Deep Classifiers with Label Noise Modeling and Distance Awareness
Authors: Vincent Fortuin
Mark Collier
Florian Wenzel
James Urquhart Allingham
Jeremiah Zhe Liu
Dustin Tran
Balaji Lakshminarayanan
Jesse Berent
Rodolphe Jenatton
Effrosyni Kokiopoulou
Venue: Accepted by TMLR ()

Abstract: Uncertainty estimation in deep learning has recently emerged as a crucial area of interest to advance reliability and robustness in safety-critical applications. While there have been many proposed methods that either focus on distance-aware model uncertainties for out-of-distribution detection or on input-dependent label uncertainties for in-distribution calibration, both of these types of uncertainty are often necessary. In this work, we propose the HetSNGP method for jointly modeling the model and data uncertainty. We show that our proposed model affords a favorable combination between these two types of uncertainty and thus outperforms the baseline methods on some challenging out-of-distribution datasets, including CIFAR-100C, ImageNet-C, and ImageNet-A. Moreover, we propose HetSNGP Ensemble, an ensembled version of our method which additionally models uncertainty over the network parameters and outperforms other ensemble baselines.

Published in Transactions on Machine Learning Research (07/2022)
Deep Classifiers with Label Noise Modeling and Distance Awareness
Vincent Fortuin"i
vbf21 @commac.m..ccc.m.mmm uk
Deportment of Computer Science, ETH Zürich Deportment of Engineering, University of Cambridge
Mark Collier
markeollier@ggonll..coo
Google Research
Florian Wenzel, James Allingham, Jeremiah Liu, Dustin Tran, Balaji Lakshminarayanan, Jesse Berent, Rodolphe Jenatton, Effrosyni Kokiopoulou Google Research
Reviewed on OpenReview: /ttp::///opereveteee
id- Id7ATt78FV
Abstract
Uncertainty estimation in deep learning has recently emerged as crucial area of inter- est to advance reliability and robustness in safety-critical applications. While there have been many proposed methods that either focus on distance-aware model uncertainties for out-of-distribution detection or on input-dependent label uncertainties for in-distribution calibration, both of these types of uncertainty are often necessary. In this work, we propose the HetSNGP method for jointly modeling the model and data uncertainty. We show that our proposed model affords favorable combination between these two types of uncertainty and thus outperforms the baseline methods on some challenging out-of-distribution datasets, including SIFAR-100C, ImageNet-C. C, and ImageNet-A. Moreover, we propose HetSNGP En- semble, an ensembled version our method which additionally models uncertainty over the network parameters and outporforms other ensemble basclincs.
1 Introduction
While deep learning has led to impressive advances in predictive accuracy, models often still suffer from overconfidence and ill-calibrated uncertainties (Ovadia et al. 2019). This is particularly problematic in safety-critical applications (e.g., healthcare, autonomous driving), where uncertainty estimation is crucial to ensure reliability and robustncss (Filos et al. 2019 Duscnberry al., 2020).
Predictive uncertainties generally come in two flavors: model uncerlainly (also known epistemic) and dala uncertainty (also known aleatoric) [Murphy] 2012). Model uncertainty measures how confident the model should be based on what it knows about the world, that is, how much it can know about certain test data given the training data it has seen. Data uncertainty measures the uncertainty intrinsic in the data itself, for example due to fundamental noise in the labelling process. Good model uncertainty estimates allow for out- of-distribution (OOD) detection, that is, for recognizing data examples that are substantially different from the training data. On the other hand, good data uncertainty estimates allow for in-distribution calibration, that is, knowing which training (or testing) data examples the model should be more or less confident about. Fig. 1 demonstrates model uncortainty for synthetic datasct, while Fig. 2 shows the effect of modeling the data uncerlainty arising from noisy labels. Ideally would like single model which offers in-distribution uncertainty modeling while reverting to uncertain predictions when faced with OOD examples.
"The research done during an internship Google Research.
Equul contribution.

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Many recently proposed models for uncertainty estimation excel at one the other of these uncertainty types. For instance, the spectral-normmalzedd Gaussian process (SNGP) (Liu et al. 2020) uses latent Gaussian process to achieve distance-aware model uncertainties and thus affords excellent OOD detection. Conversely, the hetcroscodastic classification method (Collier et al. 2020 2021) offers superb in-distribution calibration and accuracy by modeling impul- and class-dependent label noise the training data. However, there have been few attempts to combine the complementary benefits of these two types of uncertainty modeling (e.g., Depeweg al., 2018) (see related work in Section 4
In this work, propose the heteroscedastic SNGP (HetSNGP), which allows for joint modeling of model and data uncertainties using hicrarchical latent variable model. We show that HetSNGP gives good in- distribution and OOD accuracy and calibration, yielding a model with uncertainties suitable for deployment in critical applications.
Our main contributions are:
We propose new model, the heteroscedastic speetraa-normmalieed Gaussian process (HelSNGP), with both distance-aware model and data uncertainties.
We describe an efficient approximate inference scheme that allows training HetSNGP with a com- putational budget comparable standard neural network training.
We introduce a new large-scale OOD benchmark based on ImageNet-21k. We hope this benchmark will prove useful for future research in the field.
We show empirically on different benchmark datasets that HetSNGP offers a favorable combina- tion of model and data uncertainty. It generally preserves the SNGP's OOD performance and the hetcroscedastic in-distribution performance. It even outperforms these basclincs on some datasets, where both OOD and heteroscedastic uncertainties are helpful.
We propose an ensembled version of our model, the HetSNGP Ensemble, which additionally accounts for uncertainty over the model paramoters and outperforms other ensemble basclines.
2 Background
2.1 Model uncertainty and spectral-normalizer Gaussian process
Model uncertainty (or epistomic uncertainty) captures all the uncertainty about whether a given model is correctly specified for a certain lask, given the training data. Before any training data has been observed, this uncertainty depends only on the prior knowledge about the task, which can for instance be encoded into distributions over the model parameters or into the architecture of the model (i.e., the model class) (Fortuin 2021). After training data has been observed, one should expect the model uncertainty to decrease within the support of the training data distribution, that is, on points that are close to the training data in the input space. We will generally call these points in-distribution (ID). Conversely, on points that are far away from the training points and thus out- diloo--iittibuttin (OOD), we should not expect the model uncertainty to decrease, since the training data points are not informative enough to make any assertions about the correctness of the given model on these points. In this sense, we would like the uncertainty be distance-aware, that is, it should grow away from the training data (Liu al. 2020). While the term OOD is used in many different contexts in the literature, we will generally use it to refer to data points that could not have plausibly been generated by the data distribution, as opposed to data points that are just unlikely under the data distribution. For instance, when assossing OOD gencralization for a dataset of photos of cats and dogs, OOD points in our nomenclature might be carloon depictions of cats of dogs. They still can plausibly be classified into the same two classes, but they could not have been generated via the assumed generative process of taking photos of real animals. For the purposes of OOD detection, we might also consider data points that cannot be assigned any of the existing classes in the training set, for instance, photos of rabbits.
