Noise-Contrastive Variational Information Bottleneck Networks
Authors: Jannik Schmitt
Stefan Roth
Venue: ICLR 2022 Submitted (2022)
Keywords: uncertainty estimation
variational information bottleneck

Abstract: While deep neural networks for classification have shown impressive predictive performance, e.g. in image classification, they generally tend to be overconfident. We start from the observation that popular methods for reducing overconfidence by regularizing the distribution of outputs or intermediate variables achieve better calibration by sacrificing the separability of correct and incorrect predictions, another important facet of uncertainty estimation. To circumvent this, we propose a novel method that builds upon the distributional alignment of the variational information bottleneck and encourages assigning lower confidence to samples from the latent prior. Our experiments show that this simultaneously improves prediction accuracy and calibration compared to a multitude of output regularization methods without impacting the uncertainty-based separability in multiple classification settings, including under distributional shift.

Under review as a conference paper at ICLR 2022
NOISE-CONTRASTIVE
VARIATIONAL INFORMATION BOTTLENECK NETWORKS
Anonymous authors
Paper under double-blind review
ABSTRACT
While deep neural networks for classification have shown impressive predictive performance, e.g. in image classification, they generally tend to be overconfident. We start from the observation that popular methods for reducing overconfidence by regularizing the distribution of outputs or intermediate variables achieve better calibration by sacrificing the separability of correct and incorrect predictions, another important facet of uncertainty estimation. To circumvent this, we propose a novel method that builds upon the distributional alignment of the variational information bottleneck and encourages assigning lower confidence to samples from the latent prior. Our experiments show that this simultancously improves prediction accuracy and calibration compared to multitude of output regularization methods without impacting the uncertainty-bart based separability in multiple classification settings, including under distributional shift.
INTRODUCTION
Deep neural networks (DNNs) have become the standard tool for challenging classification tasks, e.g. image classi- fication semantic segmentation, due to their excellent predictive accuracy. However, predictions of DNNs often tend to be overconfident, leading miscalibration (Guo et al. 2017). This problem is amplified in the presence of distributional shift in the test data, such as from image corruptions (Ovadia et al. 2019). Multiple methods to regularize the output distribution of DNNs during train- ing have been proposed (Joo & Chung 2020, Joo et al.) PPrrerra et af.) 2017]] [Szegedy [Zegeey et al. al.] (2016) to obtain well-calibrated models. We note, however, that evaluating in-domain uncertainty quantification in terms model calibration is not sufficient, it does not indicate how well correct and incorrect predictions can be discerned based on the predictive uncertainty (Ding al. 2020). As Fig. shows, methods that indiscriminately regularize the confidence to improve calibration perform significantly worse at separating correct from incorrect predictions.
+
Delammersii, AUNIIC 0,0053
NCMIBNN AUMRC: 0.0057
VIN, AUMMCc 0,0082
Misclespiiication rate
Dirihictt Motching,
0.04
Legit Regulartzation
Entrrppy Requiarization, AUMRC, 0.0149
0.05
Laoel Smoothing, AUMRC: 0.0157
0.60
0.0
0.2
01.4 C0 0.8
10
Fraction of examples rejected
Figure 1: Ability to separate correct and incorrect predictions based on the pre- dictive entropy CIFAR-10: misclassi- fication rate under rejection uncertain high-entropy examples and the area un- der the resulting curve. Models trained with explicit output distribution regular- ization perform substantially worse than a deterministic network. Plain varia- tional information bottleneck networks (VIBN) show a similar tendency.
To address this, we turn deep variational information bottleneck networks (VIBN; Alemi et al. 2017). Similarly to output regularization methods, they benefit generaliza- tion and calibration (Alemi et al.) 2018) but. as we show
empirically, suffer from the same separability problem. However, unlike these methods, VIBNs allow us to utilize the distribution matching of the variational approximation to define a noise-contrastive loss term, which can overcome the problem of insufficient separation of correct and incorrect predic- tions based on the uncertainty while retaining the benefits of the VIBN in terms of generalization and calibration. To this end, we propose a novel model, the nonse-contrastive variational information bottleneck network (NC-VIBN), which builds upon the VIBN to improve uncertainty estimation. Instead of using distribution matching as the primary source of regularization, our model utilizes it define loss term that explicitly encourages high predictive entropy only for uninformative

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Under review as a conference paper at ICLR 2022
samples from the latent prior. Additionally, we account for weight uncertainties in the decoder and use Lg nonmalizaaiion before computing the latent embeddings to further alleviate the described problems while improving the calibration and generalization capabilities of the model.
We make the following contributions: (i) We empirically show that models that explicitly regularize the prediction confidence make it harder to distinguish between correct and incorrect predictions based on the estimated uncertainty. (ii) We link the VIBN to these methods and find that it suffers from the same behavior due to the implicit Le-reuularization through the latent KL-divergence term. (iii) We circumvent these ill effects regarding separability of correct and incorrect predictions by proposing a moise-contrastie combined with architectural refinements, leads to improved separability, calibration, and accuracy. (iv) Our results show that our proposed model also leads to improved accuracy and calibration in the presence of distributional shift introduced by image corruptions.
loss term that utilizes distribution matching in the latent space and,
2 RELATED WORK
Information Bottleneck. The information bottleneck principle has been proposed by Tishby et al. (1999) as means to analyze generalization for deep neural networks (Tishby & Zaslavsky| [2015] and relies on computing the mutual information of input, output, and intermediate variables. Since these quantities are general intractable, variational approximations of the mutual information terms have been introduced (Alemi et al. 2017 AAchlle & Soatto 2018). These variational approximations are also able to overcome problems of the information bottleneck objective for deterministic represen- tations (Amjad & Geiger 2019) and share some of the benefits of Bayesian models (Alemi et al.) 2020). Further, as shown by Aleemi et al. (2018), variational information bottleneck networks can improve calibration, which we want to further improve with method.
consa-con estimation. Noise-contrastive estimation (Gutmann & Hyvärinen, 2010) is an estimation method for parameterized densities, which is based on discriminating between real and artificial data by logistic regression using the log-density functions. Inspired by noise-contrastive estimation, Hafner al.] (2020) propose noise-contrastive priors as data space priors that encourage uncertain predictions at the boundary of the training data for regression tasks by minimizing the KL-divergence between high-variance Gaussian and the predicted output distribution for perturbed training data points. We focus on classification instead.
Uncertainty estimation. Bayesian neural networks (BNNs) are a popular and theoretically well- founded tool for uncertainty estimation. A multitude of methods have been proposed to approximate
the intractable weight posterior, including variational inference (Blundell et al. 2015 Graves 2011 :
Markov chain Monte Carlo methods (Welling & Teh 2011). Laplace approximation (MacKay 1992
Ritter et al. 2018). and assumed density filtering (Hernández-Lobaal
& Adams (015)) as well as
approximate variational inference methods based on the dropout (Srivastava et al. al.] 2014) regularization scheme (Gal Ghahramani] 2016). In the last years, these methods have been successfully scaled to larger models (Dusenberry al. 2020 Heek & Kalchbrenner [2019] Maddox al. 2019 Osawa ct al. 2019; Zhang et al. (020). Deep ensembles (Lakshminaray
et al. 2017) have also been used for uncertainty estimation and can be interpreted Bayesian model averaging. Since model averaging with respect to the approximate posterior requires multiple forward passes, BNNs incur substantial computational overhead. To lighten this, there has been an interest in Bayesian last layer approaches (van Amersfoort et al. [2021 Kristiadi et al., 22200: Liu et al. 2020: Riquelme et al.
2018, (nookk al. [015 Wilson et al.) (016)) Our approach similarly employs Bayesian treatment only for the last (few) layers, however combining it with the information bottleneck principle and a noise-contrastive loss. An alternative approach for estimating uncertainty in classification networks
is parameterizing the more expressive Dirichlet distribution (Gast & Roth 2018 Joo et al.) 2020
Malinin Gales, 2018 Sensoy al., 2018) instead of the categorical distribution at the output layer. We instead gain additional expressiveness by modelling the latent space distributions.
Related methods in -ittr--distribution detection. Lee et al. (2018) proposed to train confidence- calibrated classifiers by using a generative adversarial network (GAN) that learns to generate samples at the data boundary. The generator is trained to generate data points, which are hard to separate from in-distribution data by the discriminator while given an almost uniform labeling by the classifier. In contrast, Srriharann & Srivastava (2018) train the generator to produce low-entropy in-distribution samples while requiring the classifier maximize the entropy of those samples.
