Bayesian Neural Networks with Variance Propagation for Uncertainty Evaluation
Authors: Yuki Mae
Wataru Kumagai
Takafumi Kanamori
Venue:  (2021)
Keywords: uncertainty evaluation
sampling-free method
variance propagation
LSTM
out-of-distribution

Abstract: Uncertainty evaluation is a core technique when deep neural networks (DNNs) are used in real-world problems. In practical applications, we often encounter unexpected samples that have not seen in the training process. Not only achieving the high-prediction accuracy but also detecting uncertain data is significant for safety-critical systems. In statistics and machine learning, Bayesian inference has been exploited for uncertainty evaluation. The Bayesian neural networks (BNNs) have recently attracted considerable attention in this context, as the DNN trained using dropout is interpreted as a Bayesian method. Based on this interpretation, several methods to calculate the Bayes predictive distribution for DNNs have been developed. Though the Monte-Carlo method called MC dropout is a popular method for uncertainty evaluation, it requires a number of repeated feed-forward calculations of DNNs with randomly sampled weight parameters. To overcome the computational issue, we propose a sampling-free method to evaluate uncertainty. Our method converts a neural network trained using the dropout to the corresponding Bayesian neural network with variance propagation. Our method is available not only to feed-forward NNs but also to recurrent NNs including LSTM. We report the computational efficiency and statistical reliability of our method in numerical experiments of the language modeling using RNNs, and the out-of-distribution detection with DNNs. 

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BAYESIAN NEURAL NETWORKS WITH VARIANCE PROPAGATION FOR UNCERTAINTY EVALUATION
Anonymous authors Paper under double-blind review
ABSTRACT
Uncertainty evaluation is technique when deep neural networks (DNNs) are used in real-world problems. In practical applications, we often encounter unex- pected samples that have not seen in the training process. Not only achieving the high-prediction accuracy but also detecting uncertain data is significant for safety- critical systems. In statistics and machine learning. Bayesian inference has been exploited for uncertainty evaluation. The Bayesian neural networks (BNNs) have recently attracted considerable attention this context, as the DNN trained using dropout is interpreted as a Bayesian method. Based on this interpretation, several methods to calculate the Bayes predictive distribution for DNNs have been devel- oped. Though the Monte-Carlo method called MC dropout is popular method for uncertainty evaluation, requires a number of repeated feed-forward calcu- lations of DNNs with randomly sampled weight parameters. To overcome the computational issue, we propose a sampling-free free method to evaluate uncertainty. Our method converts neural network trained using dropout the corresponding Bayesian neural network with variance propagation. Our method is available not only to feed-forward NNs but also to recurrent NNs including LSTTM. We report the computational efficiency and statistical reliability of our method in numeri- cal experiments of language modeling using RNNs, and the outto--distribution detection with DNNs.
INTRODUCTION
Uncertainty evaluation a core technique in practical applications of deep neural networks (DNNs). As example, let consider the Syber-Physicall Systems (CPS) such as the automated driving system. the past decade, machine learning methods are widely utilized to realize the environment perception and path-planing components in the CPS. In particular, the automated driving system has drawn a huge attention as a safety-critich and real-time CPS (NITRD CPS Senior Steering Group, 2012 Wing 2009). In the automated driving system, the environment perception component is built using baNeb predictive models.
In real-world applications, the CPS is required to deal with unexpected samples that have not seen in the training process. Therefore, not only achieving the -ree-preeiition accuracy under the ideal environment but providing uncertainty evaluation for real-world data is significant for safety-critical systems (Henne et al., 2019). The CPS should prepare some options such as the rejection of the recommended action to promote the user's intervention when the uncertainty is high. Such an teractive system is necessary to build fail-safe systems (Varshney & Alemzadeh 2017 Varshney 2016).
On the other hand, the uncertainty evaluation is useful to enhance the efficiency of learning algo- rithms, i.e., samples with high uncertainty are thought to convey important information for training networks. Active data selection based on the uncertainty has been studied for long time under the
name of active learning (David et al. 1996. Gal et al. 2017 Holub et al.) 2008 Li Guo! 2013 Shui al. 2020).
In statistics and machine learning, Bayesian estimation has been commonly exploited for uncertainty evaluation (Bishop| 2006.). In the Bayesian framework, the prior knowledge is represented as the prior distribution of the statistical model. The prior distribution is updated to the posterior distribu- tion based on observations. The epistemic model uncertainty is represented in the prior distribution,

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and upon observing data, those beliefs can be updated in the form of posterior distribution, which yields model uncertainty conditioned on observed data. The entropy or the variance is represen- tative of uncertainty measures (Cover & Thomas 2006). For complicated models such as DNNs, however, a direct application of Bayesian methods is prohibited as the computation including the high-dimensional integration highly costs.
In deep learning. Bayesian methods are related to stochastic learning algorithms. This relation is utilized approximate the posterior over complex models. The stochastic method called dropout is a powerful regularization method for DNNs (Srivastava et al. 2014). In each layer of the DNN, some units are randomly dropped in the learning using stochastic gradient descent methods. Gal & Ghahramani (2016a) revealed that the dropout is interpreted as the variational Bayes method. Based on this interpretation, they proposed a simple sampling method of DNN parameters from the approximate posterior distribution. Furthermore, the uncertainty of the DNN-based prediction is evaluated using the Monte-Carlo (MC) method called MC dropout.
While the Bayesian DNN trained using dropout is realized by simple procedure, the computational overhead is not ignorable. In the MC dropout, dropout is used also at the test time with a number of repeated feed-forward calculations to effectively sample from the approximate posterior. Hence, the naive MC dropout is not necessarily relevant to the system demanding the real-time response.
In this work. we propose sampling-free free method to evaluate the uncertainty of the DNN-based prediction. Our method is computationally inexpensive comparing to the MC dropout and provides reliable uncertainty evaluation. In the following, we will first outline related works. Section is devoted to show the detailed formulae of calculating the uncertainty. In our method, an upper bound of the variance is propagated in each layer evaluate the uncertainty of the output. We show that the our method alleviates the overconfident prediction. This property is shared with scaling methods for the calibration of the class pooss--robbbility on test samples. In Section 4 we study the relation between our method and scaling methods. In Section [5, we demonstrate the computational efficiency and statistical reliability of our method through some numerical experiments using both DNNs and RNNs.
2 RELATED WORKS
The framework of Bayesian inference is often utilized to evaluate the uncertainty of DNN-based pre- dictions. In Bayesian methods, the uncertainty is represented by the predictive distribution defined from the posterior distribution of the weight parameters. MMcKKy (1992) proposed a simple approxi- mation method of the posterior distribution for neural networks, and demonstrated that the Bayesian method improves the prediction performance on classification tasks. Graves (201T) showed that the variational method efficiently works to approximate the posterior distribution of complex neural network models.
There are many approaches to evaluate the uncertainty of modern DNNs (Alex Kendall & Cipolla 2017 Choi et al. 2018:; Lu et al., 2017 Le et al. 2018). We briefly review MC-based methods and sampling-free methods.
