09:0.00
007:00000
0.0700000
0.070:0000
0:07:00000
Table 11: Test NLL of auNN and DGP for different values of M (number of inducing points) and D (number of hidden units) as the depth increases from L = 2 to L 4. Mean and one standard error over independent runs on the UCI Power dataset are shown. The lower the better.
BuNN-KBF
mcNN-TRI
DGP
D
M
15
100
10
50
75
100
10
50
75
100
5
2,86 2.000
2.00
2.84:0.022
2.84:00022
2.4.1:0.02
2.81:10.02
2.8
2.83:10.22
282:100.2
2.84:00022
2.891:0.03
2.81-00.02
2.87:0.02
2.85100022
2.80-0002
2.82:10.02
281-0.02
2.81:10.02
10
2.841:0.02
2.83:0.02
2.2220.02
2.111002
280:00002
2.811:0022
2.84
2.82:00.22
2.80:10002
2:80000002
2.110000
2.81
2.87:0002
2.86:0002
2:60::0000
2.83:10002
2.82-0.02
2.81:20.02
25
2.3---0.02
2.81
2.0020002
28140002
2814-0002
2.804-0.2
2.83-0002
2.80--0.22
2.78-0002
2.78-1-0.2
2.78-0002
2.79
2.8-00022
2.85400022
283 +0002
2.824-0022
281=0.02
2.81-10.22
2.821:0000
2.61
2.1220002
2.00:00022
28110000
2.8110022
2.82:0002
2.00:00022
2.71::0022
2.761:0022
2.761:0.02
2.061:0:33
287:10002
2500
2.83-00022
2.2:0.02
2.81:10.22
2.841::000
2:0300002
2.3:20002
28310003
28310000
2.831:0022
2.8410002
28320002
2:801000
2:
2:85:000
2.2200022
2.8620.02
2:8:10002
2801000
2:791:000
2.76-0022
2.77:1011
0.00
28020002
200
200:0002
2.80::002
2,82:000
2.0020002
2.79:002
2
2.7820.22
27720002
2.862002
283:10022
2
2.81::003
2.82-002
2.11:0022
25
2.80
2.79
2.7720.022
2.77:0022
2.77::0002
2.77
2.79200.22
2.77:0002
2.7420022
2.72550000
2.7420.0
2.742008
2.8620002
288520022
2.000002
2.82::002
2.8100012
2.81220022
50
2.78::0002
2.78:0.02
2.77:20.22
2.76:0002
2.761:0000
2.761:0003
2.78:0000
2.75:00.22
2.71::0022
2:000
2.7020001
2.70
2.87:0002
2.87:10022
2.8400002
2.82:10002
2.82:0.22
2.8120.02
5
2.84::0.02
2.
2.222.02
2834002
280:0001
2.83-20.02
3.6920.35
2.3320001
2.822002
2.
2.83-002
2.
2:86
2.
2:502:02
2.79
2.78-0.02
2.77
2811:0022
28020002
2.0220022
200
2824000
20002
288
20:1
20702001
2.79:00022
2:79::002
0.
2.020.02
284
200
2.79:0022
282=0.02
2.1220022
277900002
2.78-00.2
277=0002
2775+002
277+-002
2.764-0.22
2.000001
2.78-00:22
2775+0002
2.74-0002
2775-000
2.75-0002
2.6-00022
285-+0002
2
2.82-0002
282-0.02
281--0022
50
2.79::0m
2.00
2.75
2.77.0033
2.75
2.74
2.79-00.1
2.7720001
2.712:0022
2.74
2.7420002
2.75
2.8720002
2.8522002
2.84:0022
2.82:00022
2.82-0.02
2.81-20022

---

Published as a conference paper at ICLR 2021
Table 12: Test RMSE of auNN and DGP for different values of M (number of inducing points) and D (number of hidden units) the depth increases from L
L 4. Mean and one standard error over independent runs the UCI Power dataset are shown. The lower the better.