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The spectral-normalized Gaussian process (SNGP) Liu et al. uncertainties by specifying Gaussian process prior Rasmussen & Williams 2006) over the latent data rep- resentations in the penultimate layer of the neural network. Distance-awareness is ensured by using spectral normalization on the pro-logit hidden layers (Behrmann et al., 2019), which encourages bi-Lipschitzness of the mapping from data to latent space. This approximately preserves the distances between data points in the latent representation space by stopping the learned features from collapsing dimensions the input space onto invariant subspaces (Liu al. 2020). Note however that while the Euclidean distance in the high-dimensional input space might not always be semantically meaningful, the neural network still has some freedom to transform the data into feature space where distances are more meaningful, and indeed it has an incentive to do so in order to improve predictive accuracy and uncertainty estimation. Indeed, Liu et al. (2020) showed that this bi-Lipschitaness leads to learning semantically meaningful representations in practice. Note also that this approach only partially captures the model uncertainty, namely in form of uncertainty over the latents. It does not however capture the uncertainty over the model pa- rameters, such for instance Bayesian neural networks (MacKay 1992] Neal 1993) or ensemble met [aaksmiiaraaaaaa et al. 2017). This motivates our HetSNGP Ensemble (presented later in the paper, see Section 5.5
2020 provides such distanco-aware model
2.2 Data uncertainty and the heteroscedastic method
As opposed to the model uncertainty described above, data uncertainty intrinsie in the data, hence irre- ducible with more training data. In the case of continuous data (e.g., regression problems), data uncertainty often comes in the form of random noise on the measurements. For discrete data (e.g., classification prob- lems), it usually arises as incorrectly labelled samples. This label noise can be class- and input-dependent (Beyer et al.] 2020). For instance, the ImageNet dataset. (Deng et al.) 2009). contains 100 different classes of dog breeds, which are often hard for human labelers to distinguish from each other. Modeling this type of data uncortainty can improve the calibration and robustness of predictive models (Collier et al.. 2021).
A model that does explicitly handle the input- and class-dependent label noise is the heteroscedastic method (Collier et al. 2021). The heteroscedastic method models input- and class-dependent noise by introducing latent multivariate Gaussian distribution on the softmax logits of standard neural network classifier. The covariance matrix of this latent distribut is a function of the input (hederoscedastic) and models inter-class correlations in the logit noise.
3 Method
3.1 Setup and Notation
Let us consider a dataset D = {ax, y:) of input-output pairs, where æ; E gd and yi E (I,..., K}, that is, a classification problem with K classes. The data examples are assumed to be sampled i, from some true data-generating distribution ye) ~ (x,
3.2 Generative process
To jointly model the two different types of uncertainty, we propose a hicrarchical model of two latent random variables, which we denote by f and u. f is latent function value associated to the input æ (as in the Gaussian process literature) and is designed to capture the model uncertainty, while u is latent vector of logits (or utilities) that capture the data uncertainty, similar to the setup in Collier et al. (2020), which was inspired by the econometrics literature (Train, 2009). Similarly to Liu et al. (2020), we place a latent Gaussian process (GP) prior over f, as p(f) GP k)),,)),, where ki is a parameterized kernel function with parameters 6. Note that, in our case, this kernel is parameterized by the learned neural network, so it allows for an expressive feature mapping that can learn to represent rich semantic features the data. Evaluating this kernel on all pairwise combinations of data points in æ yields the kernel matrix (z(x, We then define a (heteroscedasticalll) noisy observation of f.
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Stacking the variables across the whole dataset gives the matrices F,U RNNK We refer to their respective rows as fú, u, E R&K and their columns as fc: ue E TRN. The columns are independent under the GP prior, but the rows are not. Conversely, the columns are correlated in the heteroscedastic noise model, while the rows are independent. A hicrarchical model using both uncortaintics therefore leads to logits that are correlated across both data points and classes. While this joint modeling of model and data uncertainly will not necessarily be useful on all tasks (e-g., in-distribution prediction on data with homoscedastic noise), we expect it to lead to improvements on out-of-distribution imputs from dalasets with class-dependent label noise. We will see in Section 3 that this is actually the case in practically relevant settings, such as when training on Imagenet-21k and predicting on Imagenet. We now give formal description of the model.
We assume the full generative process is
fe ~ N(0, Ko(m,a))
(1)
u, ~ NNs: (x;; v))
(2)
ply: = cl uu) = 1 = arg max Uikk
(3)
k
We can compute the marginal distribution, that is pvy x) Eulp(y =]]= S ply u) p(u [x) du. Intuitively, f captures the model uncertainty, that is, the uncertainty about the functional mapping between æ and y on the level of the latents. It uses the covariance between data points to achieve this distance-awareenss, namely it uses the kernel function to assess the similarity between data points, yielding an uncertainty estimate that grows away from the data. On the other hand, u captures the data uncertainty, by explicitly modeling the per-class uncertainty on the level of the logits. & in (2) can learn encode correlations in the noise between different classes (e.g., the dog breeds in ImageNet). It does not itself capture the model uncertainty, but inherits it through its hierarchical dependence on f, such that the resulting probability p(y u) ultimately jointly captures both types of uncertainty.
In practice, we usually learn the kernel using deep kernel learning (Wilson et al. 2016), that is, we define it as the RBF kernel Kg (Z,, 2,) = KRBF (h,, h2) = exp(11h; - h2 ll2/A) with length scale À and h, = h( 8) e Rm with h(:;)) being neural network model (c.g., ResNet) parameterized by 0. This kernel is then sharod between classes. Moreover, following Liu et al. (020)), we typically encourage bi-Lipschitzness of using spectral normalization (Behrmann et al. 2019). which then leads to an approximate preservation of distances between the input space and latent space, thus allowing for distance-aware uncertainty modeling in the latent space. Additionally, [(: (p) is usually also neural network parameterized by s. To case notation, we will typically drop these parameters in the following. To make the model more parameter-effiiinnt we will in practice use h; as inputs for E(-), so that the network parameterizing the GP kernel can share its learned features with the network parameterizing the heteroscedastic noise covariance. This usually means that we can make Σ(-) rather small (e.g., just single hidden layer). Note that the prior over fc (and thus also ue) has zero mean, thus leading to a uniform output distribution away from the data. This is also reminiscent of multi-task GPs (Williams et al.] 2007), where separate kernels are used to model covariances between data points and tasks and are then combined into Kronecker structure.
3.3 Computational approximations
The generative process above requires computing two integrals (i.e., over and f) and inferring an exact. GP posterior, making it computationally intractable. We make several approximations to ensure tractability.
Low-rank approximation. The heteroscedastic covariance matrix E(z; () is a K x K matrix which is function of the input. Parameterizing this full matrix is costly in terms of computation, parameter count, and memory. Therefore, following Collier et al. (2021). we make a low-rank approximation Σ(z;;) = Tx. Here, V(x;) is a K x R matrix with R < K and (xx,) is a K dimensional vector added to the diagonal of ((x.) (x:)).
V(x.) ((x.)) +
Also following Collier et al. (2021), we introduce a parameter- efficienn version of our method enable scaling to problems with many classes, for instance, ImageNet21k which has 21 343 classes. In the standard version
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of our method, V(x,) is parameterized an affine transformation of the shared representation h,. In this way, HetSNGP can be added as single out layer on top of a base network. For the parameterr-eficient HetSNGP, we parameterize (æ:)) = v(x,)1R R @ V where v(x;) is a vector of dimension K, 1 is a vector of ones of dimension R, V is a K R matrix of free parameters, and @ denotes clement- wise multiplication. This particular choice of approximation has shown a favorable tradeoff between performance and computational cost in the experiments conducted by Collier et al. (2021), , but naturally other approximations would be possible. For instance, one could use Kronecker-factored (KFAC) approximation (Martens & Grosse| 20155 Botev et al. 2017) k-tied Gaussian parameterization (Swiatkowski et al. 2020 Exploring these different options could be an interesting avenue for future work.
