Bayesian Neural Network Priors Revisited
Authors: Vincent Fortuin
Adrià Garriga-Alonso
Sebastian W. Ober
Florian Wenzel
Gunnar Ratsch
Richard E Turner
Mark van der Wilk
Laurence Aitchison
Venue: ICLR 2022 Poster (2022)
Keywords: Bayesian deep learning
Bayesian neural networks
Priors

Abstract: Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, it is unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. To find better priors, we study summary statistics of neural network weights in networks trained using stochastic gradient descent (SGD). We find that convolutional neural network (CNN) and ResNet weights display strong spatial correlations, while fully connected networks (FCNNs) display heavy-tailed weight distributions. We show that building these observations into priors can lead to improved performance on a variety of image classification datasets. Surprisingly, these priors mitigate the cold posterior effect in FCNNs, but slightly increase the cold posterior effect in ResNets.

Published as a conference paper at ICLR 2022
BAYESIAN NEURAL NETWORK PRIORS REVISITED
Vincent Fortuin'
Adrià Garriga-Alonso"
ETH Zürich, Switzerland fortuingins wtw.......thhhhh.c.......
University of Cambridge, United Kingdom ag 90ccm ac. .k
Sebastian W. Ober
Florian Wenzel
University of Cambridge, United Kingdom swo250cam, uk
Google AI Berlin, Germany
Florianwenzei com
Gunnar Rätsch
Richard E. Turner
ETH Zürich, Switzerland raetsch@ing www.eeiif.ehhh...mmmmm
University of Cambridge, United Kingdom ret260eng cam. ac
ethz. ch
Mark der Wilk!
I Laurence Aitchison
Imperial College London, United Kingdom m. vdwiiill 1 imperial. a...uu...
University of Bristol, United Kingdom laurence wwtcchion@brritteeeeei.oooo.. uk
ABSTRACT
Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, it unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. better priors, study summary statistics of neural network weights in networks trained using stochastic gradient descent (SGD). We find that convolutional neural network (CNN) and ResNet weights display strong spatial correlations, while fully connected networks (FCNNs) display heavy-tailed weight distributions. We show that building these observations into priors lead to improved performance on a variety of image classification datasets. Surprisingly, these priors mitigate the cold posterior effect FCNNs, but slightly increase the cold posterior effect ResNets.
find
INTRODUCTION
In a Bayesian neural network (BNN), we specify a prior p(w) over the neural network parameters, and compute the posterior distribution over parameters conditioned training data, p(w]2; y) p(y|u, ()p(w)/((y(x)) This procedure should give considerable advantages for reasoning about predictive uncertainty, which is especially relevant in the small-data setting. Crucially, to perform Bayesian inference, we need to choose prior that accurately reflects our beliefs about the parameters before seeing any data (Bayes) 1763 Gelman et al. 2013). However, the most common choice of prior for BNN weights is the simplest one: the isotropic Gaussian. Isotropic Gaussians are used across almost all fields of Bayesian deep learning, ranging from variational inference (e.g..
Hernández-Lobato & Adams 2015 Louizos & Welling 2017 Dusenberyyr et al.) (2020), sampling- based inference (c.g., Neal [922]Zhang et al.] 22019), and Laplace's method (c.g.. Osawa et al. 2019)
Immer et al. 2021b), even infinite networks (e.g., Lee et al. 2017; Gllriga-Alonso al. 2019).
is troubling that no alternatives are usually considered, since better choices likely exist.
Indeed, despite the progress more accurate and efficient inference procedures, in some settings, the posterior predictive distribution of BNNs using Gaussian priors still leads to worse predictive performance than baseline obtained by training the network with standard stochastic gradient descent (SGD) (e.g., Zhang et al. 2019
Heek & Kalchbrenner 2019 Wenzel et al. 2020a). Surprisingly.
Equual contribution. I Equal contribution.

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these issues largely be fixed by artificially reducing posterior uncertainty using "cold posteriors" (Wenzel al., 2020a). 'The cold posterior is p(w|z, y) for temperature 0 < T < where the original Bayes posterior would be obtained by setting T =1 (see Eq. [ Using cold posteriors can be interpreted overcounting the data and, hence, deviating from the Bayesian paradigm. This should not happen if the prior and likelihood accurately reflect our beliefs. Assuming inference is working correctly, the Bayesian solution, T really should be optimal (Gelman et 2013). Hence,
raises the possibility that either the prior (Wenzel et al. 2020aa) or likelihood (Aitchison 2020b) (or
both) are misspecified.