soNN-RBF
s:NN-TRE
75
100
4.20-20.09
4.14=0008
4.12400088
4.13::0008
4.134:0077
4,1420009
4,1620088
4.09:00088
406-20.09
4.124:09
4.32400113
4.1240009
4.24:0.100
4.19:0099
40840000
4.05-0008
4,15000
4.0170007
4.10=0.10
4.2400.10
69
25
0.0920.08
4.01 0:088
400:
401
4.00::000
40220008
4.040008
3.9220077
1.00:20008
200
3.89:20000
3.9040007
4.24:0009
4.18:000
4.100009
4.06::0008
4032000
4.01
50
0.06
0.00
4 0007
4.01
3:992:005
4,0
3.93:0.07
38862000
3.3300088
5.80
3.8100010
0.24:0.10
4.2400.100
4.18::0009
4.11
40620009
4.05=20.08
4,14--0.09
4.08
4.11+:006
4.19:-.11
4.114:0.09
4.09:0009
4.10=0.09
4.10-0.07
4.024:0000
4.04--0006
4.15+00.08
4.04--0.99
4.22-0.09
4
4.0--00009
3.92--0.07
3.90-0006
3.86--0.05
4.02::0.08
0.02:00.88
1.0010007
4:00
4:001:0.07
0.99
4,01
3.95
392110077
3.9210008
3.901:0.07
3.90
0.20:00.09
4.10:10.66
398.1.0
3:991:00077
0.05
4.03-10.08
3.96+0.088
399=2007
387400077
38770007
3.83-0077
3.88-008
380-0008
3.76+0077
3.674-066
3775
423-20.10
4.11::00099
40340.0
3.89-50.08
3.88-0000
3885
3.30::0.99
3.8-+-0007
3.80
3.86-0008
3.77:20.09
3.62:400066
+-0-8-0088
3.59-3-0.09
3.60
4.24:00.00
4.24:00110
4.12:::0.09
4,07 +:000
4.04--0.08
4.03-10.08
4.14:1:.11
0.10::0:08
4.08:10.08
4.11:10.08
4,00:1:0.5
0.09
32.00
4.00::0.07
4.06:1:0:09
4.07:1:007
4,09:1:008
4.07:10.08
0.22:10.09
4.10:10.88
3.97110.07
3:90-1:0007
3.88:10.07
3.85::0.07
10
4,01 0.008
0.08
3.99:10.07
1:99::007
4.05:1:0:07
4.01 00006
4.03:20.09
3.99
3.94220006
3.941:0077
3.92:10.07
3.93:10.08
4.20:20.09
4.12:10.08
4.8.1:0010
3,94::0000
4.0620.06
4.01:20.08
3.93---0.09
3.91::0:00
3.87--0007
5.006
3.84--0.07
3.82+-0.07
0.007
3.85
3.76-1-0088
1.751-008
3.77
3.78-1-0.9
0.24
4.18-10.09
4.11:1-0.09
4.06-1-0.08
0,04-:0.08
4,05-10..88
0.96
3.78.10.11
3.80:10.07
3.701:0.07
3.90
3.82:10.06
3.71::0.09
3.73:1:0099
3.75::0.08
3.771:0.05
4.21:20.99
4.12:1:009
4.01::0.08
4.03:10.08

---

Published as a conference paper at ICLR 2021
W
(1D)
W2
GP (1D)
42
12
(a) auNN
GP (2D)
GP (2D)
(b) DGP
GP (1D)
add
GP (1D)
add
f
(c) DGP-add
Figure 11: Representation two hidden layers (with two units per layer) for auNN (a), DGP (b), and DGP-add (c).
D COMPUTATIONAL COST SUMMARY
Table [5] shows the training computational complexity for the methods compared in this paper. Moreover, in order to evaluate the computational cost in practice, the table also shows the actual running time for the experiment factorized Gaussian for the approximate posterior. Although IBNN has the same theoretical complexity, the Spectral Stein Gradient Estimator (Shi et al. 2018) is used to compute the KL divergence gradients. Moreover, GP prior is specified at function space, for which GP must be trained as a previous step. DGP and auNN have the same theoretical complexity. In practice, auNN is typically faster because it requires fewer inducing points, recall Section 3.3 and Table The running time in Table 15 is very similar for both because the same amount of inducing points (M = 10) is used in this simple experiment.