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3 VARIATIONAL APPROXIMATION OF THE INFORMATION BOTTLENECK
We begin by recapitulating two variational approximations of the information bottleneck, the deep variational information bottleneck (Alemi et al. 2017) and information dropout (Achille Soatto 2018), which we will use to explain the behavior of such models and to build our own model upon.
Deep variational information bottleneck. The information bottleneck was first introduced (Tishby et al. 1999| Tissby& & /Zaslassyy 2015) to find a low-complexity representation Z depending on feature vector X that maximizes the mutual information with a target variable Y. To constrain the complexity of Z, the mutual information between X and Z is bounded, resulting in maximization problem with inequality constraints. Alemi et al. (2117) 7) proposed to use a variational approximation of the mutual information terms of the Lagrangian with Lagrange multiplier B. resulting in the objective
min 2..0
(x log (u(nlk)) + 3DxL [po(<le)]]](]))
N
1=
where the stochastic encoder p (=(xx) and decoder (p(()) are modeled by neural networks, parame- terized by @ and y respectively, and r(z) is variational approximation of the marginal distribution
p(z)
pa (()) dæ of z. To draw parallel to the variational inference literature (Kingma &
Welling 2014), r(z) is also referred to as the latent prior. Alemi et al. (2017) assume r(z) to
standard Gaussian and model the distribution of the latent encodings as Gaussians with diagonal covariance, resulting in
(o[[p((([e)]]](e)) = - log logo 31e. + +2 + 12 -1]
Σ
where Hgilz, and o² are the component-wise mean and variance of the latent embedding (o(2(zn) of In, estimated by the encoder network. The decoder is softmax classification network high predicting class probability vectors from 2, i.c. (u(u)2) 2)) = (y
Information dropout (IDO). Achille & Soatto (2018) proposed an alternative approach, where the encoder network 9o predicts non- non-negative vector μt as well as ag, parameterizing the log-nowe lor-normal distribution log [/ o The distribution of the latent encodings is modeled as Py O c with ~ log LogNN(O) a They show that if Man is the output of ReLU units and the latent prior r(z) is chosen to be a mixture of the delta distribution at and a log-uniform distribution, i.e. r(z) xx goo(z) + c/2,
the KL-divergence is given
DkI.ppo((<))]11()) {
log - Pa. (log c)] + log
fx =
Mz 0,
where the entropy term H(pp (log c)] is given by log ag for c ~ log log.N(0, (0.00,))) a.)) up to an additive constant. In the original formulation of /Achille & Soatto (2018), the mean of grows with 0g., resulting in higher level of saturation of the softmax outputs, hence in overconfidence. Note that if e is log-normal distributed, log is normal distributed and the entropy does not depend on its mean. Therefore, we here instead employ the mean-corrected log-normal distribution log. N/ ja2 a so that Ep., (c) O E] Me without changes to the KL-divergence.
4 UNCERTAINTY QUANTIFICATION UNDER OUTPUT DISTRIBUTION REGULARIZATION
frequently used metric to assess uncertainty estimation the expected calibration error (Guo et al. 2017) or related calibration metrics, which measure how well the prediction confidence coincides with the prediction accuracy. Methods that achieve better calibration by output distribution regularization include label smoothing (Müller et al. 2019 Szegedy et al. 2016), regularization of the predictive entropy (Pereyra et al. 2017) for the categorical distribution, or evidential deep learning (Sensoy et al. 2018) and belief matching (Joo al. 2020) for the Dirichlet distribution. Altematively. the model can be regularized function or distribution spaces prior to the final softmax layer (Joo et al. 2020), and in the simplest case results in norm penalties for the predicted logits.
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Correctly classified Micllssified Priar samples
(a)
(h)
Figure 2: Embeddings and samples from the latent prior (a) and assigned predictive entropy (b) for a NC-VIBN model with a two-dimensional information bottleneck trained on CIFAR-10. Darker areas correspond to lower entropy. By employing the proposed noise-contrastive loss, embeddings of incorrectly classified examples, which are harder to distinguish from samples of the latent prior, are assigned higher entropy, allowing for better separability of correct and incorrect predictions based on the predictive entropy.
However, these calibration metrics actually do not indicate how well correct predictions can be separated from incorrect ones based on some uncertainty estimate like the predictive entropy. In fact, model predicting with confidence equal to the expected accuracy on in-domain data for both correct classifications and misclassifications would be perfectly calibrated according to the expected calibration error, but the confidence of prediction would give no information about an example being correctly incorrectly classified (Ding al. 2020). A suitable method examine how easily correct and incorrect predictions can be discriminated, is to analyze how the predictive performance behaves under the rejection uncertain predictions (Ding et al., 2020, Nadeem et al. 2009). As can be seen in Fig. [ models that trained with an objective that regularizes the output distribution perform sigmificantly worse at assigning incorrect predictions a comparatively higher uncertainty and rejecting them early than the deterministic model. We argue that this is because these methods indiscriminately regularize the confidence of correct and incorrect predictions. More experimental evidence is presented in 6.1
5 NC-VIBN - NOISE-CONTRASTIVE VARIATIONAL INFORMATION BOTTLENECK NETWORKS
We can observe in Fig. that, similarly to the regularization methods described in the previous section, standard VIBN models also suffer from worse separability of correct and incorrect predictions. However, we can make use of distribution matching in the latent space to remedy this problem while still benefiting from the improved generalization and calibration of VIBNs. To improve uncertainty estimation with variational bottleneck networks, here propose noise-contrastive loss term that encourages the decoder network to assign high predictive entropy to samples that are closely aligned instance-level I--regularization before computing the
the latent prior. We further propose parameters of the latent distribution to reduce the sensitivity to the magnitude of input variables and also learn the uncertainty of the decoder parameters using variational inference.
5.1 LEARNING WEIGHT UNCERTAINTIES FOR THE CLASSIFIER
To improve the ability of VIBNs to quantify uncertainty, first propose learn weight distribution q(v10) for the decoder via mean field variational inference. A similar approach for variational autoencoders has been explored by Daxberger & Hernández- of generative models by approximating the decoder posterior with Markov chain Monte Carlo methods. We assume that we have dataset D (2n. such that
(2019), who learn an ensemble
log p(ynl Zn: 1b) Epp(s)e))) log qu (yn[=)] + const
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and weight prior p(e) N (1 Now, minimizing the KL-divergence between the mean field Gaussian variational approximation and the true posterior is equivalent to solving
N
1 N DKL (KkL((((()llppe))
min
(=1a)))))) [- log go (vn [uml=)]]
+
(5)
1=1
This loss function will substitute the first term in the VIBN objective in equation
5.2 THE NOISE-CONTRASTIVE LOSS TERM
Next, we will utilize the distributional alignment of the variational information bottleneck to formulate a noise-constrastie loss in the latent space. This has multiple benefits as opposed to formulating the noise-contrastive loss in the input space: We do not have to approximate the complex distribution of images in the high-dimensional input space and only have to evaluate the smaller decoder network to obtain predictions for samples from this distribution.
To motivate how we can improve the separability of correct and incorrect examples by the predictive entropy, consider Fig. 2a We can observe distinct clusters for the latent embeddings of correctly classified examples, while incorrectly classified examples for the most part are embedded in-between these clusters, resembling samples from the latent prior. Moreover, we note that embeddings of hard examples tend be aligned with the latent prior, since the KL-term of the VIBN objective (equation]) D. which regularizes the distance of the latent embedding from the prior, is class-agnostic and thus easier to minimize than the negative log-likelihood term for such hard examples. To now prevent overconfident predictions of the decoder network in these regions of the latent space, we propose noise-contrastive loss that maximizes the predictive entropy for samples generated from the latent prior, resulting in the objective
max E (wlo)
(r(()) HIgs (ul (vl=)]]]
(6)
where the predictive entropy of the categorical distribution estimated by the classifier is given by H (y]=)) - (y c/2) log (v c/2).