In this work, we study empirically whether isotropic Gaussian priors are indeed suboptimal for BNNs and whether this can explain the cold posterior effect. We analyze the performance of dif- ferent BNN priors for different network architectures and compare them to the empirical weight distributions standard SGD-trained neural networks. We conclude that correlated Gaussian priors are better in ResNets, while uncorrelated heavy-tailed priors are better in fully connected neural networks (FCNNs). Thus, we would recommend these choices instead of the widely-used isotropic Gaussian priors. While these priors eliminate the cold posterior effeet in FCNNs, they slightly increase the cold posterior effect in ResNets. This provides evidence that the cold posterior effect arises due to misspecification of the prior (Wenzel et al. 2020a) in FCNNs. In ResNets, it is difficult to draw any strong conclusions about the cold posterior effect from our results. Our obser- vations are compatible with the hypothesis that the cold posterior effect arises large-scale image models due to misspecified likelihood (Aitchison) 20205) or due to data augmentation (Izmailov) al., 2020), but there could of course be a prior that we did not consider that improves perfor- mance and eliminates the cold posterior effect. We make our library available on Github inviting other researchers to join us in studying the role priors BNNs using state-of-the-art inference.
1.1 CONTRIBUTIONS
Our main contributions are:
Weight histogram Gaussian Q-Q
Laplace Q-Q
An analysis of the empirical weight distribu- tions of SGD-trained neural networks with different architectures, suggesting that FC- NNs learn heavy- -taled weight distributions (Sec. 3.I), while CNN and ResNet weight distributions show significant spatial corre- lations (Sec. 3.2).
FCNN
emp.
emp. W
CNN
Experiments in Bayesian FCNNs showing that heavy-tailed priors give better classi- fication performance than the widely-used Gaussian priors (Sec.
-0.25 0.00 0.25
ResNet early
emp.
0
Experiments in Bayesian ResNets show- ing that spatially correlated Gaussian priors give better classification performance than isotropic priors (Sec.
weight value (w)
theoretical
theoretical w
Figure I: Empirical marginal weight distributions of layer of FCNNs and CNNs trained with SGD on MÄIST, and an early layer several ResNets trained on CIFAR-10. 10. We show weight histograms (left) and quantile-quantile (Q-Q) plots with differ- ent distributions (right). The empirical weights are clearly hcavier-tailed than a Gaussian (green line), and better fit by Laplace (orange line).
Experiments showing that the cold posterior effect can be reduced by choosing better, heavy-tailed priors in FCNNs, while the cold posterior is slightly increased when using better, spatially correlated priors in ResNets (Sec.
MIT licensed.
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2 BACKGROUND: THE COLD POSTERIOR EFFECT
When performing inference in Bayesian models, we can temper the posterior by positive temperature T, giving
1 log logp(u) y) + - T llllogg p(y/ur + logp(w)] + ((T)
for neural network weights w, inputs x regression targets or class-labels y, prior p(w), likelihood p(y] w., .7), and normalizing constant (T"). Setting I = 1 yields the standard Bayesian posterior. The temperature parameter can be easily handled when simulating Langevin dynamics, as used in molecular dynamies and MCMC (Leimkuhler & Matthews) 2012))
In their recent work. Wenzel et al. (2020a) have drawn attention to the fact that cooling the posterior in BNNs (i.e., setting 7 < 1), often improves performance. Testing different hypotheses for potential problems with the inference, likelihood, and prior, they conclude that the BNN priors (which were Gaussian in their experiments) are misspecified- at least when used in conjuction with standard neural network architectures on standard benchmark tasks-whiee could be one of the main causes of the cold posterior effect (c.f., Germain al. 2016 van der Wilk al.) 2018). Reversing this argument, we can hypothesize that choosing better priors for BNNs may lead to less pronounced cold posterior effect, which we can use to evaluate different candidate priors.
EMPIRICAL ANALYSIS OF NEURAL NETWORK WEIGHTS
As we have discussed, standard Gaussian priors may not be the optimal choice for modern BNN architectures. But how can we find more suitable priors? Since it is hard to directly formulate reasonable prior beliefs about neural network weights, we turn an empirical approach. We trained fully connected neural networks (FCNNs), convolutional neural networks (CNNs), and ResNets with SGD on various image classification tasks to obtaín an approximation of the empirical distribution of the fitted weights, that is, the distribution of the maximun posteriori (MAP) solutions reached by SGD. If the distributions over SGD-fitted weights differ strongly from the usual isotropic Gaussian prior, that provides evidence that those features should be incorporated into the prior. Hence, can use our insights by inspecting the empirical weight distribution to propose better-suited priors.
Formally, this procedure can be viewed approximate human-in in-thi--ttthee loop expectation maximization (EM). In particular, in expectation maximization, we alternate expectation (H) and maximization (M) steps. In the expectation (E) step, we infer the posterior p(w) jr, y, 0.-1) over the weights, w, given the parameters of the prior from the previous step, 6,-1- our case, we approximately infer the weights using SGD. Then, in the maximization step, we compute new prior parameters 0,, by sampling weights w from the posterior computed in the E step, and maximizing the joint probability of sampled weights and data. As y is independent of the prior parameters if the weights are known, the M-step reduces to fitting prior distribution to the weights sampled from the posterior, that is,
(0) = F.p log p pr w) + log p(gp((())) (0)]
E Ep(ulx,000 loo p( 0)] + const
0. arg max (0)
(3)
Intuitively, this procedure allows the prior (and therefore the posterior) to assign more probability mass to the SGD solutions, which are known work well in practice. This is also related ideas from empirical Bayes (Robbins] 1992). where the (few) hyperparameters of the prior are fit to the data, and to recent ideas in Bayesin theory, where data-dependent priors have been shown to improve generalization guarantees over data-independent ones (Rivasplata et al. 2020 Dziugaite et al. 2021). While such approaches introduce a certain risk of overfitting (Ober et al., al.]]202 (021). we would argue that standard BNNs typically thought to be underfitting (Neal 1996] Wenzel et al. [2000a] Dusenberry ctt al. 2020) and that we do not directly the prior parameters, but merely draw inspiration for the choice of prior family from the qualitative shape of the empirical weight distributions.