Section 3. BNN is the fastest algorithm, since it utilizes
auNN-RBF
DGP-add
2.85
5
2,80
50
10 25
75100
10 25 75100
M
Figure 12: Test NLL Power dataset for different values of D and M (the lower the better).
20

---

Published as a conference paper at ICLR 2021
auNN-RBF
DGP-add
5
4.0
50
10
75100
10 25
75100
3.8
M
M
Figure 13: Test RMSE on Power dataset for different values of D and M (the lower the better).
Table 13: Standard error for the results in Table 2 Three random train-test splits are considered.
auNN
RBF-2
RBF-3
RBF-4
TRI-2
TRI-3
TRI4
DGP-2
DGP-3
DGP-4
HIGGS SUSY
11M
28
0.0001
0.0007
0.0008
0.0003
0.0005
0.0009
0,0018
0,0016
0,0006
Brier
0.0005
HIGGS SUSY
1IM
0.0015
0.0020
0.0022
0.0010
0.0035
0.0019
0.0006
0.0004
0.0008
ECE
0.0012
0.0014
0.0005
Table 14: Standard error for the results in Table
Fifty independent considered.
auNN
DGP
RBF-2
RBF-3
RBF-4
TRI-2
TRI.3
TRI-4
DGP-2
DGP-3
DGP-4
HIGGS
0.0258
0.0325
0.0371
0.0188
0.0318
0.0378
0.0248
0.0266
0.0269
SUSY
0.0215
0.0274
0.0369
0.0202
0.0258
0.0350
0.0108
0.0126
0.0144
0.0
0.0
055
DGP
GP
Figure 14: DGP and GP trained the dataset of Section 3.1 The experimental details are analogous to those in Section 3.1 see Appendix B Whereas DGP underestimates the uncertainty for in be-between data, a simpler GP does provide increased uncertainty in the gap.
Table 15: Training computational cost for the models compared in this paper. The running time (in seconds) corresponds to the mean and one standard error over 10 independent runs the experiment in Section 3.1 More details in Appendix D
BNN
IBNN
DGP
auNN
Running time (s)
15.21 + 0.78 O(N Σ, D'D+t)
51.92 + 1.7 O(N E, D'D
22.37 0.9.7 ONN Σ, D³)
21.16 0.89 O(NA Σ, D
Complexity
21


Reviews:
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It's proposed to solve the underestimation of uncertainty for instances located in-between two clusters of training points. It also shows comparison to DGP and indicates that auNN is better than DGP by requiring less inducing points and better suited for deep architectures. There is a strong connection between auNN with DGP which is discussed in the later part of the paper which I appreciate. \n\nHowever, I think there are still a few things needed to be more explicitly stated: In page 6, \"Whereas auNN considers combinations of the same Dl 1D functions to define the (l + 1)-th layer from the l-th one, DGP-add describes Dl different 1D functions for each unit in the (l + 1)-th layer. \" This is not quite clear, why auNN combines the same functions while DGP combines different. From figure 1, it seems that the only difference between auNN and DGP is whether f1,f2,f3 are summed up into a scalar a or not. Which part is different or same? I think it might need a bit math to demonstrate that and it would be helpful to draw figures for multiple nodes in the lth layer rather than just one. Cuz from the description under eq1 and figure 1, my understanding is f1, f2 and f3 in the l-1th layer are combined together with w to get a scalar a. Then a 1D GP is constructed from this scalar a to generate a scalar f for the lth layer. When we want another f in the lth layer, we just use another w to linearly combine f1,f2 and f3 (l-1th layer) and generate f using another GP from a. While in an ordinary DGP, fd in the lth layer is from a kernel taking f1,f2,f3 (l-1th layer) all together. So presumably, fd in auNN is from different GP functions with different kernels while fd in DGP is from the same GP with the same kernel. I just think this connection/distinction is a bit confusing to me. A better way to clarify it is using mathematical equations to express the unit in auNN and DGP respectively, including what the kernel looks like, what the expression for fd is, etc. There are such definitions for auNN but not DGP, and conceptual-level description is not clear enough. \n\nMoreover, although the paper emphasized the difference between auNN and DGP, I still think auNN is a specific/simplified version of DGP. Here is the reason: in auNN, f1 is only dependent on a1 and f2 is only dependent on a2. This is saying, the joint distribution of (f1,f2) has a block diagonal covariance and the blocks are different for each d. In DGP, f1 and f2 are usually considered to be independent to each other, thus the joint distribution of (f1,f2) also has a block diagonal covariance but the blocks are the same. But people can make DGP more general by allowing different blocks (kernel functions) for f1 and f2. Another difference is the input to the kernel (It would be appreciated too if the authors can explicitly write out the kernel definitions for auNN and DGP for easy understanding). For DGP, the input to fd (lth layer) is a D-dimensional vector (f1,f2,...,fD) in the l-1th layer; different fd share the same inputs. For auNN, the input to fd (lth layer) is a scalar ad=wd*(f1,f2,...,fD), and different fd have different inputs since wd is different. This part is more like different designs of the kernel. If so, then the inference method for DGP would be just applicable to auNN (adding the optimization for w).