Note that embeddings for easier examples, in contrast, are mapped into regions of the latent space away from the high-density regions of the latent prior in order to enable the decoder to separate them from prior samples, yet prior samples are assigned low confidence due to the entropy maximizing noise-contrastive loss. This separation helps the decoder assign high confidence the embeddings of casier examples, so that the expected negative log likeliioihood term of the VIBN objective can be minimized. The effect of the interplay the VIBN objective and the noise-contrastive loss can be seen in 2bg where high entropy is assigned to the regions of the latent space that are populated with embeddings of incorrectly classified examples, while regions where the embeddings of correctly assigned examples are located are assigned lower entropy (see appendix B.I for comparison with the latent space of a standard VIBN). As a result, we can preserve the separability of correct and incorrect examples by entropy- based uncertainty estimates while still benefiting from the improved calibration and regularization VIBN networks.
Putting everything together and approximating expectations via sampling, obtain the following loss function:
1 N
L(d 0)
log qvr (vol=2.)) 3D (Do
1 DKL. (((5)))))(())))
1 Σ H H (y]=²)]
+ N
where samples 2x are drawn from p(=|æn) and th are drawn from g(v) The first loss term maximizes an empirical approximation of the expected log-likelihood, while the second loss term aligns the distribution of the latent embeddings with the latent prior. Further, the third loss term is a result of the variational approximation of the classifier's 's weight posterior distribution and penalizes divergence from the weight prior. Finally, the last loss term is our contraaste loss, encouraging high entropy for noisy latent prior samples.
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5.3 TEST-TIME NORMALIZATION
Hein et al. 2019) describe how the linear extrapolation behavior of ReLU networks can lead to overconfidence for data points that differ from the training data in terms of magnitude or, in case of image data, in pixel intensity. Similarly, for our encoder network this behavior can result in embeddings into regions that are far away from the latent prior and assigned high certainty by the decoder network for these data points. To prevent this and decouple the predicted distribution parameters from the intensity of pixel values, we add a test-time normalization step to the intermediate variables before computing the latent embeddings. Specifically, we apply Zz-normalization to the input ह of the last residual block of the encoder network, mapping
T + #/ 3 #3+4,
(8)
where 0 is small constant added for numerical stability and M is the dimension of ã.
5.4 IMPLEMENTATIO DETAILS
Encoder and classifier architecture. We adjust the ResNet architecture (He et al. 2016) for our model by using the last residual block, consisting of two convolutional layers for smaller ResNets and three convolutional layers for larger ResNets, and the fully connected layer as the classifier (decoder) network; the remaining layers form the encoder network. The channels of the last layer of the encoder are split to predict Ma and o2 or ar for the VIBN or IDO models, respectively. The width of the classifier network is accordingly reduced by half, resulting in lower total number of parameters compared to the respective (deterministic) ResNet baseline, even though additional parameters representing the weight variances of the decoder are required.
Parameterizing the latent distribution. For the VIBN, we predict log of and map it to o using the exponential function. For IDO, we follow Achille & Soatto (2018) and constrain 0₂ to be smaller than Olmax- For simplicity, we chose Ommax = 1. Instead of predicting a, directly, the unrestricted variable O₂ is predicted and mapped to ag = Ommax ((ūz), where (-)) is the sigmoid function.
Sampling from the latent distribution and prior. To sample from the latent distribution in case of the VIBN, we use 2g = P2z + d2 @ E, where ~ (O, Samples from the latent prior can be directly generated from (0, I). For IDO, again generate samples from (0,, and map them to 1rr @ exp(or @ o2/2) to sample from the corresponding latent distribution. We generate samples from truncation of the latent prior by sampling u from uniform distribution on [- and b from a Bernoulli distribution with parameter q = 0.1 and computing = = b b0es exp(u).
Weight uncertainties for the classifier. We initialize the weight variances p times the prior variance; more details can be found in appendix [A] We use the local reparameterization trick (Kingmal al. 2015) to sample from the classifier weight distributions.
Rescaling the loss terms. We do not introduce any additional hyperparameters for rescaling the loss terms except for the latent KL-divergence inherited from VIBN. Further, we apply a linear warm-up schedule to all loss terms except the expected negative log-likelihood to emulate training of a deterministic network in the early stages.
EXPERIMENTS
We begin by comparing the proposed noise-contrastive variational information bottleneck networks with fixed standard Gaussian prior (NC-VIBN) and learnable Gaussian mixture prior (NC- VIBN-MP)
as well information dropout networks (NC-IDO) their counterparts VIBN (Alemi et al.) 2017). VIBN-MP. and IDO (Achille & Soatto, 2018). as well as to deterministic network and BNN using MC Dropout (Gal Ghahramani] 2016) as baselines. For fair comparison in terms of inference time, we apply dropout only to layers that need to be resampled multiple times for the information bottleneck networks. For the mixture prior experiments, we consider a mixture of 32 Gaussian components with diagonal covariance matrices and allow learning the means, variances, and mixture weights for every component. During evaluation, we use the same number of samples (S 16) for MC Dropout and all information bottleneck networks. We train models based on ResNet18 (He et al.) 2016) on CIFAR-10, CIFAR-100 (Krizhessky & Hinton 2009), and Tiny ImageNet, a downsampled version of ImageNet (Deng al. 2009) containing 200 classes. The results can be found in Tab.

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Under review as a conference paper at ICLR 2022
Table 1: Evaluation on CIFAR-10. CIFAR-100, and Tiny ImageNet. We report the misclassification
rate (MCR), negative log-likelihood (NLLH), expected calibration error (ECE), the area under the misclasiifcation-rejees
curve (AUMRC), the misclassification rate (MR) at different rejection
levels as well as the mean confidence (MC) for correct and incorrect predictions. All values but NLLH are given as percentages.
MCR
NLLI
ECE
AUMRC
MR5%
MR10%
MR25%
MC Corr.
MC Incort
CIFAR-10
Deterministic
5.37
0,237=am:
s,22
0.53:
3,11
1.83am
0.33:aum
98.57=m04
MC Dropout
5.67=032
0.24
3,52=018
0.56=mm
3,37
1.94
0.47
98, 05
82, +00m
5.67=0re
0.238=0004
3.30=s0e
0.59:0004
3.42+025
199
0.41=00
98.41=023
81.15
VIBN
5.11
0.206
2.03
0.92 mmm
2,94
1.761uss
0.89.am
97.97.um
76.4821.
VIRN.MP
5.13gam
1.08
0,88_
3.09
1,78
0.73gam
97.03_mos
73.451087
NC-IDO (ours)
5.09
0.174
1,90
0.53
2.92an.7
1,58
0.41:nm
97.93sm05
97,9
75. 1005
NC-VIBN (ours)
4.93=2.1
0.165==00e
1.28-021
0,57=100
2.83:auu
1.52+003
0.45:aae
97, 97.48=au1
71.99+1.s
NC-VIBN MP (ours)
4.74:a30
0.164:axes
1.16ssus
0.56
2.76:
1.49
0.45
97.55aau2
72.00aa05
CIFAR-100 100
Deterministic
26.22.115
1.146
10.92.0a3
7.73.nm .m14
21. 29
20.73,crs
13,40.bas
92.28 mnm
63.39
MC Dropout
25.75
1.358
15.35
2.35
22
20.17
12.58
95.36.00
72.97
IDO
26.10ss29
1,387=00s
13.55
7.63=m0
13
20.43
12.93e 93.94=012
69.264001
VIBN
26.19
1.310
8.50=am
8.29
23.62:04
21.17
14.10=to
90,24:0as
59.50e1.32
VIBN-MP
26.37
1118
4.93:146
=.91
23.93
21.59
14.99
83.87
45.57+194
NC-IDO fours)
25.35=0m:
1.117=00
9,06
7.38-000
22.47=03:
20.01=110
12.84
91.30=o25
61.28:aaa
NC VIBN (ours)
25.16.n2s
1.09:
3.58
2.97
22.58
28
20.27
13.98
87.30
$1.331224
NC- NC-VIBN-MP (owrs)
25.07
1.044
6.72
7.83
22.45:434
20.56g
13.67
88.00.0mm
19.19
Tiny
Deterministic
43.09
2. 11
18, 18,26a22
17.93aa24
40.64-222
38, 38.20
30.63
87.64
58. 48
ImageNet
MC Dropout
41.13=0.4
1.838=0m
10.05a00
16,44sa2
38.66=as
35.94=123
28.35-14z
86.04=1.9
+7.51
ID0
43.40:a21
2.081
16.14:24
18.32
41.02:4m
38.76:013
31.10sa.