Random Fourier feature approximation. Computing the exact GP posterior requires ((N2) opera- tions (because ones needs to invert an N x N kernel matrix), which can be prohibitive for large datasets. Following Liu al. (2020), we thus use a random Fourier feature (RFF) approximation (Rahimi & Rocht, 2007), leading a low-rank approximation of the kernel matrix as Ko x) Φ (& E R'*",, with ran-
dom features .. = / 2 cos(Wh,, + b) where W is a fixed weight matrix with entries sampled from N/ 1) and b is a fixed bias vector with entries sampled from 4(0, 2x). This approximates the infnitteddimessional reproducing kernel Hilbert space (RKHS) of the RBF kernel with a subspace spanned by randomly sam- pled basis functions. It reduces the GP inference complexity to O(Nm²), where m is the dimensionality of the latent space. We can then write the model as a lincar model in this feature space, namely
u, = f: + d(z,) O CK + V(xe)en with f cp and p(3.) No Im): p(ex) = NO Ix); p(en) = (O,
(4)
Here, D.3. is a linear regressor in the space of the random Fourier features and the other two terms of u, are the low-rank approximation to the heteroscedastic uncertainties (as described above). Again, m can be tuned as hyperparameter to trade off computational accuracy with fidelity of the model. Since most datasets contain many redundant data points, one often finds an m & N that models the similarities in the data faithfully.
Laplace approximation. When using a Gaussian likelihood (i.e., in a regression setting), the GP posterior inference can be performed in closed form. However, for classification problems this not possible, because the Categorical likelihood used not conjugate to the Gaussian prior. We thus need to resort to approximate posterior inference. Again following Liu et al. (2020), we perform a Laplace approximation (Rasmussen & Williams 2006) to the RFF-GP posterior, which yields the closed-form approximate posterior for B.
N
= P(B:ID)=N(BB.
Ec) with Σ-1 = Im + Σ Pr. B..)
(5)
1=1
where Pt.e is shorthand for the softmax output p(y; = cl (z)) as defined in Eq. 3 where dg.c = T. 9.. The derivation of this is deferred to Appendix A.2 Here, Σ-1 can be cheaply computed over minibatches of data by virtue of being sum over data points. Moreover, B. is the MAP solution, which can be obtained using gradient descent on the unnormalized log posterior, - logp(P | (D) x - llggg((D) DDD (3) ||/112 where the squared norm regularizer stems from the standard Gaussian prior on B. Using our likelihood, the training objective takes the form of ridge-regularized cross-entropy loss. We also use this objective to train the other trainable parameters 0 and o of our model.
Note that this approximation is necessarily ignoring aulti-modalite of the posterior, since it can by definition only fit one mode around the MAP solution. Nonetheless, recent works have shown that it can achieve suprisingly good practical performance in many settings and offers a good tradcoff between performance and computational cost (Immer et al. a0)
Monte Carlo approximation. Finally, we are ultimately interested in En u)] J p(y () p(u] [2) du, which again would require solving high-dimensional integral and is thus com- putationally intractable. Following Collier et al. (2020), we approximate it using Monte Carlo samples.
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Moreover, we approximate the argmax in Eq. 8 with a softmax. To control the trade-off between bias and variance in this approximation, we introduce an additional temperature parameter into this softmax, leading to
p(u: mi) 1
5 Σ plus = d ==)== 1
= Σ
exp(uf,0/T)
(6)
5
2k=1 exp(uf,x/T)
K
with -h. = 2 B d(x)) 3 e + (<x)een
and ~ ~(Be D) and EK ~O( and En ~ ~O,,
It should be noted that we only use Eq. 66 at test time, while during training we replace B with its mean Be instead of sampling it (see Section 3.4) Note also that, while a larger number S of Monte Carlo samples increases the computational cost, these samples can generally be computed in parallel on modern hardware, such that the wallclock runtime stays roughly the same. Again, the number of samples S can be tuned as a hyperparameter, such that this approximation can be made arbitrarily exact with increased runtime (or, as mentioned, increased parallel computational power).
3.4 Implementation
Algorithm HetSNGP training
Require: dataset D = [[(x;, W)) Initialize 0, W, b, E for train step 1 to max step do Take minibatch (X,, v.) from D for s = to S do
Algorithm 2 HetSNGP prediction
Require: test example x²
4- V 2/ cos(Wh (Wh(("") + b)
C ~ ~N, Ik), ) cR ~ NNO
for 8 = 1 to S do
ur,c- T Be + d( ② c + V(n)ee
8 p(8 (D)
end for
c ~ N(, Ix), CR NO.
C= S E.= logg(o yi u³) 11812
1
" - Ba d(x x²)) O c + V(x*)<
Update (0, p, p} via SGD L if final epoch then
end for
exp(u2.
ply" =c ==")=} E Eiz exp(u
-ik Compute [__ J= as per Eq. 5
Predict y² arg max, ply" = =clx")
end if end for
The training of our proposed model is described in Algorithm and the prediction in Algorithm
Intuition. The distance-aware uncertainties of the SNGP modeled through b which itself depends on the latent representations h, which are approximately distance-preserving thanks to the bi-Lipschitzness. This means that when we have an input Uaar that is far away from the training data, the values of the RBF kernel exp(Iih hise Il2 will be small, and thus the uncertainty of fe will be large. The resulting output uncertainties will therefore be large regardless of the heteroscedaastic variance E(r,)) (since they are additive) allowing for effeetive OOD detection. Conversely, we have an in-distribution input, the kernel values will be large and the model uncertainty captured in fa will be small. In this case, the additive output uncertainty will be dominated by the heteroscedastic variance E(x;), allowing for effective in-distribution label noise modeling.
While at first it might seem straighuforward to combine the two types of uncertainly, it should be noted that the proposed hierarchical model is by no means an arbitrary choice. Rather, it is the only design we found that robustly achieves disentanglement of the two uncertainty types during training. When using other architectures, for instance, using the heteroscedastic uncertainty directly the likelihood of the GP using two additive random variables with respective GP and heteroscedastic distributions, the two

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Table Comparison to related work. We see that our proposed HetSNGP model is the only one that captures distance-awaro model uncertainties and data uncertaintics in a scalable way.
MODRL UNCERTAINTY
SCALING
ENSEMBLE (SIZE M) APPROX. BAYESIAN INFERENCE
DETERMINISTIC
DEEP
O(M)
SNGP
MIMO
X
X
X
POSTERIOR NETWORK
X
# CLASSES
HET.
X
X
X
' O(M)
HETSNGP
HETSNGP
HETSNGP MIMO
/
models will compete for the explained uncertainty and the heteroscedastic prediction network will typically try explain as much uncertainty possible. Only in this hicrarchical model sctting, where the SNGP adds uncertainty to the mean first and the heleroscedastic uncertainly is added in the second step, can we robustly achieve correct assignment of the different uncertainties using madient-bassed optimization.
Challenges and limitations. As outlined above, the main challenge in this model is the inference. We chose an approximate inference scheme that is computationally efficient, using a series of approximations, and modeling the distance-aware and heteroscedastic variances additively. If we wanted to make the model more powerful, at the cost increased computation, we could thus consider several avenues of improvement: (i) use inducing points or even full GP inference for the GP posterior; (ii) use a more powerful approximation than Laplace, for instance, variational approximation; or (ii)) model the variances jointly, such that we
do not only have covariance between data points in f and between classes in u, but have full covariance between different classes of different data points in u. All of these ideas are interesting avenues for future research.