Monte-Carlo methods based on Stochastic Learning: The randomness in the learning process can be interpreted as a prior distribution. In particular, the dropout is landmark of stochastic reg- ularization method to train DNNs (Srivastava et al.] 2014). Gal & Ghahramani (2016a) proposed a simple method to generate weight parameters from the posterior distribution induced from the prior corresponding the dropout regularization. The predictive distribution is approximated by the MC dropout, which compute the expected output over the Monte-Carbe sampling of the weight parameters. Gal & Ghahramani| (2016b) reported that the MC dropout efficiently works not only for feed-forward DNNs but for recurrent neural networks (RNNs). Another sampling based method is the ensemble-based posteriors with different random seeds (Takshminarayanan et al. (0177). How- ever, the computation cost is high as the bootstrap method requires repeated training of parameters using resampling data.
Sampling-free methods:
Though the MC dropout is simple and practical method evaluate the uncertainty, a number of feed-forward computations are necessary to approximate the predic- tive distribution. Recently, some sampling-fres methods have been proposed for the uncertainty
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evaluation, Probabilistic network is a direct way to deal with uncertainty. The parameters of the probabilistic model, say the mean and the variance of the Gaussian distribution, are propagated in probabilistic neural networks. Then, the uncertainty evaluation is given by a single ferward calculation. [Choi et al. (2018) used the mixture of Gaussian distributions as a probabilistic neu- ral network Wang al. (2016) proposed natural-parameter networks as a class of probabilistic neural networks based on exponential families. For a given input vector, the network outputs the parameters of the distribution. For the recurrent neural networks, [Wwang et al. (2019) proposed variant of the natural-pant parameter networks. Instead of parameters of statistical models, Wu et al. (2019) developed a sampling-fres free method propagate the first and second order moments of the posterior distribution.
Sampling-free methods can evaluate the uncertainty with a one-pass computation for neural net- works. However, specialized learning algorithms are required to train the prohabilistic networks. Our method is applicable to DNNs and RNNs trained by common learning methods with the dropout. Postels al. (2119) and Shekhovtsovv & Flach (2019) proposed similar methods that prop- agate the uncertainty of the network to the output layer. Differently from the past works, our method takes the upper limit of the correlations among the inputs at the affine layer into account when the uncertainty is evaluated. In addition, we show that our method efficiently works even for RNNs.
3 UNCERTAINTY EVALUATION WITH VARIANCE PROPAGATION
In this work, we assume that we can access to the weight parameters in the DNN and the dropout probability in the training process. As the variance is common measure of uncertainty, propose a variance propagation algorithm for the trained DNN.
Implementation of our method called 2vpbnn is presented in Section A the appendix. In method, we need only the DNN or RNN trained using dropout. Unlike various kinds of probabilistic NNs, we do not need any specialized training procedure evaluate the uncertainty. This is great advantage for our implementation. Furthermore, the representative values of the predictive distribu- tion, i.e. the mean and variance, are obtained by a one-path feed-forward calculation. Hence, we can circumvent iterative Monte-Carlo calculations.
3.1 UNCERTAINTY IN AFFINE LAYER
Let us consider the output of the affine layer y = Wx + b for the random input x, where W (Wi)) fx and b (b:) E R. Suppose that the random vector has the mean vector E[x] and the variance covariance matrix (Ex)is = Cov xz) for i.j -,...., m. Then, the mean vector Eyy] and the variance covariance matrix Ey of y are given by [yyy WE[x b and Σ, WT.
As the estimation of the full variable-covariincee matrix is not necessarily reliable, we use only the variances of each x; and an upper bound of the absolute correlation coefficient to evalu- ate the uncertainty. For W = (Wi)), the variance Var[y,] is Varly,] = Σ, W Var[x,] + Σ E Wi Wiy Cov xy). Suppose the absolute correlation coefficient among X1...., Xm is bounded above by Ps 0 S p < 1. Using the relation between the correlation and variance, we have
Varly,] S Σ Varlxil+
Σ 55''
WollWasly Var(x;) Var(x;)
= p) Σ W 2 Var[x,]
Σ (iily V
i= i=,,..,,f
Under the independent assumption, i.c., = 0, the minimum upper bound obtained. The predic- tion with a small variance leads to overconfident decision making. Hence, the upper bounding of the variance is important to build fail-safe systems. A simple method of estimating p is presented in Section 3.5
Using the above formula. the mean and an upper bound of the variance y are computed using the mean and upper bound of the variance of In this paper, such computation is referred as
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the Variance Propagation or VP for short. Let define the variance vector of the m-dimensional random vector x = (X1,...,, .m.....))............. ) E IR" by Var[x] = (Var[x]].... [..]...... Var[xm] E R"". Furthermore, denote the concatenated vector of the mean and variance of its approximation as ((z), i.e., U(z)) = (E|2), Var(z)). The VP at the affine layer is expressed by the function Taft.
U(y) = (m, v) = Tan (L(()))
(2)
where m W + b E I and each element of v E m is defined by equation
The average pooling layer, global average pooling layer (Lin al. 2013). and the batch normaliza- tion layer (loffe& & Szegedy 2015) are examples of the affine layer. Hence, the VP of the affine layer also works to evaluate the uncertainty of these layers.
The distribution of is well approximated by the univariate Gaussian distribution if the correlation among x is small (Wang& & Manning, 2013 Wu et al. 2019). Based on this fact, the uncertainty of y, can be represented by the univariate Gaussian distribution (E Var[y,)). In our method, the variance Varly,] of the approximate Gaussian is given by the variance vo in equation 2
3.2 OUTPUT OF DROPOUT LAYER
Let us consider the uncertainty induced from the dropout layer (Srivastava et al. 2014). The dropout probability is denoted by p. In the dropout layer, the m-dimensional random imput vector x (X1...., Xm) is transformed by the dement-wise product the i.i.d. Bernoulli random variables, i.e., Bernoull(p). As x and d are independent, the VP in the dropout layer is given by (E(z), Var(zl) Tarop(U (xx),, where [E[z] pE|xx and Var[z] Var[[x] + p(1 p){"
xd, where d (d1,..., .....d)) is
According to the Bayesian interpretation of the dropout revealed by Gal et al. (2017), the approx- imate posterior distribution of the output from the affine layer trained using dropout is given by the distribution of the random variable y, = Σø= Wixxydz + b,, di...., dm ~ Bernoulli(p). The mean and the variance of y, satisfy Ely] = p> +b and Varly,] S (1-p) E, [Wa)) [Vr[[xdd]] Since the stochastic input and the weight parameter in the dropout layer
O(E, [Wiy/ / /Var[x,d]) are independent, one can exactly calculate the variance of the product using each expectation and variance. The VP at the affine layer with the dropout is given by the composite function,
(m, v) Taft o Tarro (U(x).
The uncertainty of y; then represented by the Gaussian distribution N(m,, ez). similar formula is found in the uncertainty evaluation of the LSTM unit in Section 3.4 with the explicit expressions.
3.3 UNCERTAINTY VIA ACTIVATION FUNCTIONS
The nonlinear activation function is an important component of neural network models in order to achieve high representation ability and accurate prediction (Cybenko) 1989). The ReLU, sigmoid function, and their variants are common activation functions. In several works, the expectation and the variance of the output from activation functions have been calculated (Frey & Hinton, 1999. MacKay 1992. Daunizeau 2017). Let us introduce the transformed distribution by the Rel.U and sigmoid function.