85.73
$5.78
VIBN
43.23.aa
2.092.000
11,99
18.49193
40.84..10
38.46.e3
31,27
1036
82.26
51.03
VIBN-MP
42.83
1.843
6,67=aax
18.1
40.47:03
38. 38.10
30,78aa3e
78.50.0os
44 23
NC-IDO (ours)
42.09=aas
1.784=a0e
4,90=a64
17.62+43:
39.65+a33
=250
30.01
42.10 77.78=um
NC-VIBN (ours)
41.82:a12
1.756:am
6.60s22
17.57amm:
39.45sa34
37.1
29.85
79.61=141
44.01
NC-VIBN-MP (ours)
41.46
1.78
9,35
16.91
39.08s022
36.76st35 29.26
82,39
47.36
Table 2: Comparison to output regularization methods and DUE (yan Amersfoort et al. 202/ on C0. Regularization methods are more accurate and better calibrated than the deterministic model, but perform worse at separating correct and incorrect predictions via their predictive entropy, resulting in higher AUMRC, Our NC-VIBN clearly outperforms these previous methods.
MCR
NLLI
ECE
AUMRC
MR5%
MRI0S:
MR25%
MC Corr.
MC Incom:
Deterministic
5.37am
0.237
3.22an
0.53
3,11
183ann
0.33
98. 98.57
82. 11 anss
Entropy Regularization
5,20
0.201
2.01
1.49
3.11
1.97
1.39=010
94. 94.84
33.00
Logit La- Regulaaarizztinn
5.33sum
0.208-000
1.74502
1.35
3,11
1.85
0,98
97.40aas4
78.45
Label Smoothing
1,15
0.209.wox
1,28.
1.57
3.16.1
1.97ams
1.14
96.731a13
78.10
Dirichlet Relief Matching
5.15
0.197.mm
1.47
1.07
3.03
1.7620ss
0.88
96.73111
77.71
DUE
5,27
0.195
2.24
0.76aum
3.06
1.83
0.68
97.92.as
78.791um7
NC-VIRN (owes)
4.93
0.165 moo
1.28
0.57.005
2.83
1.52
0.45
97.48.
71.99 os
Next, we compare against several baseline approaches for regularizing the output distribution of DNNs in Tab. We report the average performance and standard deviation of four training runs.
We use the misclassification rate (MCR) to assess how well a method works as a regularizer and improves generalization. To evaluate the uncertainty quantification, likelihood (NLLH), the expected calibration error (ECE; Guo et al. [207)). the area under the misclasifiiation--eecccoo ceraaiiiiniicattoteeeeeteooooon eurve (AUMRC; Nadeem al.) 2009), and the misclassfication rate at certain rejection rates. Training details, a detailed description of the evaluation metrics (Sec. A as well as further experiments (Sec.B) B can be found in the appendix.
report the negative log-
6.1 IN-DISTRIBUTION CLASSIFICATION ACCURACY AND UNCERTAINTY QUANTIFICATION
Regularization of the output distribution and intermediate variables. We first look in detail at methods that regularize the categorical distribution, such as entropy regularization (Pereyra et al.) 2017) and label smoothing (Müller et al. 2019: Szegedy et al. 2016), or the Dirichlet output distribution, in particular Dirichlet belief matching (Joo et al. 2020). well methods that regularize the intermediate variables, such as logit Lg-regularization TJo & Chung 2020). Note that the KL-divergence term of the VIBN also implicitly regularizes the Lz-norm of the latent variables. From Tab. 2 we observe higher accuracy compared to the deterministic baseline, which shows that these baselines indeed work well as regularization and improve generalization. By controlling the
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regularization of the output distribution or intermediate variables, calibration can be improved and the confidence for incorrect predictions is reduced. However, the confidence of correct predictions is decreased well. Hence differentiating between correct and incorrect predictions based on the predictive entropy becomes harder, as indicated by the MR-curve (Fig. T While this effect is more pronounced at higher rejection rates, it still significantly impacts the area under the MR-curve, increasing it from 0.005 for the deterministic baseline on CIFAR-I0 to above 0.010 for these models. The benefits of our NC-VIBN become apparent when comparing the mean confidence on correct and incorrect predictions to output regularization methods (Tab. 2) Even though these methods using explicit regularization can reduce the confidence of incorrect predictions as desired, our NCBN still shows lower confidence on incorrect predictions while not impacting the confidence of correct predictions as much. As a consequence, it is better calibrated than output regularization methods without the increase in AUMRC, while being more accurate at the same time.
Next, we compare our results with DUE (van Amersfoort et al., 2021). There has been a recent interest in uncertainty estimation methods that use Gaussian processes and RBF kernels on the network output, for example SNGP (Liu et al. 2020), DUQ (van Amersfoort et al. 2020), and the aforementioned and most recent DUE. These methods enforce Lipschitz constraints the neural network to ensure that the feature distance in the output space contains information about feature distance in the input space, avoid feature collapse, and improve the ability to detect in-distribution data, as can be seen in Tab. DUE offers benefits regarding the misclassification rate, NLLH, and ECE, although not as significant as our NC-VIBN models. Moreover, the ability of DUE to detect misclassifications based on the uncertainty assigned using information about the output space distance as measured by the AUMRC appears to be considerably worse when compared the deterministic baseline. Our proposed NC-VIBN outperforms the recent DUE across all metrics.
data points. For
Comparison of different latent priors. We compare three different variational approximations of the information bottleneck objective, IDO, VIBN, and VIBN-MP, in Tab, [ to understand which of the observed effects are specific to the variational distributions. We find that IDO provides only minor benefits to classification accuracy and no significant benefits to uncertainty quantification as indicated by NLLH and ECE. Yet, we do not see the same rise in AUMRC for IDO as we see for VIBN or VIBN-MP. Since for the variational approximation by IDO these effects are not present, we can conclude that the information bottleneck does not inherently benefit model calibration or harm separability of correct and incorrect predictions. This suggests that the observed behaviour of the VIBN and VIBN-MI arises from the chosen variational distribution and the implicit L -eggullaaiizttinn in the latent KL-divergence term, see equation 2 Note, that the adaptable Læ-regularization present for VIP while consistently improving calibration, does not avoid the separability problem, only slightly reducing the AUMRC on CIFAR-10 and Tiny ImageNet while increasing it on CIFAR-100.
Benefits of the NC-VIBN. As can be seen in Tab. D our NC-VIBN improves generalization but also offers a whole number of benefits regarding the uncertainty evaluation metrics, including a significantly better description of the data, as can be seen by lower neg- ative ligeeoo...ood. On CIFAR-10, our proposed model reduces the NILH of 0.24 for the baseline and 0.21 for VIBN to 0.17. Similar improvements can be obtained on Tiny ImageNet, where the NLLH is decreased from 2.19 and 2.09, respectively, to 1.76. We further see significant improvements regarding calibration (cf. also Fig. 3), lowering the ECE on CIFAR-10 from 0.032 for the deterministic baseline and 0.020 for VIBN to only 0,013, while on Tiny Im- ageNet the ECE decreased from O. 183 and 020, respectively. to 0.066. Only VIBN-MP with more complex learnable mixture prior offers slightly better calibration on 100 and -100, while our NC-IDO achieves the best calibration on Tiny ImageNet. Further, we almost recover the AUMRC of the deterministic network on CIFAR-10 and -100, with the misclassification rate only lagging behind at high rejection levels (Fig. I while improving it on Tiny ImageNet, achieving the 2d lowest AUMRC. The same modifications can also be applied to IDO to obtain significant improvements across all metrics.
Deterministic, ECE: 0.0322
VIEN, ECE: 0.0203
NC-VIEN, ECE: 0.0128
0.5
Accuracy
0.5-
0.0
0.2
0.4
0.6 0.3 Lo
Confidence
Figure 3: Calibration curves for the deter- ministic, VIBN, and NC-VIBN networks on CIFAR-10. For our NC-VIBN, confidence and accuracy are better aligned.
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Table 4: Image corruptions on CIFAR-10 (top) and CIFAR-100 (bottom) with three different levels of severity. The NC- VCBN model is consistently more robust to corruptions and offers better uncertainty estimates compared to the deterministic and VIBN baselines.
Comupptin Level 1
Corruption Level
Comupttion Level 5
MCR
NLLH
ECE
MCR
NLLH
ECE
MCR
NLLH
BCE
Deterministic
17.03-a39
0,941
11.7640.