\n\nIn sum, this design of kernel and independent fd is interesting but the overall novelty is not strong enough from both the methodology perspective and inference perspective. \n\nA few other things:\n1. In the intro, it says \"First, it has been recently shown that BNNs with well-established inference methods ... underestimate the predictive uncertainty for instances located in-between two clusters of training points. Second, BNNs do not extrapolate sensibly to out-of-distribution (OOD) data. Both issues are illustrated graphically in Figure 3.\" I don't see both issues in Figure 3. Or more precisely speaking, what's the difference between these two points?\n\n2. Define d when it first shows up in the paper. \n\n3. Define the dimensions for mud, Kd.\n\n4. It shows that auNN requires less inducing points but the number of parameters is actually more than DGP due to w. It would be helpful to have such a discussion too. Since people know that the optimization for DGP could be tricky even with two parameters (e.g., length scale and marginal variance in RBF) in the kernel. Now the kernel has more parameters, how it would behave during the optimization and at the optimum? I think the larger variance of auNN across layers might be due to w, so comparing auNN with DGP-add might a bit unfair. \n\n----------------------------------------------------\nI raise my score to 6 after the rebuttal given the clarification the author added to answer my confusion.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "pAz1Rbb6VYS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nIn this paper, the authors contribute to the recent trend of replacing uncertainty in the weight space in favor of uncertainty in the function space. Their main contribution, of implementing activation-level uncertainty, is a valuable one and will inspire future research. \n\nOverall, this paper is well-motivated, clear-to-read, novel, and interesting. The mathematical reasoning is sound and the Figures are of excellent quality: in particular, Figure 2 is a very good illustration of the paper's core concepts. Figure 6 is also clear, thought-provoking, and informative. \n\nTo improve the work, the authors could devote more time to discussing the limitations of the proposed method. In particular, the authors make a point of the scalability: however, the datasets used in the network, while having many training samples, are quite small in their dimensionality (D=28 for HIGGS and 18 for SUSY). While the work compares extensively against deep GPs, a broader set of comparisons would improve the paper. Work such as 'Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning' shows that Bayesian inference can be scaled to work on ImageNet, using resnet-50 — although this work is not cited in the paper, it does show that existing approaches can scale to large networks and large datasets. Finally, the computational complexity of the proposed method should be outlined more explicitly — although there is some content in a footnote, a table comparing different models and their requirements would be of interest. \n\nIn terms of the modeling, I would also appreciate some discussion of the 'independence between units' assumption in the probabilistic model. The assumption that the uncertainty of a given unit depends only on the value of its activation seems to me to be a fairly strong one, so some more explicit discussions of the possible consequences of this choice would improve the work. \n\nOne structural change I would recommend is to move the 'related work' section forward: it currently precedes the conclusion, whereas I feel it would be better earlier in the manuscript to set the scene more thoroughly. \n\nThere are a couple of more minor points to address: \n\n* I find the 'Introduction' section of the work to be poorly-researched. All of the underlying motivations of modeling uncertainty have references to deep-learning papers from the last 5 years or so. It has the unfortunate effect of implying that people in these fields had never thought about these issues prior to the current deep learning boom. Using the cited papers would be fine