The ReLU function is defined by y = max(x, 0). For X; ~ (E(x,), Var(x:]). the ex- act expectation and variance of y are expressed by the probability density d and the cu- mulative function & of the standard Gaussian distribution (Frey & Hinton| 1999 Wu et al. 2019 : Eyy] = (Ex] Var(x]) /arrxx) + /Vaa(x]o(B[xx// / //ar[x)) and Varly) = [E[x]² + Var(x] (E|x)) /E/xi//Varixx) /arl(x)
+ E[x] /Var(x|j(Eix]// (itxs//(Ei////aarrra /Var[x]) - Ely? using the element wise opera- tions for the two vectors E[x] and Var(x].
The sigmoid function is defined by y, s(x;) = 1/(1 + xx)). For N(E[x,). Var(x;]), MacKay (1992) and Daunizeau (2017) derived the approximate expectation and variance of y,
Ely] R3
E(x]
Varly) R s( V
Ex (1 s( Vi+evarx
Ex (1 VI+eVar(x where the
V1+eVarix x)
constant e depends the approximation method. The common choice is T/8 2s 0.393, while
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Daunizeau 2017) found 0.368 based numerical optimization. In the same way, one cal- culate approximate expectation and variance of tanh(y). The VP at the activation layer is expressed by Uly] Tact (2([x)). where the operation Tact depends on the activation function. The output U[y] is defined by the above expectation and variance.
In the multiclass classification problems, the softmax function is commonly used the last layer in DNNs. However, the expectation of the softmax function does not have analytic expression under the multivariate Gaussian distribution, (2017) utilized the approximate expectation of the sigmoid function to approximate the expected softmax output. However, the variance of the softmax function was not provided. In this paper, we interpret the multiclass classification problem as the multi-label problem and at the last layer, we use the sigmoid functions as many as the number of labels. Given the transformations 7 m s(7e), = 1,...,G G at the last layer for the classification with G labels, the prediction is given by the label that attains the maximum value of s.z2). The advantage of this replacement is that the reliable evaluation of the uncertainty is possible for the sigmoid function shown above. In numerical experiments, we show that the multi-label formulation with several sigmoid functions provides a comparable prediction accuracy as the standard multi-class formulation using the softmax function, while it also gives reliable uncertainty evaluation.
3.4 LSTM UNIT WITH DROPOUT
The uncertainty evaluation of the Recurrent Neural Networks (RNNs) is important task as the RNNs are widely used in real-world problems. This section is devoted to the uncertainty propagation in the LSTM unit when the dropout is used to train the weight parameters (Gal & Ghahramani, 20166). According to Greff et al. (2017), the standard form of the LSTM unit is defined by
U Wi
Ü
ÜR
U
(i f g o) (s 8 tanh 8) (hi-
xx)
Wr
W g
W.
h = o (c)) C, = fc--1 + ii
using the sigmoid function 8, where the multiplication of two vectors is the element-wise operation and is the composition of the linear layer and the activation function, i.e., i shh,- + x, Wi), = tanh(h,-11g x, Wg), etc. The matrices W's and Ú'S are the input weights and recurrent weights. respectively. The vectors, i,f. g. and denote the input gate, forget gate, candidate vector, and output gate. The cell state c and the hidden state h, retain the long and short term memory.
Here, U. and W, are regarded as random matrices distributed from the posterior distribution in- duced from the dropout using Bernoull(pp). Hence, each row of U. and W. are set to the null row vector with probability 1 - p. When the tied dropout is used for LSTM, the same rows of all U, randomly dropped and the same rule is applied to W.. On the other hand, in the untied dropout layer, the dropout is separately executed for each U_ and W.. Detail of the tied and untied dropout is found in Gal & Ghahramani (2016b).
Let us consider the map from U(;,-, 1t-1) to (h, c.). The map depends on the data x,. Since the computation in the LSTM with the dropout is expressed as the composite function of the dropout layer, affine layer and the activation function, we have
U(i, f.g.g.. g.g. = Tact o Taft o Tarop (M(,,
Hence, the mean and variance vectors of h, and C2 are obtained from those of i, g, o and c--1- This computation is shown below. We need an appropriate assumption to calculate EEcr] and Var as we do not use the correlations. The simplest assumption is the independence of random vectors. When f, ct-1, i and g are independent, we obtain
Elca] = E[fc,-1] + E E(ig] = E[ E[j
(3)
Var[c:] = [ar[f]Var[cr_]]] + [ar[f] + Var(c--
Var(i]
(4)
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This is the VP for the cell state vector Ct-1 in the LSTM. Likewise, the VP for h, is obtained. The above update function to compute the uncertainty of h, and c, from i, f. g, o and C₁-1 is denoted by Teell- As a result, we have
U(,, ce) Teall (i(i o), o), Teell Teell(acs Tatt (U(: ((c,)(()))))
This is the VP formula from (ce-1, -(1 to (c,, he). Repeating the above computation with the observed sequence {xr xx one can evaluate the uncertainty of the cell state vectors and the outputs (yr)?=1, where y: h,, t 1,...,7.
Let us consider the validity of the above independence assumption. For given h the conditional independence of i, f, g. and - holds when the untied dropout is used to train the LSTM unit, i.e., the equality p(i, f, g, O, C--1 ct-1(h,-1) 1) - p(c,-1 ( (i.f.go)) p(s(h;-1) holds for the posterior distribution. The randomness comes from the Bayesian interpretation of the untied dropout. Here, constant without uncertainty. Then, equation (atitoonnnan a hold
the observation x; is regarded by replacing the mean and variance with conditional expectation and the conditional variance under the condition of h If the variance of h is small, the independence assumption is expected to be approximately valid. When the uncertainty of h is not ignorable, the sampling from the Gaussian distribution representing the uncertainty of h,-1 is available with the formulae E(c,] En, [E(c,]ht- and Varle,] = En, [Var[c,] [1] to compute E(c,) and Var[cr] approximately.
3.5 ESTIMATION OF CORRELATION PARAMETER
When we evaluate the uncertainty in the affine layer, we need to determine the correlation param- eter P in equation T If the correlation of input to the affine layer is not ignored, the upper bound of the variance with an appropriate is used to avoid overconfidence. Hence, the estimation of the parameter p is important. A simple method of estimating appropriate p is to use the validation set as follows.
1. For each candidate of p. execute the following steps.
(a) For each data (x,, yi) the validation set, compute the mean vector m. and variance vector U, of the output of the network for given o; using VPBNN.
(b) Compute the predictive log-likelihood on the validation set,
LLp
Σ logr(y; mi; v:),
E:validation set
where p(y: m, v) is the probability density of the uncorrelated normal distribution with the mean m, and variance Vy for each element.
2. Choose P that maximizes LLp-
Though need to prepare several candidates of the correlation parameter, the computation cost still lower than MC dropout. To evaluate the uncertainty on Ntest test samples, MC dropout with T samplings requires Nusa? feed-forward calculations. The VPBNN with adaptive choice of p needs approximately 2N KNoatKoor + 2Ntest feed-forward calculates, where Nval is the number of the validation computation of both mean and variance. Usually, Keor is much less than T and Nval is not extremely large in comparison to Nuest- In practice, we do not need a large validation set to estimate p. as shown in numerical experiments. Hence, VPBNN with adaptive p is computationally efficient than MC dropout. If distinct correlation parameters are used in each affine layer, the computation cost becomes large. In numerical experiments, we find that the uncertainty evaluation using the same p in all the affine layers works well.
and Keor is the number of candidates of p. The factor comes from the
4 SCALING METHODS FOR CALIBRATION AND VARIANCE PROPAGATION
The variance propagation is regarded as a calibration based on the uncertainty. There are some scaling methods for the calibration. Platt scaling (Platt 1999) is classical calibration method for multiclass classification problems, and the temperature scaling (Guo et al. 2017 Ji et al. 2019 is a simplified method of the Platt scaling.