31.90 31.90ssss
2.232-014
23.12+000
49.98 19.984024
4.049=:34
38.05=121
VIBN
16.02:094
0,790
8.84
30.32
=.791
18,73:00
48.62
3.193
32. 18:0s
VIBN
15.0
0.545
6.44.15 1.06
29.31
Luus
15.54.1.
48.16.
2.266
29.45.1m
Deterministic
40.65 m
2.047ances
18.33
55, g5.74
3.3362nms
27.76an75
71.31
4,955
38.73
VIBN
41.64=a
16.39=000
56.52=s38
4.103=015:
24.14
72.07=031
6.025=1344
35.77
NC-VIBN
39,37:au
1.768:am2
54.39 10.24:
2.764:aue11
17,51=177
70.23
4.0000
27.02:221
The ablation study in Tab. 3 shows that both the deterministic and VIBN baseline models bene-
Table 3: Ablations for the NC-VIBN on CIFAR- 10. We compare the effect of Lg-normalization and learning weight uncertainties for the last layers via variational inference with the noise- contrastive loss and the full NC- VIBN setup.
fit from last-layer variational inference and T.2- normalization with respect to classification accu- racy and calibration but that they do not substan- tially improve the separability of correct and incor- rect examples as measured by the AUMRC. The noise-contrastive entropy maximization term its own leads to a far more pronounced reduction in AUMRC while also benefiting the calibration the cost of a minor drop in accuracy. However, when combined with the other two components the drop in accuracy can be avoided and further significant improvements regarding all metrics can be gained.
MCR NLLH
ECE
AUMRC
Deterministic
5.37
0.237
3.22
0.53
Det. + LL. VI & L2 Ig-norm.
5.03
0.170
1.61
0.56
VIBN
5.11
VIBN+ Noise-contrastive loss
5.23
0.187
1.41
0.74
VIBN+ Dec. VI & Lo-nomm.
4.95
0.179
1.55
0.87
NC. VIBN
4.93
0.165
1.28
0.57
6.2 ESTIMATION QUALITY UNDER DISTRIBUTION SHIFT THROUGH IMAGE CORRUPTIONS
To evaluate robustness and uncertainty quantification in the presence distributional shift, we use the CIFAR-10- 10-C and CIFAR- CIFAR-100-C datasets (Hendrycks & Dietterich 2019). which feature 15 different types of common image corruptions at five levels of severity. Since distributional shift will likely cause the latent embeddings to be less aligned with those computed on the clean training data, they may be less distinguishable from samples from the latent prior. In this case, the noise-contrastive entropy loss helps assigning higher entropy to examples misclassified due distributional shift. The results in Tab. 4 confirm that our proposed NC-VIBN performs better under distributional shift than deterministic and VIBN baselines. It is more robust these image corruptions, with lower misclassification rates across all levels of corruption severity. Even stronger benefits can be observed for the metrics evaluating uncertainty quantification. The lower NLLH at 2.27 for NC-VIBN compared to 4.95 for the deterministic and 3.19 for the VIBN baseline on the most severe corruptions on CI0 with similar reductions for other corruption levels for both CIFAR-10 and CIFAR- show that our NC-VIII VC-VIBN offers a better explanation for hold-out image data even when corruptions are present. Further, the improved model calibration is retained as the corruption severity increases, e.g. resulting in a significant decrease of the calibration error from above 0.350 for the most severe corruptions on CIFAR-100 to 0.270.
CONCLUSION
We propose to improve uncertainty estimation in classification models by building upon the distribu- tional alignment of deep variational information bottleneck networks. Our experiments identify problem of prior work, being significantly worse at separating correct and incorrect predictions based on the prediction uncertainty, behavior that can be found in VIBNs as well as other models that regularize the distribution of output or intermediate variables. We address this specifically using a cousse-onttastive entropy maximization term, well Lę Lz-nommalization and weight uncertainty in the decoder. Experiments show that our NC- VIBN model can improve generalization and uncertainty estimation at the same time, both in-domain data and under domain shift by image comupttins.

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ETHICS STATEMENT
While our approach can empirically improve the reliability of deep learning systems, failure cases may still arise. However, this consistent with other black-box uncertainty estimation methods in deep learning. We, therefore, believe that it is important to inform possible users of our as well as other models predicting uncertainty that uncertainty estimates cannot blindly trusted so that their availability does not induce false sense of security.
REPRODUCIBILITY STATEMENT
We include training and evaluation code for the noise-contrastive information bottleneck models and baseline models in the supplementary material. Details on hyperparameter selection can be found in appendix
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A TRAINING AND EVALUATION DETAILS
A.1 EVALUATION METRICS
For our experiments, we compare the uncertainty estimates of different models with regard to the following metrics:
Negative log-likelhhood. A low negative log-likelihood (NLLH) indicates that the model of plyla) assigns high probability (density) to the correct labels. The metric encourages high confidence for correct classifications but also penalizes assigning high confidence to the incorrect predictions.
Expected calibration error. The expected calibration enror (ECE) (Guo et al. 2017) is a measure for model calibration, i.c. how well the estimated confidence agrees with the accuracy of predictions of a similar confidence level. We divide the interval [0, 1] into M 20 bins and assign the N predictions into these bins based on their confidence value. Within each bin Bj. we compute the prediction accuracy acc(j) and the average confidence conf(j). The expected calibration error is then given by Σee N [acc(j) conf(j))-
Area under the miiscassiftcation-reectiin curve. To evaluate how well a model is able to re- duce the misclassification rate by rejecting uncertain predictions, compute the area under the misclassification-ree rejeccion curve (AUMRC) (Nadeem et al. 2009). which is the step function obtained by sequentially excluding the example with the highest predictive entropy and adding point to the curve where the x-coordinate is the fraction of rejected examples and the y-coordinate is the misclassification rate among the remaining examples. We also report the misclassification rate at certain rejection thresholds. We chose this metric for evaluating if incorrect predictions are assigned higher uncertainty than correct predictions instead of using the AUROC or AUPR for misclassification detection, because latter metrics are not suited for comparing models with different levels of accuracy DDing et al. 2020).
A.2 DATASETS
For CIFAR-IO and CIFAR-100 Krizhevsky & Hinton 2009). we split 000 images from the training set, which consists of 50000 images, to use for validation; we report results on the 100000 test images. For ImageNet (cf. Sec. (.4). we use the ILSVRC2012 (Russakovsky et al., 2015) training split, containing 1.3 million images, train the networks and evaluate on the 50000 validation images. Similarly, we use the 100 000 training images from the Tiny ImageNet dataset for training and report the results on the 10000 validation images.
A.3 DATA AUGMENTATION
We normalize the RGB values of the images for all datasets. For CIFAR- 10 and CIFAR- 100, we apply random shifts by up 16 and 8 pixels, respectively, in addition to random horizontal flips. For Tiny ImageNet, we randomly rescale the images by factor between 0.8 and 1.0 as well as changing the aspect ratio by a factor between 0.75 and 1.33 (both uniformly distributed the given range) and randomly crop the image to the original size. For ImageNet (Sec.[B.4). we first rescale the images such that the smaller dimension measures 256 pixels. Before randomly cropping 224 x 224 pixel patches from the images, we apply random rescaling by factor between 0.1 and 1.2 and changes to the aspect ratio by a factor between 0.75 and 1.33. During evaluation, a 224 x 224 pixel patch is cropped from the center of the image.
A.4 TRAINING SETUP
For the experiments on CIFAR-IO, CIFAR-I00, and Tiny ImageNet, we use a single Nvidia GTX 1080 Ti GPU. Our method results in a minimal computational overhead. On CIFAR- 10, for example, the training time per epoch of 38 seconds for the deterministic baseline is increased to 40 seconds for the VIBN.BB. We use four Nvidia GTX TITAN X for the ImageNet experiments in Sec. [B.4 The last layer setup adds no computational overhead, resulting in a training tine of 56 minutes per epoch for the deterministic baseline and the NC-VIBN.
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Table Overview of additional hyperparameters introduced by different methods.