if they were phrased more like \"deep learning has applied uncertainty estimation to the following critical applications\", but if the authors want to support the statements they've actually written, they should do more research. A possible starting point would be Bishop, §1.5.3. \n* The authors refer often to 'reversing' when they mean 'reverting' (to revert is to return to a previous condition, which is exactly the meaning the authors want to convey). \n* The paper https://openreview.net/pdf?id=B1l08oAct7 could also be referenced and discussed. \n", "extra_scores": {}, "extra_text": {}}, {"review_id": "luKKF6PAsN", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Either putting the uncertainty on the weights (e.g., Bayes by BP) or on the activation (e.g., fast dropout or variants of natural-parameter networks [2,3] or Bayesian dark knowledge [4]) or both [1] have been investigated before. The idea of moving the uncertainty from the weight to the activation function is not new. One could argue that VAE-style parameterization or local reparameterization trick is also a kind of methods that put uncertainty in the activation function. In fact the proposed method does involve the reprarameterization trick in each layer as shown in Eq. 7.\n\nThe hypothesis that introducing stochasticity in the activation functions is effective in handling uncertainty estimation has also been investigated before [1,2,3].\n\nIn terms of uncertainty estimation OOD data, note that BDK [4] and natural-parameter networks (NPN) [1] also claimed satisfactory performance in uncertainty estimation OOD data (see figures in both papers on the toy regression dataset). Therefore it would be interesting to make proper comparison. \n\nThis work is heavily built on the DGP work with help from the reparameterization trick. The author mentioned their differences at the end of Sec. 2, emphasizing that DGP define the function on D^{l-1} dimensions while the proposed method is on 1D. Looking from another perspective, the proposed method combined with the previous linear layer can also be seen as ‘defining the function on D^{l-1} dimensions’. In this sense, I would guess that the different is really on the factorization, i.e., how they factorize the linear plus nonlinear operations.\n\nGiven that this is built on GP, a complexity analysis on running time (both for training and inference) is provided. I am wondering what is the scale for M, i.e., the number of inducing points, for the method to work properly. Also, it would be good to provide and compare the actual running time.\n\nExperiments are either on toy datasets or two very small data UCI datasets. It would be more convincing to evaluate on larger/higher-dimentional data. \n\nIn Figure 3, it is mentioned that BNN underestimate uncertainty between two clusters. I was wondering whether this is true for all types of BNNs. As an example, see Figure 1 of [1], where areas with fewer points do result in higher uncertainty from the PBP and NPN models (which also handles uncertainty in the activation). This comes back to my concern on the experiments. It would also have been stronger if more baselines are included to make a more convincing case.\n\nAnother important experiment that is missing is to include DGP in Figure 3. Would DGP perform similarly or even better than auNN?\n\nAnother problem with the toy dataset in Figure 3 is that: it is hard to say which way of extrapolation is ‘more correct’. Should the model extrapolate like BNN does or like fBNN does. Given that the input is in a continuous space and that inside each cluster the output is indeed increasing with the input, I would say extrapolating as BNN does also makes sense.\n\nThe presentation may be improved. For example, it would be helpful to make it clear what U^l denotes from the beginning. It would also be good to better structure Sec. 2.\n\n[1] Natural-Parameter Networks: A Class of Probabilistic Neural Networks, NIPS 2016\n[2] Feed-forward Propagation in Probabilistic Neural Networks with Categorical and Max Layers, ICLR 2018\n[3] Sampling-free Epistemic Uncertainty Estimation Using Approximated Variance Propagation, CVPR 2019\n[4] Bayesian Dark Knowledge, NIPS 2015\n\n---------\nAfter rebuttal\n\nAfter the discussion phase, I agree that with R3 that in cases such as with periodic patterns, auNN would be a better way to incorporate such knowledge. In this case, the paper would be stronger if related experiments on such patterns are included, to highlight what the key difference between auNN and typical BNN.