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Let us consider the conditional probability function Pr(y|z) = ctv E e (21...., 2m) E Th and y = 1, ,.... m. The temperature scaling of Pr(y[z) is given by Pr(ylz/T), where T > 0 is the temperature parameter. Usually I'' is greater than one, and it softens the class probability. The Platt scaling of Pr(y)=) =)) is defined by the scaling of such that Pr(y W b). where W is a diagonal matrix and b is a m-dimmesional vector. The Platt scaling is the coordinate- wise calibration, while the temperature scaling the homogeneous scaling for the feature vector z. Another expansion of the temperature scaling is the bin-wise temperature scaling (Ji et al.) 2019). In the bin-wise scaling, the sample space are divided into K bins. The label probability is calibrated by =r(s)=/Tk(s) in which k(z) is the index of the bin including the sample z, and Tx >> 0 is the temperature for the calibration at the k-th bin. Each T3 is determined from validation data in the k-th bin. Intuitively, an extremely large label probability tends to yield an overconfident prediction. At the point z having the large maximum probability, the large scaling parameter Tę is introduced to soften the overconfidence at the region including 2.
for
The calculation of the VP at the sigmoid activation layer for the multi-label classification is given by s(2)) م s (E[zz)/ /I+ eVar(z,)), j = L...., m. When the uncertainty of the random vector is not taken into account, the prediction using (E[zy])), Comparing to s (E[z,]). the uncertainty defined by the variance Var[z,] works as a coordinate- -wise calibration like the Platt scaling. If the variance is isotropie, the scaling does not change the ranking of the label probability like the temperature scaling.
1, 1..... is apt to be overconfident.
In the variance propagation using the Taylor approximation (Postels al. 2019), the class prob- ability is calculated as the standard manner without calibration, while the variance is propagated along the layers in DNNs in order evaluate the uncertainty. Hence, the calibration effect is not incorporated into the naive Taylor approximation method.
EXPERIMENTS
5.1 NUMERICAL EXPERIMENTS ON SYNTHETIC DATA
We assess the uncertainty computed by the VPBNN and the Taylor approximation using synthetic data. Let us consider the uncertainty evaluation for regression problems. The target function is the real-valued function on one-dimensional input space shown Figure [ We compared three methods; MC dropout with 1000 samples from the posterior distribution associated with the dropout, Taylor approximation, and VPBNN with adaptive p and several fixed p's. The architecture of the NN model is shown in Figure 4 of the Appendix. The results are presented in Figure [ The Taylor approximation and the VPBNN with p = 0 tends to be overconfident and the VPBNN with p 0.15 gives similar result to the MC dropout. We find that the adaptive choice of p can avoid the overconfidence, while providing a meaningful result.
In Section C in the appendix, we present the uncertainty evaluation for RNNs. Likewise, find that the appropriate choice of p relaxes the overconfident prediction and that the adaptive p provides meaningful result well MC dropout.
Overall, the VPBNN with appropriate p provides similar results to the MC dropout. As the VPBNN needs only the one- mne-path calculation well as the VP using Taylor approximation, the computation cost is much less than the MC dropout. The adaptive choice of using the validation set efficiently works produce similar result as MC dropout.
Further numerical results are presented in Section B of the appendix. The Taylor approximation for the uncertainty evaluation proposed by Postels et al. 2019) also leads to a computationally efficient single-shot method to compute the uncertainty. However, we find that Taylor approximation tends to lead overconfident result compared to our method in the present experiments.
5.2 RNN FOR LANGUAGE MODELING
We report numerical experiments of language modeling problem. The problem setup is the same as the problem considered by Zaremba et al. (2014) and Gal & Ghahramani (2016b). We use Penn Treebank, which is a standard benchmark in this field. In the experiments, the LSTM consisting of two-layers with 650 units in each layer is used. The model architecture and most of hyper parameters
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MC mate 1900
0.35
anad
100
MC dropout
VPBNN (p = 0.35 adaptive)
Taylor approx.
Indiper berbiroce
Upponnede -00
Uppanmode
1 mm
VPBNN (p- = O fixed)
VPBNN (p 0.15 fixed)
VPBNN (p = 1 fixed)
Figure 1: The uncertainty for ferwado NNs is evaluated. The solid line is the target function and the training samples (x) are plotted. For each method, the uncertainty is depicted as the confidence interval.
Single model perplexity for the Penn Treebank language modeling task is presented. The asterisk (+) denotes the best perplexity on the test set for each dropout setting. The asterisk (o) means the perplexity reported in (Gal & Ghahramani] 2016b).
Table 1:
untied weights
tied weights
Validation
Test
Validation
Test
MC d
78.6 + 0.1
79.0 = 0.1
standard dropout approx. Taylor approx. VPBNN: =
81.9 + 0.2
79.7 + 0.1
81.8 + 0.2 82.63
79.7 + 0.1
82.65
79.34
79.67
81.30
78.05*
81.09 78.20"
set to those used by Gal Ghahramani (2016b). Figure 7 the appendix shows the RNN and the converted VPBNN. The weight decay parameter is set to 10-7 according to the code in Github provided by the authors of Gal & Ghahramani (20165), the parameter was explicitly written in their paper.
The results are shown in Table The prediction performance evaluated by the perplexity on the test set. In the table, the standard dropout approximation propagates the mean of each approximat- ing distribution as input to the next layer (Gal & Ghahramani 2016b). As the Taylor approximation computes the mean of the output without using the variance, it must provide the same result as the standard dropout approximation. In our experiment, both methods produced almost identical per- plexity scores. This result means that we approximately reproduced the numerical results reported in the past papers. The MC dropout and the VPBNN with o = achieved a lower perplexity than the others. Our method using only one-path calculation can provide almost the same accuracy as the MC dropout that requires more than 1000 times fced-forward calculations the output values.
Note that the VPBNN is not the approximation of MC dropout. Both MC dropout and VPBNN an approximation of the posterior distribution, though MC dropout with a sufficient number of feed- forward calculations tends to provide satisfactory result. The numerical experiments indicates that the number of feed-forward calculations in MC dropout is not sufficient for this task.
5.3 OUT-OF-DISTRIBUTION DETECTION
Let us consider the out-of-die d--istribuutin detection problem. The task is to find samples whose distri- bution is different from that of the training samples. The uncertainty of samples is evaluated for this task. First of all, the neural network is trained using Fashion-MNIIS dataset (Xiao et al. 2017). Then, several methods for uncertainty evaluation used detect samples from non-training
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Results of out-of-distribution detection are presented. "Test accuracy" is the prediction accuracy computed on the test set of Fashion-MNISTI For each pair of training domain (Fashion- MNIST) and non-training domain (MNIST, EMNIST, Kannada or Kuzushiji), the averaged AUC score computed using 30 random seeds is shown with the standard deviation. The asterisk ** (resp.