Method
Required hyperparameters
MC Dropout
Dropout rate
babel-smooohhngg
Label-amoothing rate a
Outpul regulatization
Weight of the regularization term
Variational approximations the IBN objective
Lagrange multiplier /
Noise-contrastvve entropy regularization
No additional hyperparameters required
Decoder variational inference
Prior precision r and initial dampening factor p
A.5 ARCHITECTURAL DIFFERENCES
On ImageNet (see Sec. B.4). we use the original ResNe150 architecture (He et al. 2016). i.e. the first convolutional layer has stride 2 and max- max-pooling applied its output. For all other datasets, we use architectures based on Re Due to the differing image resolutions, the initial feature extraction layer of the ResNets varies for these datasets. The stride of the first layer is reduced to I for Tiny ImageNet, where the input resolution is 64 x 64 pixels. For CIFAR-10 10 and CIFAR- --00, where the input resolution is 32 x 32 pixels, the first convolutional layer has stride I and no max-pooling is applied.
A.6 HYPERPARAMETER SETTINGS
We use SGD with Nesterov momentum with a momentum parameter of 0.9 for all experiments. Further, apply gradient clipping if the co-norm of the gradient is greater than 0.1. We tune the optimizer byperparameters on the deterministic baseline and apply the same settings to all other models, only adjusting the model hyperparameters specific them (see below). I.g-regularization with a weight of 10-1 applied to the network's ' parameters for all experiments. For all information bottleneck and noise-contrastive information bottleneck models, the additional loss terms are scaled up linearly starting from 0.0 to 1.0x (i.e. their reported value, see below) until the first learning rate drop. For the variational information bottleneck networks with learnable Gaussian mixture priors, we set the number of mixture components to 32, sample the initial means from a centered Gaussian with variance 0.01 and initialize the variance of the mixture distributions 1. This choice for the number of mixture components leads to consistent improvement in negative log-likelihood for the VIBN models across all datasets. We tried to increase the number of mixture components further, but were not able to gain any additional benefits.
To select the hyperparameter 3 for VIBN and NC-VIBN as well as for the baseline models with latent priors, use a simple search over 8 E {10-1,10- 10-2, 10-3, 10 1004, 10105, 10-6), choosing the value with the highest validation accuracy, respectively. If two 3 values have similar accuracy, the one with lower validation NLLH is preferred. The B found for NC-VIBN tends to be lower than that for VIBN as the noise-contrastive entropy loss term offers an additional source of regularization. For learning the weight uncertainties of the decoder via variational inference, examine possible prior precision values E {10-1 100 1 1 with an initial dampening factor 1, see Sec. 5.4, and scale down by 10 if training is not stable during the first few iterations. We tune 7 and p after fixing the Lagrange multiplier 33 to the value found as described above, because the effect of the stochastic decoder settings is less than that of the weighting of the latent KL-term. Since over-regularization for many of these settings becomes apparent during the first few training epochs, effectively only a small subset of these hyperparameter settings has to be validated.
To similarly tune the remaining baseline methods we compare to, we search for the weight of the regularization term for Dirichlet Belief Matching (Joo et al. 2020) and logit I-g-regularization (Too) 10 for the dropout rate (Gal & Ghahramani 2016) and entropy regularization weight (Pereyra et al. 2017) in {0.05, 0.1, 0.53, and for label smoothing (Szegedy et al. 2016) in {0.01, 0.02, 0.05, 0.
& Chung 2020) in {10-1 10 10 10 10
An overview summarizing which hyperparameters are required by different methods can be found in Tab.5] Our NC- VIBN approach uses variational approximation of the information bottleneck objective,
entropy regularization, and decoder variational inference. We detail the hyperparameters determined this way in the following sections.
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Correctly Misclessiied Prior samples
(a)
(b)
Figure 4: Embeddings and samples from the latent prior (a) and assigned predictive entropy (b) for VIBN model with a two-dimensional information bottleneck trained on CIFAR- 10. Darker areas correspond to lower entropy.
CIFAR-10. The networks are trained for 200 epochs with batch size of 128. The learning rate of the optimizer is set to 0. and reduced by a factor of 0.2 at epochs 60, 120, and 160. For MC dropout, the parameter search yielded a suitable dropout rate of 0.2. The Lagrange multiplier 3 scaling the latent KL-divergence term was identified (see above) as 1 for VIBN (Alemi et al. 2017), IDO (Achille & Soatto 2018), and NC-IDO. while the search yielded 10 for NC-VIBN. The prior's precision for the decoder weights was determined as 10-1 for NC-VIBN and NC-IDO. For dampening, the initial variance scaled down by a dampening factor of p = 10-2
In the learnable mixture prior experiments, the parameter search yielded 3 10 for the VIBN-MP and 3 = -10 1, and p = 10-1 for the NC-VIBN-MP network. In the baseline experiments with output regularization methods, the parameter search yielded label smoothing of 0.02 (Szegedy et al. 2016) and scales for the regularization term of Entropy Regularization (Pereyra et al. (017) given as 0.2, for Dirichlet Belief Matching (Joo et al. 2020) as 10-1, and for I-g-regularization tried vary the number of inducing points see if we can improve the in-distribution uncertainty estimation performance, and found that in the set {10, 20, 50, 100} the models using 10 inducing points performed best.
the logits (Joo & Chung 2020) as 10-3, respectively. For DUE,
CIFAR-100. For CIFAR-100, the parameter search yielded the same hyperparameter settings as for CIFAR-10.
Tiny ImageNet. We train the models for 60 epochs with batch size of 64, reducing the initial learning rate of 0.1 by factor of 0.1 epochs 30, 40, and 50. For the MC Dropout baseline model, the parameter search yielded dropout rate of 0.1. The weight 3 of the latent KL-divergence term was identified as 10- for VIBN and 10-6 for NC-VIBN, while the prior precision of the decoder weights was determined by the search as 1 The initial variance dampening parameter was identified as 10-1 For IDO we found B 10-3, Similarly, for NC-IDO we found 3 = 10-3 =10,, and p 10-1
For the mixture prior experiments, we determined parameters B = 10-3 for VIBN-MP and B = 10 10-5 1, and p 10 1 for NC- NC-VIBN-MP.
B ADDITIONAL RESULTS
B. LATENT SPACE OF VIBN MODEL
Similarly to Fig. we show the predicted embeddings and assigned predictive entropy of a VIBN network trained on CIFAR-10 10 with a two-dimensional bottleneck in Fig. While the embeddings of correctly classified data points still form clusters, which can be separated from the embeddings of incorrect examples that are closer to the latent prior, this effect is far less pronounced compared to NC- VIBN network. Further, without the moise-oontrastive loss the high-entropp regions much more condensed, and many embeddings of incorrectly classified examples fall within regions with lower entropy, leading to poor separability.
16

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Under review as a conference paper at ICLR 2022
Table Additional results for modified last layer semup on CIFAR-IO, CIFAR-I0O, Tiny ImageNet, and ImageNet.
MCR
NLLH
ECE
AUMRC
MR5%
MR10%
MR25%
MC
MC
CIFAR-I0
5.12
0.203
2.71
1.08
2.85
0.83
98.50
NC- NC-VIBN (ours)
4.94
0.176
1.59
0.76
2.69
1.42
0.65
97.63
76.56
CIFAR-100
7,92
23.44
20.95
NC. V.C.IBN (ouzs)
24.26
1.098
5.92
7.30
21.50
28.97
12.28
87.00
46.44
VIBN NC-VIBN (ows)
43.41
2.031
11.73
18.10
41.00
38.57
31.07
83.26
48.86
ImageNet
42.74
1,910
9.99
18.03
40.38
38.11
30.73
82,56
46,78
ImageNer
Deterministic
24.35
0.971
3.74
7.38
21.53
19.08
12.50
87.48
58,31
NC. VIBN (ours)
24.34
0.974
2.28
7.56
21.59
19.22
12.87
86.13
51.79
Table 7: Ensembling results on CIFAR-10, CIFAR-100, and Tiny ImageNet.
MCR
NELH
ECE
AUMRC
MR5%
MR10%
MR25%
MC Cor:.