4 Related Work
Uncertainty estimation. There has been lot of research on uncertainty estimation in recent years (Ovdiaa et al.) 2019), including the carly observation that one needs data and model uncortaintics for successful use real-world applications (Kendall & Gal 2017/. There have also been attempls to directly optimize specialized loss functions to improve model uncertainty (Tagasovska Lopez-Paz, 2019 Pearce et al. 2018). However, one wants to directly implement prior knowledge regarding the OOD behavior
of the model, one usually needs access OOD samples during training (Yang et al. 2019 Hafner et al,
2020 Another popular idea is to use a hierarchical output distribution, for instance Dirichlet distribution
[Milios et al.] 2018 Sensoy et al. 2018 Malinin & Gales 2018 Malinin et al.) 2019 Malinin & Gales 2019 Hobbhahn et al.] 2020 Nandy et al. 2020). such that the model uncertainty can be encoded in the Dirichlet and the data uncertainty in its Categorical distribution samples. This idea was also used in our
Posterior Network baseline (Charpentier et al. 2020 While this allows to capture both model and data uncertainty, the model uncertainties are necessarily distance-aware, which has been shown to be crucial for effective OOD detection (Liu et al. 2020). Moreover, the Dirichlet data uncertainties in these models do not generally allow for correlated class-dependent uncertainties, such as in model and Collier et al.) (2021). Similarly to our idca of Icarning separate variances, it has also been proposed to treat the output variance variationally and lo specify a hierarchical prior over (Stirn & Knowles 2020). Finally, single-pas uncertainty approaches such as DUQ (Van Amersfoort et al. 2020 and DUE (van Amersfoort et al.) 2021) also capture model uncertainty in a scalable way, but do not additionally capture the data uncertainty. They could possibly be used as drop-in replacement for the SNGP component in our hierarchical model, which is interesting avenue for future work.

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(a) Deterministic
(b) Heteroscedastic
(c) SNGP
(d) Posterior Net
(e) HetSNGP
(f) Deterministic
(g) Heteroscedastic
(h) SNGP
(i) Posterior Net
(i) HetSNGP
Figure 1: Estimated model incertainties synthetic datasets. The red cluster of points unseen OOD points, while the other colors represent different classes of training data. The background color shows the predictive uncertainty (maximum class probability) for the different methods. The deterministic and heteroscedastic methods are overconfident on the OOD data, while the other methods offer distanco-awaro uncertainties. The uncertainties of the Posterior Network do not grow quickly enough away from the data, such that only the SNGP and proposed HetSNGP provide effective OOD detection.
Bayesian neural networks and ensembles. The gold-standard for capturing model uncertainty over the parameters are Bayesian neural networks (Neal 1993 MacKay 1992). but they are computationally
quite expensive, require a careful choice prior distribution (Fortuin et al. 2021b Fortuin 2021 and
require specialized approximate inference schemes, such as Laplace approximation (Immer et al. 2021b
variational inference (Hinton Van Camp 1993 Graves 2011
2015 or MMrkovv Chain Monte Blundell et al.
Carlo (Welling & Teh 2011 Garriga-Alonso Fortuin 2021
Fortuin et. al.
2021a] A more tractable model
class deep ensemble methods (Lakshminarayanau et al. 2017
Ciosek al. 2019 D'Angelo et al. 2021
D' D'Angelo & Fortuin
2021), although they are computationally still expensive. There are however some ideas
to make them less expensive by distilling their uncertainties into simpler models (Malinin et al. 2019 Tran
ct al.
2020 Havasi et al. 2020 Antorán et al. 2020
SNGP and heterosccedastic method. The models most relevant to our approach are the SNGP [Liu et al. 2020) and the heteroscedastic method (Collier et al.) 2020: 2021). SNGP yields distance-aware model uncertaintics, which grow away from the training data. The heteroscedasticc method models input- and class- dependent label noise data uncertainty inside the training distribution. Heleroscedastic data uncertainly modeling can also improve segmentation models (Monteiro et al. 2020) and has been linked to the cold posterior effect in Bayesian neural networks (Wenzel et al., 2020 Adlam et al., 2020: Aitchison, (220)). Prior work has primarily focused on modeling data and distance-aware model uncertainty separately. But safety- critical practical applications require both types of uncertainty, and we see Table [ that our proposed HetSNGP mothod is the only one which offers joint modcling of distancc-aware model uncortaintics and data uncertainties, while still being scalable.
5 Experiments
We conducted experiments synthetic data and standard image classification benchmarks. As baselines, compare against standard deterministic model (He et al.) 2016 Dosovitskiy et al. 2020), the het- croscodastic method (Collier et al.) (021) and SNGP (Liu et al. 2020). We also compare against the Posterior Network model (Charpentier et al.) 2020). which also oflers distance-aware uncertainties, but it only applies to problems with few classes. Note that our main motivation for the experiments to assess whether combining the SNGP and heteroscodastic method can successfully demonstrate the complementary benefits of the two methods. Since not all tasks and datasets require both types of uncertainty at the same time, do not expect proposed model to outperform the baselines all tasks. We would however

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Figure 2: Synthetic data task with label noise. From left to right: clean labels, noisy labels, deterministic predictions (0.804 accuracy), SNGP predictions (0.841 accuracy), heteroscedastic predictions (0.853 accu- racy), and HetSNGP predictions (0.866 accuracy). We see that the proposed method performs on par with the heteroscedastic method and that they both outperform the other baselines thanks to their label noise modcling capabilities.
expect that, depending on the particular task and required uncertainty type, it would generally outperform one of the two baselines. Our non-synthetic experiments are developed within the open source codebases petrraiintybasselinng
(Nado et al., 2021 and robustness, tetettttssssmiiccssssssss metrics (Djolonga et al. 2020) (to assess
the OOD performances). Implementation details are deferred to Appendix A.1 Our implementation of the HctSNGP is available as a layer in edward2 httpp:///ggttbbb.ccoo///g/////ae
fensorfloo//aayrrr//sstaagg... and the experiments are implemeented in uwwertainty. mucerraaintyyybaeeliinee (e.g..
To mcasure the predictive performance of the models, we use their accuracy (Acc) and negative log-likelihood (NLL). To measure their uncertainty calibration, we use the expected calibration error (ECE) (Naeini et al. 2015). To measure OOD detection, we use the area, under the prciver-operatoo-chaaacttt using the predictive uncertainty as a score (ROC) and the false-positive rate (rate of classifying an OOD data point as being in-distribution) at 95% recall (FPR95) (Fort et al., 2021).
curve when
5.1 Synthetic experiments
Synthetic OOD data Following the SNGP paper (Liu et al. 2020 to assess the distance-awarenoss property of HetSNGP in a visually verifiable way, we performed experiments on two-dimensional synthetic datasots: (i) two moons and (ii) a Gaussian mixture. We see Fig. | that neither the deterministic nor heteroscedastic models offer distance-aware uncertainties and are therefore overconfident on the OOD data (red points). While the Posterior network offers uncertainties that seem mildly distance-aware, on these datasets, those uncertainties grow more slowly when moving away from the training data than in the case of the SNGP methods. As result, the SNGP methods enable a more effective OOD detection, especially on near-OOD lasks (close to the training data), where the Posterior network seems to be overconfident. Moreover, we will see later that scaling the Posterior network to datasets with many classes is too expensive to be practical. Only the SNGP and our proposed HetSNGP are highly uncertain on the OOD points, thus allowing for effective OOD detection, while the same time being computationally scalable to larger problems with many more classes.