Table 2:
+) denotes the highest (the hechne-highest) AUC for each non- non-training dataset.
AUC score for each non-training domain
Test accuracy
MNIST
EMNIST
Kannada
Kuzushiji
MC100
0.923 + 0.002
0.904 0.019
0.928 = 0.012
0.897 0.005
0.965 0.006
MC2000
0.923 + 0.002
0.916 = 0.022*
0.937 + 0.013*
0.909 + 0.015*
0.971 + 0.006
Taylor approx.
0.923 + 0.002
0.775 = 0.028
0.833 + 0.017
0.766 + 0.023
0.860 + 0.017
VPBNNadaptive
0.923 + 0.002
0.923 + 0.026
0.946 + 0.016"
0.916 + 0.020
0.981 + 0.005
datasets. In this experiments, we use MNIST (Lecun et al., 1998), EMNIST-MNIST (Cohen et al.) 2017). Kannada (Prabhu 2019), and Kuzushiji (Clanuwat al. 2018) non non-training datasets. The detection accuracy of each method is evaluated by the AUC measure on the test dataset.
We compared MC dropout with 100 sampling (MC100) 2000 sampling (MC2000), Taylor ap- proximation, and VPBNN with adaptive p. The network architecture is the CNN shown in Figure B of the Appendix. At the output layer, the multi-label sigmoid function is used. Numerical results for the softmax function are reported in Section of the appendix. In the training process, Adam optimizer with early stopping on validation dataset is used. The 60k training data was divided into 50k training data for weight parameter tuning and 10k validation data for hyper-paramete tuning.
We confirmed that all methods achieve almost the same prediction accuracy the test data of The result is shown in the "Test accuracy" column in Table 2 The prediction is done by using the top-ranked labels. Though MC dropout and VPBNN tend relax the overcon- fident prediction, the calibration does not significantly affect the label prediction accuracy of this problem.
For the uncertainty evaluation, we used two criteria. One is the entropy computed from the mean value of the output, [H[[ -1 E [Ely,] log Ely, + +1 - (1-1-By,) 1og(1- Ely,)), for the output y (y..... Ymm of the NN, and the other is the mean-standard deviation (mean- std) (Kampffmeyer et al. 2016 Gal et al. 2017) that is the averaged standard deviation, i.c., (y)) = m Din / Vaarly.l In Section of the appendix, we report the results of other uncer- tainty measure using not only sigmoid function but the softmax functions. Overall, we find that the mean-ster standard deviation outperforms the entropy measure in the dittr--distributionn detection. This is because the uncertainty is rather related to the variance than expectation of the output value. Table 2 shows the results of the mean-standart deviation. In the adaptive VPBNN of this task, the estimated correlation parameter P approximately ranges from 0.0 0.0005. Hence, VPBNN with the independent assumption, i.c., = O, also works well. Taylor approximation method fails detect the sample from non-training distribution, This is because the estimation accuracy of the variance is not necessarily high as shown in Section B of the appendix.
Let us consider the computation cost. our experiments, the computation time of MC dropout with 100 sampling is 15.6(sec] = 28.3 [ms] on average for the uncertainty evaluation of 10K test samples. For MC dropout with 2000 sampling, the computation cost is approximately 20 times higher since it is proportional to the number of sampling. For the adaptive VPBNN, the computation time is 175(ms) + 31.7 [ms] including the adaptive choice of p from 10 candidates when 10K validation samples used. VPBNN with adaptive e provides comparable performance to MC dropout using sufficient number of sampling while keeping much less computation cost. As discussed in language modeling, the number of feed-forward calculations in MC dropout is considered not sufficient for this task.
6 CONCLUSION
We developed a sampling-free free method for uncertainty evaluation. Our method requires only a one- path calculation of DNNs or RNNs, while the MC-based methods need thousands of ferd-forward calculations to evaluate the uncertainty. In the numerical experiments, we show that our method provides more reliable results than existing sampling methods such the Taylor approximation.

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A IMPLEMENTATION OF NN2VPBNN
Using Keras (Chollet et al., 2015). we implemented nn2vpbnn that converts the DNN or RNN trained using dropout to the corresponding Bayesian neural network with variance propagation, i.e., VPBNN. Each layer of the VPBNN propagates the mean and the variance from the input layer to the output layer. The dropout layer in the VPBNN the operation defined by Tarop- The output of VPBNN provides the mean and variance of the predictive distribution for the original BNN, In each layer, the VPBNN with zero variance provides the output without uncertainty. Figure 2 is an example of the VPBNN for the LSTM model. In the VPBNN, the input and output size is doubled after the dropout layer. This is because information on the uncertainty is added as the variance. Similarly, nn2vpbnn works to convert CNN models to the corresponding VPBNN.
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At the affine layer, our method allows the users tune the degree of the dependence among input variables by setting the parameter p. This parameter is determined according to the balance between the validity of the independence assumption and the safety required to the system. On the numerical experiments shown in Figure 6, setting = O tends to produce an over-confident prediction, while the parameter setting with p corresponds to the least-confident prediction.
In the BNN, the dropout layer is the main source of the uncertainty. Besides the dropout layer, the batch normalization layer also has Bayesian interpretation (Teye al., 2018). Gaussian Dropout layer and Gaussian Noise layer also yield the uncertainty to the output of the layer, while these layers do not necessarily have the Bayesian interpretation. One can easily implement the variance propagation for these layers as part of mn2vpbmn
In our method, we utilize neural networks trained with dropout. Unlike various kinds of probabilistic NNs, we do not need any specialized training procedure to evaluate the uncertainty. This is great advantage for our implementation. Furthermore, the representative values of the predictive distri- bution, i.e. the mean and variance, are obtained by the one-path feed-forward calculation. Hence, we can circumvent iterative Monte-Carlo calculations. These advantages are shared with the Taylor approximation method by Postels et al. (201)).
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Figure Left: LSTM. Right: the corresponding VPBNN.
B SUPPLEMENTARY ON APPROXIMATION ACCURACY
Let us consider the numerical accuracy of VPBNN and Taylor approximation. The target is to compute the mean and variance of output value y f(Wx + b) for random input vector x. The ReLU or sigmoid function is used the activation function f. These two methods are compared with the Monte-Carle method with sufficiently many samples. The distribution of input variable is the Gaussian distribution the uniform distribution. The mean (resp. the standard deviation) of input variable varies in the interval [-10, 10] (resp. [0.1, 10)) for both Gaussian distribution and the uniform distribution.
The absolute error of VPBNN and Taylor approximation from the MC method is shown in Figure 3 The horizontal axis and vertical axis denote the variance and mean of the input distribution, respec- tively. In numerical experiments, VPBNN achieved higher accuracy than Taylor approximation. We find that Taylor approximation tends yield extremely large variance even when the sigmoid func- tion is used as the activation function. Overall, VPBNN has a preferable property compared to the Taylor approximation.
The architectures of the NN and RNN used in Section 5.1 are shown in Figure and
C NUMERICAL EXPERIMENTI ON SYNTHETIC DATA: RNN
Numerical experiments of RNNs using synthetic data are also conducted. The RNN model is shown in Figure 5 of the Appendix. We evaluated the uncertainty of Bayesian RNN trained with dropout,
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Gaussian imput to ReLU function
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The mean and variance of the output from the ReLU or sigmoid function for the Gaus- sian or uniform inputs are computed using VPBNN and Taylor approximation. Absolute errors of VPBNN and Taylor approximation the MC method presented.