MC Incom:
CIFAR-10
Deterministic
5.37
0.237
3.22
0.53
3.11
1.83
0.33
98.57
82.11
NC-VIBN
4.93
0.165
1.28
0.57
2.83
1.52
0.45
97.48
71.99
Ensemble
4.10
0.141
0.76
0.33
2,06
1,03
0.59
97.18
67.75
NC. VCRV Ensemble
3.73
0.125
1.32
0.29
2.05
0.79
0.15
96.17
63.37
CIFAR- 100
Deterministic
26.22
1.146
10.92
7.73
23.32
20.73
13:40
92.28
63,39
NC-VIBN
26.19
1.310
8.50
8.29
23.62
21.17
14.10
90.24
59.50
Ensemble
22.29
0,850
1.96
6.00
19.28
16.75
10.42
87.19
50.76
VCBN Fisemble
21.31
0.793
4.56
6.06
18.66
16.37
10.76
83.08
43.49
Tiny
Deterministic
43.09
2.184
18.26
17.93
40.64
38.20
30.63
87.64
58,70
ImageNet
NC-VIBN
41.82
1.756
6.60
17.57
39.45
37.15
29.85
79.61
44.01
Ensemble
37.93
1.675
4.07
14.38
35.26
32.58
24.81
79.65
43.99
NC. VIBN Ensemble
37.32
1.517
3.39
14.48
34.76
32.26
24.96
73.76
35.99
Deterministic
NC-VIBN
Connect
Correct
Correct
025
0250
0.75
3.58
3.55
2.0
0.90
3.20
3.5
0.35
3.00
1.25
1.50
1.70
2.30
0.62
0.56
0.00
0.75
1.00
1.35
Entropy
Entropy
Entropy
Figure 5: Relative densities of the empirical predictive entropy distribution of correct and incorrect predictions for deterministic, VIBN, and NC-VIBB network CIFAR-10.
B.2 EMPIRICAL DISTRIBUTION PREDICTIVE ENTROPY VALUES
We show the empirical distribution deterministic, VIBN, and NC- VIBN networks on CIFAR-10 in F5g For all networks, the predictive entropy of correct predictions close to zero. The distribution of the predictive entropy of the misclassifications exhibits also peak near zero for the deterministic and VIBN network. This peak far less pronounced for our NC- NC-VIBN network, showing a more uniform distribution of predictive entropies for incorrect classifications. The NC-VIBN network is therefore less likely to assign low entropy to incorrect predictions.
B.3 INFLUENCE OF THE LAGR ANGE MULTIPLIER |
To study the impact of the Lagrange multiplier 3
Table Influence of the Lagrange multiplier B3 for our full NC-V NC-VIBN model from the main paper trained on CIFAR-1O.
controlling the weight of the latent KL-divergence term on the VCBV models, we vary 3 over range from 5 10-4 to 2 10-5 see Tab For all choices of 3, NC-VIBN performs better than VIBN (cf. Tab. 1) with respect to all evaluation metrics, showing that the improvements achieved by our NC-VIBN are robust regarding the choice of B. Further, we can observe a trade-off between the
5 5-10-
2-10-4
1-10-
5 10-
10-
MCR
4.90
4.89
4.93
5.02
5.03
NILH
0.172
0.173
0.165
0.167
0.170
ECE
1.43
0.74
1.28
1.64
1.89
AUMRC
0.62
0.59
0.57
0.55
0.56
ECE and AUMRC. Increasing 3 to 2 10-4 reduces the calibration error but increases the AUMRC
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Under review as a conference paper at ICLR 2022
and NLLH. Setting the value of B even higher to 5- 10 overregularizes the confidence of the model and results in a higher ECE as well as NLLH and AUMRC. On the other hand, decreasing & to 5 10-5 slightly decreases the AUMRC while increasing the ECE.
B.4 LAST-LAYER NC-VIBN
We additionally report results for modified setup, where the information bottleneck is placed before the last layer and the variance of the latent embeddings is fixed, see Tab. This simplification allows us to easily scale our approach to large-scale tasks, here ImageNet object classification.
Hyperparameter settings. For CIFAR-10, 10, the hyperparameter search above) for this additional last-layer setup identified suitable 3 as 10-4 for VIBN and 10-6 for NC- XC-VIBN and found 10- suitable for the decoder prior precision т. On CIFAR- 100, the parameter was found as 10 for VIBN and the prior precision for NC-VIBN is changed to 10-2, For TinylmageNet, 3 was found as 10-4 and the NC-VIBN decoder's 's prior precision as 10
ImageNet. The ImageNet models for this additional experiment trained for 90 epochs with batch size of 256 and a learning rate of 0.1, which is reduced by factor 0.1 at epochs 30, 60, and 80. The last layer NC-VIBN model scales the latent KL-divergence term with weight prior with precision 10 to the decoder's weights, and dampens the initial variance by factor of 10-3. To account for the higher sampling variance caused by the larger latent space dimension, samples are rescaled by factor of 0.75.
10 applies
Results. The last layer setup of NC-VIBN matches the deterministic baseline ImageNet regarding MCR, ECE, and AUMRC, while substantially improving the calibration by reducing the ECE from 0.037 to 0.023. For CIFAR-IO. CIFAR-100, and Tiny ImageNet, we test both the baseline last-layer VIBN as well as our proprosed coise-contrastive VIBN. For this setup again, our NC-VIBN models consistently perform significantly better than the respective VIBN models on all datasets with respect to all evaluation metrics.
Comparing this additional last- setup to the full setup in the main paper, see Tab. we find that the last-layer setup tends to be already competitive in terms of accuracy, but that our full setup has clear benefits terms of uncertainty estimation, where performs strictly better regarding the ECE and NLLHH. The benefit of the last-layer setup is that it easily scales to large-scale datasets such as ImageNet.
B.5 ENSEMBLING
We compare our NC-VIBN models with deep ensembles (Lakshminarayanan et al. 2017 CIFAR- 10, CIFAR-100, and Tiny ImageNet, see Tab.7 While an ensemble of four deterministic networks performs better than a single NC-VIBN network, it constitutes an almost fourfold increase of parameters and computation. A more appropriate comparison in terms of computational effort and memory requirements is thus to compare an ensemble of deterministic networks an ensemble of NC-VIBN networks. Here, the NC-VIBN ensembles are significantly more accurate, achieve lower NLLH and comparable AUMRC for all datasets. However, ensembles of NC- networks are not calibrated as well as deterministic ensembles. This is because our NC-VIBN models are designed to achieve good calibration with a single model and averaging the predictions of multiple NC-VIBN models leads to underconfidence.
REFERENCES
Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115 1-252, 2015.