Synthetic heteroscedastic data retains the in-distribution heteroscedasticc modeling property of the heteroscedastic method. We use the noisy concentric circles from Berthon et al. (2021) ; where the three circular classes have the same mean, but different amounts of label noise. We see Fig. 2 that the heteroscedastic model and our proposed method are able to capture this label noise and thus achieve a better performance, while the deterministic baseline and the SNGP are not. The intuition for this advantage is that when capturing the class-dependent label noise, these models can ignore certain noisy data examples and therefore, when in doubt, assign larger predictive probabilities to the classes that are less noisy in the data, which in expectation will lead to more accurate predictions than from the other models, which are overfitting to the label noise. We see this visually in Fig, 2 where the predicted labels for the heteroscedastic and HetSNGP models capture the true geometric structure of the clean labels, while the deterministic and SNGP methods crroncously fit the shape of the noisy labels. Note that Appendix C in Collier et al. (2021) also offers theoretical explanation for this behavior, using Taylor series approximation argument. Based on these two synthetic experiments, it becomes apparent
We now wish to verify in a low-dimensional setting that HetSNGP also
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Table 2: Results on CIFAR-100 with Places365 as Far-OOD and CIFAR-10 as Near-OOD datasets. We
report means and standard errors over 10 runs. Bold numbers are within one standard error of the best performing model. Our model outperforms the baselines in terms of accuracy on corrupted data and is on par with the best models on OOD detection.
Method
1ID Acc
LID NLL
1Corr Acc
[Corr NLL
1Near-ROC
LNeur-FPR95
fFa--ROC
LFar-FPR955
Det.
0.808 0.200
0.791 0.02
0.455 0.001
2.890
0,506
ooce
0.963
0,00s
0.142
onto
0.935
0.00
Nest
0.728 5.00.
1.603 0.024
0.411 0.00.
3.086
nocc
0.472
0.013
1.000
-..co
0.518
1.000
0.00
Hot.
0.807 5.00:
0.782 0.0
0.447 a n.rot
3.130
a: nots
0.496
0.00s
0.957
0.04
0.420
0.02s.
0.958
0.000
SNGP
0.797 sam.
0.762 comz
0.466
+ som
2.339
+ c.00
0.493
0.961
micto
0.518
nurr
0.919
HetSNGP (ours)
0.799 2.00:
0.856 0.000
0.171 0.000
2.565
1. 0.0c7
0.499 0.020
0.955
0.005
0.525
8.03s
0.910
c.ook
Table 3: Results ImageNet. We used ImageNet-C as near-OOD and ImageNet-A A as far-OOD, We report the mean and standard error over 10 runs. Bold numbers are within one standard error of the best model. HetSNGP performs best in terms of accuracy on ImageNet-C and ImageNet-A.
Method
11D Ace
LID NLL
LID ECE
flaC Ace
[mm NLL
JmmCC ECE
tmmd Ace
JmmA NLL
JmmA ECE
Det.
0.759 ma3
0.952
0.033
2..00
0.419
3.078
0.096
0.006
8.088
0.421
Het.
0.771 1.00
0,912 0.00
0.033
0.000
D.124
uses
3,200) à 0.0s
0.111 0.0
0,010 a.00
7,911
0,436 + 400
SNGP
0.757
0.947
0.014
0.120
axea
2.970
0.046
0.007
7,184
0.356 0.00
HaSNGP (ours)
0.700
0.927
0.033
0.428
gaom
2.597
0.085
0.016
7.113
0.401 2ow0
that our proposed HetSNGP successfully combines the desirable OOD uncertainties of the SNGP with the heteroscedastic uncertainties these simple datasets. We now proceed to evaluate these properties more challenging digeer-dimensional datasets.
5.2 CIFAR experiment
We start by assessing our method on a real-world image dataset; we trained it on CIFAR-100 and used CIFAR-10 as near-OOD dataset and Places365 (Zhou et. al. 2017) far-OOD. We measure the OOD detection performance in terms of area under the coeetver-opraatorcccaaceeeee positive-rate at 95% confidence (FPR95). We also evaluated the methods' generalization performance on corrupted CIFAR-100 (Hendrycks & Dietterich 2019). In Table 2 the heteroscedastic and SNGP methods in terms of in-distribution accuracy, but outperforms all baselines in accuracy on the corrupted data. Moreover, it performs on par with the best-performing models on both near- and far-OOD detection. This suggests that in in-distribution, only the heteroscedastic uncertainty is needed, such that both the heteroscedastic method and our HetSNGP outperform the standard SNGP in terms of accuracy. However, on the corrupted data, which is outside of the training distribution, the SNGP outperforms the heteroscedastic method in terms of accuracy and our HetSNGP outperforms both baselines, since both types of uncertainty are useful in this setting.
curve (ROC) and false-
we see that HeSNGG performs bet wetween
Table 4: Additional ImageNet OOD results on ImageNet-R and ImageNet-V2. V2. The reported values are means and standard errors over 10 runs. Bold numbers are within one standard error of the best performing model. Our model outperforms the basclines in terms of accuracy on ImageNet-V2.
Method
fmm Ace
JmR NLL
Umm ECE
1mmV Acc
L1mV2 NLL
LmmV ECE
Det.
0.229 + 0.001
5.907 + 0.014
0.239
= 0.001
0.638
+ 0.001
1.598 0.003
0.077 + 0.001
Het.
0.235
0.001
5.761
1 0.010
0.251
1 0.001
0.648
= 0.00 001
1.581
0.002
0.084 0.001
SNGP
0.230
0.001
5.344
0.000
0.175
0.001
0.637
0.001
1.552 0.001
0.041
HetSNGP (ours)
0.232 0.001
5.452
0.011
0.225 0.002
0.647
1.564
0.080 0.001
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Table 5: Results on ImageNet-21k. The reported values are means and standard errors over runs. Bold numbers are within one standard error of the best performing model. We use standard ImageNet, ImageNet- C, ImageNet-A, ImageNet-R, and ImageNet-V V2 as OOD datasets. HetSNGP outperforms the baselines on all OOD datasets.
Method
+ID prec@1
fIm Acc
1mC Acc
1mmA Acc
1mR Acc
1mV2 Acc
Det.
0.471
0.800
0.603
0.149
0.311
0.000
0.694
0.000
Het.
0.480
0.001
0.796
0.002
0.590
0.001
0.132
004
0.300
a 0.006
0.687
0.000
SNGP
0.468
0.001
0.799
0.001
0,602 0.000
0.165
0.003
0.328
+ 0.005
0.696
0.003
HetSNGP (ours)
0.477 0.001
0.806
+ 0.001
0.613 + 0.003
0.172
= 0.007
0.336
+ 0.002
0.705
0.001
5.3
ImageNet experiment
A large-scale dataset with natural label noise and established OOD benchmarks is the ImageNet dataset [Deng ct al., 2009; Beyer et al.) 2020). The hetcroscedastic method has been shown to improve in-distribution perlormance on ImageNet. (Collier et al. 2021). We see in Table 3 that HetSNGP outperforms the SNGP in terms of accuracy and likelihood on the in-distribution ImageNet validation set and performs almost par with the heteroscedastic model. The slight disadvantage compared to the heteroscedastic model suggests a small trade-off due to the restricted parametcrization of the output layer and application of spectral normalization.
However, the true benefits of our model become apparent when looking at the ImageNet OOD datasets (Table 3 right side). Here, we still have the noisy label properties from the original ImageNet dataset, such that heteroscedastic uncertainties are useful, but we are also outside of the training distribution, such that distanco-aware model uncortainties become crucial. On ImageNet-C and ImageNet-A, we see that our proposed model makes good use of both of these types of uncertainties and thus manages to outperform all the baselines in terms of accuracy. Additional OOD results on ImageNet-R. and ImageNet- V2 are shown in Table
5.4 ImageNet-211 21k
We introduce a new large-scale OOD benchmark based on ImageNeet221k. We hope this new benchmark will be of interest to future work in the OOD literature. ImageNet-21k is a larger version of the standard ImageNet dataset used above (Deng ct al. 2009). It has over 12.8 million training images and 21,843 classes. Each image can have multiple labels, whereas for standard TmageNet, a single label is given per image. In creating our benchmark, we exploit the unique property of ImageNet-21k that its label space is superset of the 1000 ImageNet classes (class n04399382 is missing).