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Figure 4: The architecture of NN to learn the regression function Scction 5.1 ReLU is used at the Dense of the middle layer and the identify function is used at the Dense in the last layer.
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Figure 5: The architecture of RNN used in Section 5.1 The sigmoid function and tanh function used as the activation function.
MC dropout.
VPBNN (p 0.4 adaptive)
Taylor approx.
VPBNN (p 0 fixed)
VPBNN (p 0.5 fixed)
VPBNN (p 1 fixed)
Figure 6: The uncertainty of Bayesian RNN is evaluated. The solid line is the target function. For each method, the uncertainty depicted as the confidence interval. Upper panels: the target and estimated regression function with its uncertainty. Lower panels: the standard deviation of output value at each input value.
where the input is 30 length sequence. The results shown in Figure Again the Taylor approxi- mation and the VPBNN with the independent assumption (p 0) tend to yield overconfident results compared to the MC dropout. We find that the VPBNN with adaptive p provides similar results to the MC dropout.
D SUPPLEMENTARY OF RNN FOR LANGUAGE MODELING
The architectures of RNN used Section 5.2 shown in Figure
E SUPPLEMENTARY OF OUT-OF-DISTRIBUTION
Let us consider the out-of distribution detection. First of all. the neural network is trained using Fashion-MNIST (iioo et al. 2017). Then, several methods for uncertainty evaluation used to
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Figure 7: The architecture of RNN used for Language Modeling in Section 5 function and tanh function are used as the activation function.
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detect samples from non- non-training datasets, MNIST (Lecun et al., 1998), EMNIST-MNIST (Cohen al. 2017), Kannada (Prabhu 2019). and Kuzushiji (Clanuwat et al. 2018). The detection accuracy of each method is evaluated by the AUC measure on the test dataset.
The network architecture used in Section 5.3 is the CNN in Figure of of the Appendix. addition to the CNN with the softmax function provided in Keras, we implemented another CNN with multi-label sigmoid functions at the output layer.
In the training process of the NNs. Adam optimizer with early stopping validation dataset is exploited. The 60k training data was divided into 50k training data for the weight parameter tuning and 10k validation data for ayper-paramedt criteria; one is the entropy computed from the mean value of the output, and the other the mean- standard deviation (mean-std) (Kampfimeyer al. 2016
tuning. For the uncertainty evaluation,
used two
Gal et al. 2017) that is computed from
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the variance. More precisely, the entropy defined from the softmax output is
H
2 Ely, log Ely,l.
and the entropy defined from the sigmoid function for multi-label setting is given by
Hy)
EEyy, log Fly, + (1 Ely.)) log(1 EEly)))
The mean-std is defined by
(y)) 1
m m
Varly,].
The results presented in Table g in the appendix. For the out-of-distribution detection, find that the mean-std based method outperforms the other methods using entropy criterion. More- over, our method provides the uncertainty by only one-path ferd-forward calculation, while the MC dropout needs more than hundreds of feed-forward calculations. The Taylor approximation fails to detect the sample from non-tre con-training distribution. This because the approximation accuracy of the Taylor approximation is not necessarily high as shown in Section B
On the other hand, all the methods considered here achieve almost the same prediction accuracy on the test data of Fashion MNIST as shown in Table 4 The prediction is done by using the top-ranked labels. In this experiment, the calibration of the label probability does not significantly affect the ranking of the label probability.
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Table 3: AUC score of Out-of-Distribution detection for MC dropout using 2000 sampling, Taylor approximation, and VPBNN. The standard deviation of the AUC score is calculated using 30 differ- ent random sceds. The asterisk ** (resp. *) denotes the highest (the second-highen AUC for each pair of training and non-traininn non-training dataset.
Training non-training dataset
Method
Uncertainty
Activation func.
AUC
Fashion + MNIST
MC dropout
Entropy
softmax
0.866 = 0.019
sigmoid
0.796 + 0.024
Mean-std
softmax
0.934 = 0.014
sigmoid
0.916 + 0.022
Taylor approx.
Entropy
soltmax
0.718 = 0.026
sigmoid
0.682 = 0.028
Mean-std
softmax
0.728 1 0.025
sigmoid
0.775 = 0.028
VPBNN (adaptive p)
Entropy
sigmoid
0.806 = 0.025
Mcan-std
sigmoid
0.923 = 0.026*
Fashion EMNIST
MC dropout
Entropy
softmax
0.893 + 0.013
sigmoid
0.846 + 0.016
Mean-std
soltmax
0.941 = 0.011
sigmoid
0.937 + 0.013
Taylor approx.
Entropy
softmax
0.791 1 0.019
sigmoid
0.755 = 0.021
Mcan-std
softmax
0.777 = 0.017
sigmoid
0.833 = 0.017
VPBNN
Entropy
sigmaid
0.856 0.016
(adaptive
Mean-std
sigmoid
0.946 + 0.016*#
Fashion Kannada
MC dropout
Entropy
softmax
0.867 1 0.017
sigmoid
0.785 1 0.020
Mean-std
soltmax
0.931 1 0.013
sigmoid
0.909 = 0.015
Taylor approx.
Entropy
softmax
0.725 = 0.027
sigmoid
0.672 = 0.023
Mean-ste
softmax
0.738 = 0.024
sigmoid
0.766 + 0.023
VPENN (adaptive p)
Entropy
sigmoid
0.796 = 0.020
Mean-std
sigmaid
0.916 + 0.020*
Fashion Kuzushiji
MC dropout
Entropy
softmax
0.913 1 0.014
sigmoid
0.879 + 0.014
Mean-ste
softmax
0.964 = 0.008
sigmoid
0.971 = 0.006
Taylor approx.
Entropy
softmax
0.800 = 0.025
sigmoid
0.766 + 0.017
Mean-std
softmax
0.809 + 0.023
sigmoid
0.860 + 0.017
VPBNN
Entropy
sigmoid
0.893 = 0.013
(adaptive
Mean-std
sigmoid
0.981 + 0.005
Table 4: Test accuracy on the test set of Fashion-MNIST..
Method
activation func.
Test accuracy
MC dropout
softmax
0.923 + 0.003
sigmoid
0.923 = 0.002
Taylor approx.