18


Reviews:
[{"review_id": "xqmLLZ452J-", "reviewer": "Reviewer_mHTV", "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": "Strengths\n* The paper is generally easy to read.\n* The empirical comparison with baselines is rather comprehensive which includes different metrics. The proposed method leads to consistent better performance over output regularization methods. \n\nWeakness\n* The proposed method consists of three components: 1) Bayesian learning for classifier weight; 2) noise-contrastive loss; 3) test-time normalization. But 1) & 3) are not very well motivated and disconnected with the previous empirical analysis, and they are ablation-studied separately in Table 3. It could be better to reorder Sec. 5 by introducing 2) first (which is closely related to previous analysis) and then 1) & 3) with more motivating explanations (especially for 3) that seems to be hand-wavy). Furthermore, from Table 3, it seems that 1) & 3) indeed improve both the accuracy and calibration a lot, and 2) improves calibration but slightly hurts the accuracy (MCR). More detailed discussion about the effectiveness of each component needs to be included. \n* The interplay between variational information bottleneck (VIB) loss and noise-contrastive loss is not clear. The VIB loss encourages the encoder latent distribution to be close to the latent prior (by a KL divergence), but the noise-contrastive loss encourages high predictive entropy for the latent prior. Would it also encourage the encoded latents to have a high entropy, and why a separation between correctly classified latents and the prior latents would happen from a theoretical perspective? It would be great to include a more detailed discussion. \n* The claim that the noise-contrastive loss increases the separability of correct and incorrect samples is not supported by enough empirical evidence. From Table 1, it is hard to tell if VC-VIBN consistently improves the relevant metrics (including AUMRC, MRX%, MC Corr., MC Incorr.) over its VC-VIBN counterpart and the MC Dropout baseline, especially on large datasets like CIFAR-100 and Tiny ImageNet. \n\nAdditional questions & comments \n* It seems that the proposed loss only regularizes the predictive entropy of noise samples (from the latent prior) but no contrastive loss used, why is it called the noise-**contrastive** loss?\n* Since Fig. 2(a) is plotted with NC-VIBN, it is not convincing enough that the misclassified samples would get clustered with latent prior samples for the original VIBN model without the NC loss, similarly to a chicken-and-egg problem. It would be great to have a plot with the original VIBN model (i.e., Fig. 4) in the main body. A discussion for the comparison between Fig. 2 and Fig. 4 can also highlight the effect of the noise-contrastive loss.\n* Typo in Eq. (4) and above: $p(y_n | z_n, \\psi)$ -> $p(y_n | x_n, \\psi)$, $\\mathcal{D}(z_n, y_n)$ -> $\\mathcal{D}(x_n, y_n)$\nSince the empirical comparison involves several metrics, it would be more friendly for the reader to indicate for each metric whether it is lower is better or higher is better in the tables.\n", "summary_of_the_paper": "This paper first empirically observes that output regularization methods for reducing model overconfidence improves calibration but make correct and incorrect predictions indistinguishable based on predictive uncertainty.  To tackle the issue, it proposes a noise-contrastive loss and some architectural refinements based on the variational information bottleneck. Empirical results demonstrate both improved accuracy and calibration over output regularization methods. ", "summary_of_the_review": "The central proposal of the paper is the noise-contrastive loss for VIBN, but its effectiveness is obfuscated by other two proposed tricks that are not well motivated and studied. Though the proposed combined method leads to better accuracy and calibration than output regularization methods, the claim that the noise-contrastive loss increases the separability of correct and incorrect samples is not clearly backuped with sufficient empirical evidence. Thus, I recommend a ‘reject’ for the paper but will consider raising my score if my concerns are well addressed. \n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "wu9dqvVOL_Y", "reviewer": "Reviewer_r8pu", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "**Pros**\nThe authors provide an intriguing solution to a problem concerning the uncertainty estimation of machine learning models. Although the proposed method is limited to the variational bottleneck network, it is simple and practical and empirically proves that it is more effective than VIBN and IDO in the separability problem. \n\n**Cons**\nMy main concern is empirical verification. The reviewer thinks that the proposed method should be tested to determine if it works well with different convolutional networks other than ResNet. Also, the reviewer wonders the proposed method will work if we apply distribution matching methods in input space, such as augmentation strategies [1, 2], to the VIBN. In the appendix, the authors are experimenting with relatively simple augmentation policies only. It is crucial to clarify how the proposed loss term works if strong augmentation policies are used.\n\n[1] AutoAugment: Learning Augmentation Policies from Data, Cubuk et al., *CVPR* (2019)\n\n[2] Fast AutoAugment, Lim et al., *NeurIPS* (2019)", "summary_of_the_paper": "This work suggests a *noise-contrastive loss* for variational information bottleneck networks to resolve the poor performance at separating correct and incorrect predictions in regularization methods. Standard regularization methods suffer *separability problems*: models indiscriminately regularize confidence to improve calibration perform much worse at differentiating correct from wrong predictions. The authors address this problem in the variational information bottleneck, suggesting a remedy that utilizes the distribution matching in the latent space. The authors have applied the proposed loss term to the two variational approximations of the information bottleneck, the deep variational information bottleneck (VIBN, Alemi et al., 2017) and information dropout (IDO, Achille & Soatto, 2018).", "summary_of_the_review": "The paper is generally well-written and given the technical content. The reviewer considers the proposed method quite interesting, but some issues are to be addressed empirically. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "OL7ClwPTBis", "reviewer": "Reviewer_DCEH", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "Please see my detailed comments below:\n\nW1: Is there any intuition why “embeddings of hard examples tend to be aligned with the latent prior” and “is easier to be optimized” Is there any empirical evidence or reference supporting such argument? (Description and motivation above Eq. (6). It seems this is the motivation of the paper, and I think it is better to provide some empirical evidence of such claim. \n\nW2: I am not sure if the \"noise-contrastive\" loss is a proper description of the penalty Eq. (6), since it seems to be an entropy term in absence of any contrastive flavor. Please could you clarify what is what is the connection between the entropy term with the “noise-contrastive”. \n\nW3: Novelty. It seems to be that the only difference between the proposed framework and conventional VAE framework is the entropy term and the additional prior on the network weights. I think there have been many works investigating the entropy term in similar conditional generative process context though. Take for example, in paper [a] [b], the entropy loss has already been explored thoroughly in order to encorage feature seperability, when it is associated with a generative loss, where a locally-Lipschitz constraint is even further imposed to guarantee the correctness of such entropy loss in [a]. It seems to me that the minor difference here is that the generative model now belongs to a variational inference family such as VAE rather than adversarial training (i.e., GAN) in [a] [b]. Please correct me if I misunderstood anything.  \n\nW4: Empirical evidence. The paper claims that harder samples shall benefit from the proposed procedure owing to the additional entropy constraint. However, it seems to me that the empirical evidence does not fully support this claim. I noticed that on Tiny ImageNet100 which is seemingly more complex than CIFAR10/100, the proposed method could hardly beat the other state-of-art method, leaving me doubt the effectiveness of the method. \n\nIf possible, please clarify the above concerns during the rebuttal. I am happy to increase my score if the above issues or misunderstandings are properly addressed. \n\n\n[a] Shu et al., “A DIRT-T Approach To Unsupervised Domain Adaptation” ICLR 2018.\n\n[b] Miyato et al., “Virtual Adversarial Training: A Regularization Method for Supervised and Semi-Supervised Learning”., IEEE Transactions on Pattern Analysis and Machine Intelligence 2019\n", "summary_of_the_paper": "The paper proposes a new model NC-VIBN for the classification tasks. To ensure the separability between classes, the authors introduced an entropy term into the conventional VAE framework. ", "summary_of_the_review": "I cannot recommand the acceptance of the paper until the above issues and concerns are addressed. But I would be happy to increase my score if any potential misunderstandings are clarified. Please see detailed comments above regarding technical concerns. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "8RcsIfWUp9H", "reviewer": "Reviewer_dzx4", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "### Novelty\n\nThis most significant contribution is the noise contrastive loss term. This builds on the one proposed by Hafner et al. (2020), but differs by drawing samples from a prior over latents (in the information bottleneck), rather than from a distribution over inputs. This is significant and, I imagine, necessary for scaling the noise-contrastive loss to high-dimensional inputs. The L2-normalisation step is an additional minor contribution. While the use of a variational distribution over neural net weights is not novel in itself, its combination with the other components is shown to be helpful.\n\n### Strengths\n- Proposed methods make sensible design choices. \n- Thorough experiments with ablation studies showing the usefulness of each component.\n- Good performance from the proposed methods, especially in terms of the negative log-likelihood (NLLH).\n\n### Weaknesses/questions\n- A standard technique for improving generalisation and reducing overconfidence is early stopping (e.g. by saving multiple checkpoints throughout training and taking the one which is best according to some validation metric). As far as I can tell, the authors did not do this for any baselines (or their models). I would imagine that doing so might significantly improve the baselines, particularly the deterministic classifier. The choice of which metric to use to decide when to stop (e.g. MCR or NLLH) could also have a big impact. Could the authors comment on this?\n- Is the \"test-time normalisation\" step also performed at training time? If so, it looks very similar to Instance Normalisation (https://arxiv.org/abs/1607.08022, https://pytorch.org/docs/stable/generated/torch.nn.InstanceNorm1d.html) except that the mean is not subtracted during normalisation. Readers might find it helpful if the paper comments on the similarity between these methods.\n- Can the authors give guidance on (how to choose) which of NC-IDO, NC-VIBN or NC-VIBN-MP to use in practice?\n", "summary_of_the_paper": "This paper attempts to improve uncertainty estimates for neural network-based classification. To do so, they combine various techniques: the information bottleneck, a variational distribution over neural net weights, a noise contrastive loss term, and an L2-normalisation step for the last part of encoder. They present experimental results justifying these choices (including, notably, significantly improved NLL compared to the baselines).", "summary_of_the_review": "Overall, I am giving a score of 6. The novelty is not ground-breaking but the authors achieve good results for an important topic. Importantly, the design choices made are well supported by ablation studies.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a classification method that improves model calibration using variational information bottlenecks and a noise-contrastive loss.

Unfortunately, the authors' were not able to participate in the discussion of the paper with the reviewers. Given this, the reviewers raised several unaddressed concerns: First, it was argued that the different components of the proposed method required additional justification, in particular with regards to novelty. Second, reviewers argued that the paper required additional empirical validation, for example by testing if it works well with different convolutional methods such as ResNets.

Given these concerns, a consensus was reached that this paper should be rejected which is also my recommendation,
Decision: Reject