Having trained on the large TmageNeet221k training set, we then evaluate the model on the 1,000 ImageNet classes (setting the predictive probability of class n04399382 to zero). Despite now being in a setting where the model is trained on an order of magnitude more data and greater than 21x more classes, we can use the standard ImageNet OOD datasets. This assesses the scalability of our method and the scalability of future OOD methods.
In our experiments, we train VIT-B/16 (Dosovitskiy et al., 2020) vision transformer model on the ImageNet- 21k dataset (see Appendix(A.. for training details). To scale to over 21k output classes, efficient versions of the heteroscedastic and HetSNGP methods. We found that SNGP and HetSNGP underfit to the data when using the posterior p(B, ID) at test time, so instead we use the posterior mode B at both training and test time. Addition,lly, we found spectral normalization was not necessary to preserve distance awareness for the VIT architecture; hence ImageNet-22 experiments are run without spectral normalization.
the parameter-
We see in Table 5 that the parameter-efficien heteroscedastic method has the best in in-distribution preci- @1. However, its generalization performance to the OOD benchmarks is the weakest of all the methods.
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Table 6: Results on ImageNet. Ensemble size set to 4. ImageNNt--C C' is used as the OOD dataset. HetSNGP outperforms the basclines in terms of in-distribution accuracy and on all OOD metries.
Method
1DD Acc
LID NLL
LID ECE
1m Ace
|mmC NLL
JmmC ECE
Det Ensemble
0.779
0.857
0.017
0.449
2.82
0.047
Het Ensemble
0.795
0.790
0.015
0.449
2.93
0.048
SNGP Ensemble
0.781
0.851
0.039
0.449
2.77
0.050
HetSNGP Ensemble (ours)
0.797
0.798
0.028
0.458
2.75
0.044
Our method recovers almost the full in-distribution performance of the heteroscedastic method, significantly outperforming the deterministic and SNGP methods. Notably, it also clearly outperforms all other methods the OOD datasets. Similarly to the CLFAR experiment, we thus see again that in-distribution, the heteroscedastic method and our HetSNGP both outperform the SNGP, since OOD uncertainlies are not that important there, while on the OOD datasets, the SNGP and our HetSNGP both outperform the heteroscedastic method. Our method therefore achieves the optimal combination.
5.5 Ensembling experiment
Deep ensembles are popular methods for model uncertainty estimation due to their simplicity and good performance (Lakshminarayanan et al. 2017). We propose a variant of our HeLSNGP method which captures a further source of model uncertainty in the form of parameter uncertainty by a deep ensemble of HetSNGP models. HetSNGP Ensemble therefore a powerful yet simple method for capturing three major sources of uncertainty: (1) data uncertainty in the labels, (2) model uncertainty in the latent representations, and (3) model uncertainty in the parameters.
We compare the HctSNGP Ensemble to determinstic ensemble as well as ensembles of heteroscedastic and SNGP models. Wesser in Table 0 that in the case of an ensemble of size four, HetSNGP Ensemble outperforms the baselines on the ImageNet-(C OOD dataset in terms of all metrics, while also outperforming them on the standard in-distribution ImageNet dataset terms accuracy. Due to computational constraints, we do not have error bars in this experiment. However, we do not expect them to be much larger than in Table 3
In the above experiments, we have observed that the model uncertainty in the parameters, captured here by ensembling, appears to be complementary and benefits all approaches. We defer to future work the exploration of more efficient variants of HetSNGP Ensemble. In Appendix A.3 we discuss a promising extension based on MIMO (Havasi et al. 2020 and the advantages it would bring.
6 Conclusion
We have proposed a new model, the HetSNGP, that jointly captures distancc-awaro model uncortainties and heteroscedastic data uncertainties. The HetSNGP allows for favorable combination of these two comple- mentary types of uncertainty and thus enables effective out-of-distribution detection and generalization as well as in-distribution performance and calibration on different benchmark datasets. Moreover, we have pro- posed an ensembled version of our method, which additionally captures uncertainty in the model parameters and improves performance even further, and new large-scale out-of-distributtin benchmark based the ImageNet-21k. dataset.
Acknowledgments
VF acknowledges funding from the Swiss Data Science Center through PhD Fellowship, well from the Swiss National Science Foundation through Postdoc.Mobillty Fellowship, and from St. John's College Cam- bridge through a Junior Research Fellowship. We thank Shreyas Padhy, Basil Mustafa, Zelda Mariet, and Jasper Snoek for helpful discussions and the anonymous reviewers for valuable feedback on the manuscript.
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A
Appendix
A.1
Implementation details
To assess our proposed model's predictive performance and uncortainty estimation capabilities, we conducted experiments on synthetic two moons data (Pedregosa et al., 2011), a mixture of Gaussians, the CIFAR-100 dataset (Krizhevsky & Hinton 2009p, and the ImageNet dataset (Deng et al., 2009). We compare against standard deterministic ResNet model as bascline (He al.] 2016), against the heteroscedastic method (Collier al. 2020; 2021) and the SNGP (Liu et al. 2020) (which form the basis for our combined model) and against the recently proposed Posterior Network model (Charpentier et. al. (0200). which also offers distance- aware uncertainties, similarly to the SNGP. We used the same backbone neural network architecture for all models, which was a fully-connected ResNet for the synthetic data, a WidcRosNet18 on CIFAR and a ResNet50 in ImageNet.
For most baselines, we used the hyperparameters from the uncertainty 2021). On CIFAR, we trained our HetSNGP with learning rale of 0.1 for 300 epochs and used R = 6 factors for the heteroscedastic covariance, softmax temperature of 0.5 and S 5000 Monte Carlo samples. On ImageNet, we trained with a learning rate of 0.07 for 270 epochs and used R = 15 factors, a softmax temperature of r = 1.25 and S = 5000 Monte Carlo samples. We implemented all models in TensorFlow in Python and trained on Tensor Processing Units (TPUs) in the Google Cloud.
pttertaiitttbbaeeeeeeeee baseelineeessee library (Nado et al,
We train all Imagnet-2111 models for 90 epochs with batch size 1024 on Adam optimizer with initial learning rate of 0.001 using linear learning rale decay schedule with termination point 0.00001 and warm-up period of 10,000 steps. We train using the sigmoid cross-entropy loss function and L2 weight decay with multiplier 0.03. The heteroscedastic method uses temperature of 0.4, 1,000 Monte Carlo samples and R = 50 for the low rank approximation. HetSNGP has the same heteroscedastic hyperparameters except the optimal lemperature is 1.5. For SNGP and HetSNGP the GP covariance is approximated using the um scheme presented in Liu et al. 2020 with momentum parameter 0.999.
TPU slices. We train using the
A.2 Laplace approximation
In this section, we will derive the Laplace posterior in Eq. 5 The derivation follows mostly from the sections 3.4 and 3.5 Rasmussen & Williams (2006).
First. note that the log posterior of Bg given the data is
log p(Be x y) logp(y (8)) + ((g)((B))) Z
(7)
where Z is normalization constant that does not depend Be. Following Rasmussen & Williams (2006) we will denote the unnormalized log posterior as
V (Bec = log Be) + log p(S.)
(8)
Recall that the first term is the likelihood and the second term is our prior from Eq. 4
The Taplace approximation now approximates the posterior with a local second-order expansion around the MAP solution, that
pBBal x. \ A -1 with the MAP solution B. arg max,, (B.) and the Hessian A = -
(9)
(.
The MAP solution can be found using standard (stochastic) gradient descent, while the Hessian is given by
(P.( = V² logr(y B.) + V l(g)(3e)
V Vu log u) V Im
V Vu log
4 Im
T V2 log p(y u) I.