softmax
0.923 =|2 0.003
sigmoid
0.923 = 0.002
VPBNN (adaptive p)
sigmoid
0.923 0.002
18


Reviews:
[{"review_id": "oFoQZtDr0AS", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "This paper proposes a sampling free technique based on variance propagation to model predictive distributions of deep learning models. Estimating uncertainty of deep learning models is an important line of research for understanding the reliability of predictions and ensuring robustness to out-of-distribution data. Results are shown using synthetic data, perplexity analysis for a language modeling task and out-of-distribution detection performance using a convolutional network.\n\nOverall I vote for rejecting the paper. The paper proposes an upper bound to the variance estimate of predictive distributions. However, the paper does not explain how the upper bound can be ensured. Furthermore, the experiments based on real data are conducted not using the upper bound approach but assuming strong independence assumptions (\\rho = 0). In my opinion, the independence assumption needs extensive experiments to validate its performance under different scenarios. Furthermore, for further adoption of the upper bound approach, I think the authors need to provide extensive experiments to showcase the tradeoffs. For instance, as a reader of the paper I would like to get a sense of how much I will be overestimating the variance under typical scenarios. For this, I would recommend a formal analysis of the uncertainty estimates by inspecting the confidence intervals through coverage properties. \n\nMy specific comments related to this approach:\n-\tThe authors propose two approaches in Section 3.1 for estimating the variance in predictions and the remaining subsections in Section 3 build on this. \no\tThe first approach (a) assumes independence in the input. In my opinion, this is a very strong assumption. For instance, multi-collinearity in features in DNN’s is a very common usage pattern. In domains like images, by construction of the problem, you expect a spatial correlation structure. Furthermore, the multi-layer perceptron architecture by construction is susceptible to correlation among variables. I would highly recommend adding a discussion on the validity independence assumption in general.\no\tThe second approach (b) will be necessarily true if `\\rho >= \\max_{j,j’:j!=j’} \\rho_{j,j’}`. Only then this will guarantee that you will obtain an upper bound to the variance but it is very likely that you will overestimate the variance since your estimate is going to be bound by the highest correlation in your system. Again, in my opinion, this is a significant limitation of this approach and I recommend that authors highlight these points. Another minor thing to note in the manuscript is that when \\rho=0, option (b) reduces to option (a).\no\tI would suggest adding another summation term in Var(y_i) related equations to denote the double summation (indexed by j’) happening over the covariance terms (in Section 3.1).\no\tRelated to the above point, I could not follow the mathematical derivation from line (1) to line (2) in Equation (1). Could the authors provide an explicit derivation to ensure that the derivation is correct?\no\tFurthermore, author’s claim that “distribution of y_i is well approximated by the univariate Gaussian distribution if the correlation among x is small” (Section 3.1) needs justification. To my knowledge, Wang & Manning (2013)’s Gaussian approximation holds due to central limit theorem as the number of samples approaches infinity but I do not think this will be applicable in this sample-free setting.\n\n-\tRegarding the results:\no\tThe results shown in Figure 1 demonstrates that choosing an appropriate `\\rho can be challenging. We see underestimation of uncertainty with \\rho <=0.15 and overestimation with \\rho = 1. However, since these are based on synthetic data, I would suggest that the authors formally assess the fit using metrics like confidence interval widths and coverage.\no\tThe authors note that “Estimation of ρ is possible by observing the outputs of middle layers several times under the approximate predictive distribution.”. I am not convinced that this is an easy problem, I would recommend that the authors provide an example and elaborate this in depth.\n\nMinor comments:\n-\tPlease indicate the best results in the tables by highlighting them. \n-\tThe uncertainty in deep learning literature typically employs distinction between aleatory and epistemic uncertainties. I think the manuscript can benefit how this proposed approach maps to the different sources of uncertainties. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "WWzaO2n2Nv0", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "This paper proposes a variational-approximation of Neural Network uncertainty dependent on the 'bayesian interpretation' of dropout. Using this approach uncertainty measurements are provided using 'variance propagation.' Validation is provided on synthetic data, and MNIST/FMNIST.\n\nAlthough a genuine effort is made towards deriving uncertainty through 'variance propagation,' I think the proposed approach is rather ad-hoc. The proposed approach relies on a 'bayesian interpretation,' of dropout, and is of limited applicability requiring hard coded rules for propagating through each layer. Currently, rules are derived for a small set of neural network layers.\n\nThe chief issue with this work is the 'generality' of the work. The promised approach here is applicable to specifically neural networks trained with SGD using dropout. Thus although the proposed approach claims to evaluate uncertainty, it is rather estimating the uncertainty introduced by dropout using a variational approximation, and assuming this is the same as uncertainty. I find this assertion unfair to other works which attempt to estimate various types of uncertainties (e.g. epistemic) in a principled manner.\n\nThe proposed approach depends on the hyperparameter \\rho, which requires additional tuning to allow variance propagation through affine (e.g., fully connected) layers. This additional hyperparameter gives some doubt whether the proposed approach is something which can be easily integrated into existing models. \n\nIn Table 1, how do the authors address that their approach (which is meant to be an approximation to MC) appears to outperform MC? Shouldn't an approximation perform worse?\n\nAlthough the proposed approach outperforms MC sampling in detecting out of distribution sampling, I think this benchmark is rather unfair. I'd like to see comparisons to other works which quantify uncertainty. The issue is that this only compares against model uncertainty assuming the 'bayesian interpretation' of MC is valid. I would strongly like to view comparisons to other works in quantifying network uncertainty.\n\nOn the whole, the lack of generality of the proposed approach, as well as the limited scope of 'variational approximation of model uncertainty assuming bayesian interpretation of dropout,' causes me to be highly skeptical of this work.\n\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "LrY9PW6PiEF", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "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 paper proposes a sampling free approach for estimating predictive uncertainties in Bayesian neural networks trained via Monte-Carlo dropout. In particular,  given a dropout trained neural network, the paper develops a deterministic approximation to the test time predictive distribution that is otherwise approximated through Monte-Carlo simulations.\n\nThe paper is clearly written and proposes a solution to an important problem. Extracting fast and reliable predictive uncertainties from BNNs, trained via MC-dropout or other approximate inference techniques, is crucial for deploying BNNs in resource constrained and/or real time applications. Deterministic approximations like the one presented here are promising. \n\nThat being said, I have several concerns with the paper that revolve around differentiation from previous work,  missing empirical comparisons, and some curious modeling choices. \n\n* The primary contribution of the paper appears to be the amortization of the MC-dropout posterior predictive density. This is achieved by propagating  uncertainty (first and second moments of the inputs) through the network.  The technical details of such uncertainty propagation have been worked out by several authors in the past (see 1, 2, 3, and the papers cited by the authors Wu et al., 2019 and Shekhovtsov & Flach 2019). While the application to amortizing MC-dropout is interesting, it appears to be a direct application of previous work.\n* The notion of upper bounding the variance during the propagation is indeed distinct from previous work, but is neither principled nor empirically vetted to be consistently useful.  The experiments provide limited evidence in support of using the upper-bound. The derived uncertainty seems to be crucially dependent on the weighting factor $\\rho$ and it is unclear how one would select this weighting factor. In general, injecting additional noise does not guarantee better calibrated uncertainties. Neither the language modeling task nor the OOD detection task make use of the upper bound.