\n\nGap uncertainty is indeed a real problem in BNN, and I am glad that auNN performs well on this, this is also why I would like to see more comparison to other baselines (e.g., NPN) with good performance on gap uncertainty, to better gauge its improvement/capability (e.g., see https://arxiv.org/pdf/1611.00448.pdf Figure 1(right) where there is also a gap in the middle, although not as large as the example from the auNN paper).\n\nIf this paper is accepted, it would be great if the author could include R3’s summary during the discussion into the final version (see below) as well as proper BNN baselines, which I think will be quite helpful in positioning the current work:\n\n‘Typical BNNs use stochastic weights and deterministic activations, while the authors’ model (auNN) uses deterministic weights and stochastic activations. You are definitely correct in asserting that uncertainty in weights manifests as uncertainty in activation functions. Directly modeling activation function uncertainty however provides distinct advantages, has not been studied carefully before.’\n\nExperiments mentioned in the first paragraph would also be a good addition to make the paper’s point.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "0aav7Vblyqa", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors capture uncertainty in deep networks by employing stochastic activation functions but deterministic network weights.  They place a Gaussian process (GP) prior over activation functions, propose a customized GP kernel, and demonstrate that activation uncertainty leads to both more sensible extrapolations and better-calibrated interpolation uncertainties when compared to standard Bayesian neural networks (BNNs)  with weight uncertainty and deterministic activations. \n\nThe observation that BNNs have too many parameters for meaningful posterior inference is widely acknowledged; modeling activation uncertainty instead of weight uncertainty is an interesting approach for alleviating this issue.  While GP based activations have been considered before, the paper does a particularly good job of executing on the idea and clearly demonstrating its advantages. The experiments are well thought out and nicely illustrate the benefits afforded by the paper's contributions.  Overall, I  enjoyed reading this paper and only have minor quibbles.\n\n1. As with any GP based approach computational scalability is a concern. While it is reassuring that only a modest number of inducing points are needed, the complexity of one epoch: $O(NM^2(D_1 …+ D_L))$ suggests that it would be challenging to scale auNN to deeper / wider networks. Section 3.4 demonstrates scalability to large data but it does not demonstrate scalability to large architectures. It would be good to include a discussion of this issue and how the authors envision scaling the approach to more standard deep learning architectures. Table 3 should include timing numbers for both standard variational BNNs and functional BNNs (even if they are not competitive in terms of performance). Having these numbers would help guide follow up work interested in replacing GPs with (potentially more scalable) parametric function approximations.\n2. The choice of the GP kernel seems to have a large effect on the learned activations (Figure 8). It would be interesting to consider uncertainty over GP kernels as opposed to a-priori fixing all activations to be drawn from GPs with the same kernel and potentially allow different layers or different units in a layer to prefer activations drawn from GPs with different kernels.\n3. I found the sentences (section 3.1, page 6, last paragraph) highlighting differences between a deepGP with an additive kernel and auNN confusing. I think the point being made is that in auNN different nodes in layer $l+1$ use a *shared* set of distinct $D_l$ functions, with each node weighting these functions differently. While in DGP there are no shared functions. From the text it almost sounds like that in auNN functions in layer $l$  (all $D_l$ of them) are the same, which doesn’t seem to be the case. \n4. The observation that many BNN inference techniques struggle with appropriate “gap” uncertainty was concurrently made by Foong et al., and Yao et al., https://arxiv.org/pdf/1906.09686.pdf ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents  a new approach to model uncertainty in DNNs, based on deterministic weights and simple stochastic non-linearities, where the stochasticity is encoded via a GP prior with a triangular kernel inspired by ReLu. The empirical results are promising. The comments were properly addressed. Overall, a good paper.
Decision: Accept (Poster)