--W + Im
17

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Published in Transactions on Machine Learning Research (07 (07/2022)
Table 7: Runtimes of inference per sample for different methods. Runtimes are computed on ImageNet using Resnet50. We see that the methods do not differ strongly.
METHOD
RUNTIME (MS/EXAMPLE)
DETERMINISTIC
0.04
SNGP
0.047
MIMO
0.040
HET. (5000 MC SAMPLES)
0.061
HETSNGP (100 MC SAMPLES)
0.045
Table 8: MIMO performance on ImageNet (in-dist), ImageNet-C and ImageNet-A. The reported values are means and standard errors 10 runs.
1ID Ace
LID NLL
LID ECE
1m Aoe
Imc NLL
UmC ECE
1mmA Ace
Umm NLL
Umm ECE
0.772 0.901 soes 0.039 0.001
0.440 c.nos
2.979 0.000
0.101
0.013 0.01
7.777 no0
0.432 0.00s
where we used the chain rule and the fact that u = D3 and W is a diagonal matrix of point-wise second derivatives of the likelihood, that is, Wa - V2 log p(w u.) (Rasmussen Williams) 2006). For instance, in the case of the logistic likelihood, Wa = P. (1 - p.), where p, vector of output probabilities for logits u. To get the Hessian at the MAP, we then just need to compute this quantity for = $
The approximate posterior therefore
pB xz Ri N( (W (W
Imm)
(10)
where the precision matrix can be computed over data points (recovering Eq. 5 as
N
A = mm - Σ
P. 2.
(11)
A.3
Future work:
Combination with efficient ensembling
MIMO Havasi et al. 2020 is promising efficient ensemble method we could build HetSNGP upon. We tested MIMO alone to assess how promising it is, see Table g In Table [ we summarize the benefits we would have by combining HetSNGP and MIMO. In particular, we hope to preserve the gains of the expensive HetSGNP Ensemble for a fraction of the cost.
A.4 Runtime comparison
We profiled the runtimes of the different methods and see in Table 10 Imagenet-21k that the different methods do not differ strongly in their computational costs. In particular, our HetSNGP performs generally on par with the standard heteroscedastic method.
Table 9: MIMO performance on ImageNet-R and ImageNet-V2. The reported values are means and standard errors over runs.
ImR Acc
JmR NLL
HmR ECE
fmm2 Acc
JlmV2 NLL
JmV2 ECE
0.245 0.003
5.851 0.039
0.248 0.084
0.654 4 0.003
1.538 0.010
0.085 4 0.003
18

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Published in Transactions on Machine Learning Research (07 (07/2022)
Table 10: Profiling of different methods on ImageNet-21k using Vision Transformers (B/16). We report the millisoconds and GFLOPS (=109 FLOPs) per image, both training and evaluation time. All measurements are made on the same hardware (TPU V3 with 32 cores). We see that the methods do not differ strongly and that the HetSNGP performs par with the standard heteroscodastic method.
Model
ms / img (train)
GFI.OPS/ / img (train)
ms img (eval)
GFLOPS img (eval)
Det
4.72
106.61
1.08
35.31
SNGP
4.74
106.65
1.08
35.32
Het
6.09
112.17
1.37
37.79
HetSNGP
6.13
112.22
1.38
37.80
Table 11: Ablation study with different numbers of MC samples for the HetSNGP on Imagenet. We see that there are no improvements when using more than 100 samples.
# of MC samples
Accuracy
NLL
ECE
5000
0.763
0.961
0.041
1000
0.772
0.922
0.036
250
0.769
0.927
0.031
100
0.772
0.916
0.029
25
0.772
0.949
0.047
0.704
1.342
0.222
A.5 Ablation studies
We also performed several ablation studies to the effect of the computational approximations in our proposed HetSNGP method. We see in Table 11 that 100 MC samples are already enough to achieve good performance Imagenct and that increasing that number beyond 100 does not offer any further improvements. Moreover, see in Table 12 that the performance does not improve beyond rank of 7 for the low-rank heteroscedastic covariance matrix and that even a rank of 2 already performs well. What is more, we see Table 13 that softmax temperatures around 1.0, that is, standard untempered softmax, perform well practice. Finally, we see in Table 1 that MAP inference during training is sufficient to perform well and that MC sampling during training does not significantly improve the performance.
Table 12: Ablation study with different ranks for the low-rank heteroscedastic covariance matrix in the HetSNGP on Imagenet. We that after rank 7, there further improvements, and even rank 2 already works well.
Rank of covariance
Accuracy
NLL
ECE
0.772
0.910
0.032
0.774
0.907
0.035
0.772
0.922
0.036
0.767
0.941
0.033
50
0.766
0.928
0.034
19

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Published in Transactions on Machine Learning Research (07 (07/202) /202))
Table 13: Ablation study with different softmax temperatures for the HetSNGP on Imagenet. We see that temperatures around 1.0 (corresponding to standard softmax) perform well.
Temperature
Accuracy
NLL
ECE
0.4
0.763
0.973
0.051
0.8
0.766
0.929
0.037
1.25
0.772
0.922
0.036
1.6
0.770
0.934
0.036
2.0
0.772
0.941
0.031
3.0
0.767
0.983
0.026
5.0
0.729
1.392
0.352
Table 14: Ablation study with and without MC sampling for the training of the HetSNGP Imagenet. We see that MC sampling during training does not significantly improve the performance, so using MAP inference during training is sufficient.