\n* Amortization of the posterior predictive distribution is not a new idea. A popular approach is to use distillation [4, 5] based techniques to approximate the posterior predictive distribution with a second neural network. This work needs to be both discussed (advantages / disadvantages of the proposed approach over distillation)  and empirically compared against. \n* Since the goal of the paper is to sufficiently well approximate the posterior predictive density, the experimental section would benefit from including calibration metrics (ECE / Brier score; see 6). \n* (Minor) The paper at several places claims that the proposed approach “Unlike various kinds of probabilistic NNs, we do not need any specialized training procedure to evaluate the uncertainty”. While this is technically true, this is also true for other distillation based amortization techniques. Which arguably are simpler because one can just use an off the shelf network and not have to worry about propagating moments correctly. \n* (Minor) Since the categorical Gaussian integral is intractable, the authors replace the softmax layer with a series of independent sigmoids and use an approximation for the sigmoid Gaussian convolution. This is strange. If we are going down this path, why not use independent probits instead of sigmoids? The probit-Gaussian integral is exactly computable and requires no further approximations (see equation 3.82 in http://www.gaussianprocess.org/gpml/chapters/RW3.pdf). \n\n[1] https://papers.nips.cc/paper/5269-expectation-backpropagation-parameter-free-training-of-multilayer-neural-networks-with-continuous-or-discrete-weights.pdf\n\n[2] https://arxiv.org/abs/1502.05336\n\n[3] https://www.aaai.org/ocs/index.php/AAAI/AAAI16/paper/view/12391\n\n[4] https://arxiv.org/abs/1506.04416\n\n[5] https://arxiv.org/abs/2005.08110\n\n[6] Tilmann Gneiting, Fadoua Balabdaoui, and Adrian E Raftery. Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69(2):243–268, 2007. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "65ofW10Yc5T", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "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": "Thank you for the interesting paper!\n\nSummary\n\nThe authors focus on the important problem of efficient uncertainty quantification. More specifically, they propose a methodology that approximates the variance across samples from an MC dropout model with a single forward pass. To do this, they define analogs to existing layers such that they analytically (and often approximately) propagate variance from the input to the output of the layer. By repeating this for all layers, they construct a \"variance-propagation BNN\" (VPBNN) model that otherwise has the same architecture as the existing MC dropout model but is able to effectively output sample variance with a single forward pass. They demonstrate the effectiveness of their approach on (1) a simple synthetic problem, (2) a language modeling task, and (3) OOD detection.\n\nStrengths\n\n- Efficiently (from a FLOPS standpoint) propagating model uncertainty in a BNN is an important research area, particularly for compute-constrained use cases.\n- The authors focus on a relevant set of experiments to demonstrate both the ability of their method to approximate MC dropout, and the ability to perform at, or better than, existing methods on downstream tasks.\n\n\nWeaknesses\n\nAs noted below, I have concerns around the experimental results. More specifically, I feel that there is a relative lack of discussion around the (somewhat surprising) outperformance of baselines that VPBNN is aiming to approximate, and I feel that the experiments are missing what I see as key VPBNN results that otherwise leave the reader with questions. Additionally, I think the current paper would benefit from including measurements and discussion around the specifics of computational and memory costs of their method.\n\nRecommendation\n\nIn general, I think this could be a great paper. However, given the above concerns, I'm currently inclined to suggest rejection of the paper in its current state. I would highly recommend that authors push further on the noted areas!\n\nAdditional comments\n\n- p. 1: \"The uncertainty is defined based on the posterior distribution.\" For more clarity it could be helpful to update this to say that the epistemic model uncertainty is represented in the prior distribution, and upon observing data, those beliefs can be updated in the form of a posterior distribution, which yields model uncertainty conditioned on observed data.\n- p. 2: \"The MC dropout requires a number of repeated feed-forward calculations with randomly sampled weight parameters in order to obtain the predictive distribution.\" This should be updated to indicate that in MC dropout, dropout is used (in an otherwise deterministic model) at test time with \"a number of repeated feed-forward calculations\" to effectively sample from the approximate posterior, but not directly via different weight samples (as in a variational BNN). With variational dropout, this ends up having a nice interpretation as a variational Bayes method, though no weight distributions are typically directly used with direct MC dropout.\n- p. 2: Lakshminarayanan et al. (2017) presented random seed ensembles, not bootstrap ensembles (see p. 4 of their work for more info). They used the full dataset, and trained M ensemble members with different random seeds, rather than resampled data.\n- p. 4: For variance propagation in a dropout layer with stochastic input, it's not exactly clear from the text how variance from the inputs and dropout is being combined into an output Gaussian. I believe using a Gaussian is an approximation, and while that would be fine, I think it would be informative to indicate that. The same issue comes up with local reparameterization for BNNs with parameter distributions, where they can be reparameterized exactly as output distributions (for, say, mean-field Gaussian weight dists) so long as the inputs are deterministic. Otherwise, the product of, say, two Gaussian RVs is non-Gaussian.\n- p. 7: Figure 1 is too small.\n- p. 7: \"Estimation of ρ is possible by observing the outputs of middle layers several times under the approximate predictive distribution. The additional computation cost is still kept quite small compared to MC dropout.\" How exactly is $\\rho$ estimated? Is it a one-time cost irregardless of data that can then be used for all predictions from the trained model? Without details, this seems like a key component that can yield arbitrary amounts of uncertainty.\n- p. 7, 8: For the language modeling experiment, why do you think VPBNN was able to achieve lower perplexity values than MC dropout? The text generally focuses on VPBNN as an approximation to MC dropout, and yet it outperforms it. The text would greatly benefit from more discussion around this point. \n- p. 8: For the OOD detection experiment, I'm surprised that $\\rho = 0$ was the only VPBNN model used, since Section 5.1 and Figure 1 indicated that it led to overconfident models. Can you include results with other settings of $\\rho$? Moreover, from Figure 1 we see that (for that model) VPBNN with $\\rho = 0$ qualitatively yielded the same amount of predictive variance as the Taylor approximation. However, in Table 2, we see VPBNN with $\\rho = 0$ outperform MC dropout (with 100 or 2000 samples) and the Taylor approximation. Why do you think this is the case, particularly if the standard deviation was used as the uncertainty signal for the OOD decision. I see that \"This is because the approximation accuracy of the Taylor approximation is not necessarily high as shown in Section B\", but I did not find Section B or Figure 3 to be clear. I think the text would benefit from more discussion here, and from the additional experiments for $\\rho$.\n- Can you include a discussion and measurements for FLOPS and memory usage for VPBNN? Specifically, given the discussion around efficiency and the implementation that doubles the dimensionality of the intermediates throughout the model, I believe it would be informative to have theoretical and possibly runtime measurements.\n\nMinor\n\n- p. 1: s/using the dropout/using dropout/\n- p. 1: s/of the language modeling/of language modeling/\n- p. 2: s/is the representative of/is representative of/\n- p. 2: s/In the deep learning/In deep learning/\n- p. 2: s/This relations/This relation/\n- p. 5: Need to define $s$ as the sigmoid function in the LSTM cell equations.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper proposes an approach to estimating uncertainty in deep neural network models that avoids the need to make multiple forward passes through a network or through multiple individual models in a posterior ensemble. In terms of strengths, this is an important and timely topic that is of significant interest. The paper is clearly written for the most part. In terms of weaknesses, the significance of the work is low. As the reviewers note, there are multiple questions around the experimental evaluation that remain unresolved following the author feedback and discussion. In particular, the authors do not compare to baseline MCMC methods like HMC/SGHMC that can yield gold standard estimates of posterior predictive uncertainty. While not feasible for large-scale models, MCMC methods provides crucial sanity checks for uncertainty estimation on small-scale (e.g., MNIST-scale) models. Posterior distillation methods like Bayesian Dark Knowledge are also not considered in the evaluation and should be compared to where the distillation computation is feasible. There are also foundational technical correctness issues with respect to uncertainty quantification due to the fact that the paper is approximating the measure of uncertainty produced by MC Dropout, which itself only approximates the true Bayesian posterior predictive distribution under additional assumptions. This makes empirical comparissons to MCMC methods all the more important. Following the discussion, the reviewers agree that the paper is not yet ready for publication.
Decision: Reject