Method
Accuracy
NLL
ECE
MAP
0.772
0.922
0.036
MC sampling
0.772
0.939
0.053
20


Reviews:
[{"review_id": "xuaTVYvzfM", "reviewer": "Reviewer_WKjU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. Hierarchical modeling of model and data uncertainty is a novel and interesting idea. \n\n2. Paper is well-written and easy to read. The proposed model is also well motivated.\n\nWeaknesses: Please refer to the following part.", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a hierarchical model of two latent random variables (f and u) for input variables and logits to capture the model and data uncertainty respectively. These latent variables are modeled using Gaussian processes as their prior distribution. Their hierarchical modeling allows them to capture the distance aware model and data uncertainties.", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "1. Section 3.2: While the authors said that “f is a latent function value associated to the input x”, the kernel of its corresponding Gaussian prior is computed using the hidden representations k_θ(xi , xj ) = k_RBF (hi , hj ) — this makes sense as f is suppose to capture the model uncertainties. However, I feel that this description should be clearly written in the beginning of section 3.2. Otherwise, it can be confusing that “f” should capture the data uncertainty as it is associated with the input variables x.\n\n2. Related works: The authors should move the related work section in the main paper. Since TMLR has a 12 page limit and the current version of the paper contains 10 pages of main context.\n\n3. The authors often referred to the related works without properly comparing their drawbacks/ limitations in the context of their proposed method.  For e.g. in their related work, the authors said that “Another popular idea is to use a hierarchical output distribution, for instance a Dirichlet distribution (Milios et al., 2018; Sensoy et al., 2018; Malinin & Gales, 2018; Malinin et al., 2019; Hobbhahn et al., 2020), such that the model uncertainty can be encoded in the Dirichlet and the data uncertainty in its Categorical distribution samples. This idea was also used in our Posterior Network baseline (Charpentier et al., 2020).”\n\nAlso in their experiment section they mentioned that: “We also compare against the Posterior Network model (Charpentier et al., 2020), which also offers distance-aware uncertainties, but it only applies to problems with few classes. ”\n\nMy question/ suggested change(s): \na. Evidential models such as Sensoy et al., 2018; Malinin & Gales, 2019; Nandy et al. 2020 do not suffer from this problem. Then why do the authors choose only the posterior network for their comparison (Charpentier et al., 2020)? \n\nb. I also feel that these evidential models are very closely related to the proposed method and should be discussed more carefully. Also, it would be good to have some discussion on models such as DUQ. Finally, please include the missing citations.\n\n(Amersfoort et al. 2020) Uncertainty Estimation Using a Single Deep Deterministic Neural Network, ICML 2020\n(Malinin & Gales, 2019) Reverse kl-divergence training of prior networks, NeurIPS 2019\n(Nandy et al. 2020) Towards Maximizing the Representation Gap between In-Domain & Out-of-Distribution Examples, NeurIPS 2020\n\n4. Section 4: “Since not all tasks and datasets require both types of uncertainty at the same time, we do not expect our proposed model to outperform the baselines on all tasks.” – I have understood and agreed on this statement. However, I think it would be good to elaborate it up-front before going into the details to section 4.2.\n\n5. Section 4.1 and Figure 1: I think it would be good to bring Figure 1 to page 7 (near section 4.1) for better readability.\n\n6. “ The Posterior network does offer distance-aware uncertainties, which on these datasets grow more slowly away from the training data than the SNGP methods.” :-- I do not understand this statement. In particular, from Figure 1, it seems that SNGP models produced the best results. Perhaps, can you please be more specific/ increase the size of the axes to (-20, 20)*(-20*20)?\n\n7. “We see in Fig. 2 that the heteroscedastic model and our proposed method are able to capture this label noise and thus achieve a better performance, while the deterministic baseline and the SNGP are not.”\nI could not understand (or do not agree) with the authors. How capturing label noise (i.e. data uncertainty) leads to better accuracy? In fact, a classifier can produce incorrect predictions due to data-uncertainty (and we may not be able to reduce that as they are irreducible). However, it can still correctly identify the cause of uncertainty as label-noise, providing a better uncertainty estimation. Can you please explain?\n", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wpDlNTLuDX", "reviewer": "Reviewer_QcRU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n* The contributions of the paper and differences to recent related works are clearly stated in the introduction and appendix. \n* The paper is well written and was easy to understand. \n* The new data set and large-scale experiments will be of interest to the community. \n\nWeaknesses:\n* This may be a problem with the terminology used in many Bayesian deep learning papers itself, but it is not clear to me what the terms \"in-distribution\" and \"out-of-distribution\" really mean. For example, in image classification, it may be difficult to distinguish rare examples from the heavy-tail of the distribution with examples which are truly \"out-of-distribution\". So even though the examples may be sampled from the same training distribution, they can be substantially different from all the previous training examples. But should we really call this \"out-of-distribution\" or rather \"heavy-tail\"?   \n\n* In the paper it is written, \"After training data has been observed, one should expect the model uncertainty to decrease within the support of the training data distribution, that is, on points that are close to the training data in the input space.\".  But for high-dimensional data, it is hard to say when two examples are different or close. The Euclidean distance seems to mostly make sense in the 2D toy examples. For high-dimensional image data such as ImageNet, two images may be very close semantically, but far away in Euclidean distance. Therefore, the Euclidean distance does not make much sense and the use of bi-Lipschitzness of the kernel embedding seems problematic. \n\n* The technical part of the paper is of incremental nature in view of (Liu et al., 2020) and (Collier et al., 2020, 2021). \n\n* There are no ablation studies which investigate the computational approximations. For example, would more accurate approximations improve the performance? \n\n* Both f and u are parametrized using deep neural networks. It was difficult for me to understand whether two different networks are used, or the same one with different \"heads\". \n\n* While the method obviously scales very well (experiments on ImageNet21k), after reading the experiments, it is still not clear how much computational overhead the method adds over \"standard training\"?  Is it negligible? Will the code be released? ", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a latent variable approach to account for both data uncertainty and model uncertainty.  The techincal contribution is a combination of two recent approaches for model uncertainty (Liu et al., 2020) and data uncertainty (Collier et al., 2020, 2021).  The method is evaluated on several large-scale data sets and a new data set, offering improved results over standard training (SGD) and the Liu et al., Collier et al., works. \n", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "I recommend acceptance of the work. The large-scale experiments and new data set will be of interest to the (Bayesian) deep learning community. Here are some minor suggestions which may improve readability and the presentation of the paper:\n\n* Figure 2 is hard to understand for color-blind people -- perhaps a more diverse color scheme can be picked. \n\n* Perhaps a comment on differences between the heavy-tailed nature of the data distribution and \"out-of-distribution\" can be added in the introduction to clarify the differences. \n\n* The sentence about Laplace approximation being close in performance to true posterior is misleading. It should be removed or rephrased. To best of my knowledge, (Immer et al., 2021a,b) do not compare to the true posterior, which is intractable for deep neural networks. \n\n* In the paragraph before the equations (1), (2), (3) it should be briefly mentioned how the neural networks are used to parametrize f and u. For example, the reader could be pointed to the corresponding part of the appendix. \n\n* Perhaps a table could be added which compares the run-times of \"Det.\" (standard training) with \"HetSNGP\" under the same hardware setup.  As of now, the method appears expensive (GP inference), and practicioners may question whether the added computation is worth the moderate improvement over standard training.  ", "broader_impact_concerns": "The paper poses no immediate ethical concerns, and no broader impact statement is required. ", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "XmMSC59m8A", "reviewer": "Reviewer_X4fV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n+ A fully Bayesian model, namely, GP + distribution over parameters.\n+ Several approximations are proposed to make the model tractable.\n+ By using the RBF kernels, the model is distance-aware as expected.\n+ The paper is well-written and easy to follow.\n+ The experiments are convincing.\n\nWeaknesses:\n- Some choices of approximations are not fully motivated.\n- The effect of the temperature used in (6) is not fully presented.\n- Training could be done with MC samples rather than the MAP solution. I do not see any discussion on that.\n- Since the method is an extension of the SNGP, a computational comparison with this method would be beneficial.", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a new model that allows modeling both distance-aware model and data uncertainties. The resulting model, the heteroscedastic spectral-normalized Gaussian process (HetSNGP), extends the spectral-normalized GP (SNGP) by capturing the uncertainty over model parameters. As a result, a fully Bayesian model is obtained.\n\nThe presented approach is evaluated on benchmark datasets (including an OOD benchmark). Moreover, the authors propose a new OOD benchmark (ImageNet-21k). The results of the experiments are convincing.\n", "claims_and_evidence_yn": "", "claims_explanation": "", "audience_yn": "", "audience_explanation": "", "requested_changes": "1) I would like to see a better motivation for the chosen low-rank approximation. What about other approximations? Why the picked one is better than the others? A (short) discussion would be beneficial.\n2) It would be great to see the influence of the temperature in (6) on the final performance. Perhaps, on toy data to get a better insight.\n3) I would like to see a comparison between the HetSNGP and the SNGP in terms of computational complexity.", "broader_impact_concerns": "The paper does not contain any broader impact statement. However, I do not see any specific concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors propose a Bayesian approach to model simultaneously the model uncertainty and the data uncertainty in deep learning. This is illustrated on an exhaustive set of experiments, with a clear improvement on existing methods.

The reviewers all found the idea interesting, and the experimental results overall convincing and useful for the community. I agree with them. Overall, they also found the paper clear and well written. Each of them pointed out some minor problems: lack of motivation for some choices in the simulations, a few misleading comments, and a few places where the clarity could be improved. They suggested some improvements. The authors submitted a revised version where all these comments were taken into account. The 3 reviewers were satisfied and recommended acceptation of the paper.
Decision: Accept (poster)