We begin by considering whether the weights of FCNNs and CNNs are heavy-tailed, and move on to look correlational structure in the weights of CNNs and ResNets. Note that in the exploratory experiments here, we used SGD to perform MAP inference with a uniform prior (that is, maximum likelihood fitting). This avoids any prior assumptions obscuring interesting patterns in the inferred
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(a) Student-t DoF of weights by layer, ResNet20
(b) Spatial covariance of the weights, 3-layer CNN
Figure 2: (a) Degrees of freedom for Student-t distributions fitted to the weights of ResNet20 trained on CIFAR-I0. 10. The degrees of freedom get larger in deeper layers, implying that the weight distributions become less heavy-tailed and more similar to Gaussians. The layers marked with asterisks (*) are the first layers their respective ResNet blocks. (b) Spatial covariance of the weights within CNN filters for ttree-hidden layer network trained on MNIST, normalized by the number of channels. The weights correlate strongly with neighboring pixels, and anti-correlate (layer or do not correlate (layer 2) with distant ones. Each delineated square shows the covariances of filter location (marked with x) with all other locations.
weights. These patterns inspired our choice of priors, and we then evaluated these priors BNNs, showing that they improved classification performance (see Sec. 4).
3.1 FCNN WEIGITS ARE HEAVY-TAILED
We trained FCNN (Fig. D top) and CNN (Fig. middle) MNIST (LeCun al. 1998). The FCNN is a three layer network with 100 hidden units per layer and ReLU nonlinearities. The CNN is three layer network, with two convolutional layers and one fully connected layer. The convolutional layers have 64 channels and use 3 x 3 convolutions, followed by 2 x max max-pooling layers. All layers use ReI.U nonlinearities. Networks were trained with SGI) for 450 epochs using a learning rate schedule of 0.05, 0.005, and 0.0005 for 150 epochs each. We can see in Figure that the weight values of the FCNNs and CNNs follow a more heavy-tailed distribution than Gaussian, with the tails being reasonably well approximated by Laplace distribution. This suggests that "true" BNN priors might more heavy-tailed than isotropic Gaussians.
Next, we did similar analysis for a ResNer20 trained on CIFAR-10 (Krizhessky 2009) (Fig. bottom). Since this network had many layers, we quantified the degree -aivy-tailedness by fitting the degrees of freedom parameter of Student-t distribution. For - co, the Student-t becomes Gaussian, so large values indicate that the weights are approximately Gaussian, whereas smaller values indicate heavy-tailed behavior (see Sec. We found that at lower layers, small, so the weights were somewhat heavy-tailed, whereas at higher layers, became much larger, so the weights were approximately Gaussian (Fig. (h)
These results are perhaps expected if we assume that the filters have (using neuroscience terminology) "localized receptive fields". like those in Olshausen & Field (1997). Such filters contain a large number of near-zero weights outside the receptive field, with number of very large weights inside the receptive field (Sahani & Linden 2003), [Smyth al. 2003). and thus will follow a heavy-tailed distribution. As we get into the deeper layers of the networks, receptive fields are expected to become larger, so this effect may be less relevant.
3.2 CNN WEIGHTS ARE SPATIALLY CORRELATED
In the second part of our empirical inspection of fitted weight distributions, we looked at spatial correlations in CNN filters. In particular, considered 9-dimensional vectors formed by the 3 x 3 filters for every input and output channel. We studied our three-layer network trained on MNIST
and found strong correlations between nearby pixels, and lesser (layer 2) correlations at more distant pixels (Fig. (2).. We found similar spatial correlations in a ResNet20 trained CIFAR-10 10, across all layers, with correlation strength increasing move later layers
negative (layer 1)
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Figure 3: Spatial covariances for the convolutional weights of the layers a ResNet-20, normalized by the maximum variance for each layer, which is shown the bottom right. We trained the network with SGD on CIFAR-10 with data augmentation (10 times). Layer the closest to the input. The first layer of every ResNet block is marked with asterisk (*). We that there significant covariances in all layers, but that their strength increases for later layers.
(Fig. 3) We found by far the strongest evidence of correlations spatially, that is, between weights within the same convolutional filter. This could potentially be due to the smoothness and translation cquivariance properties of natural images (Simoncelli 2009). However,
also found some evidence for spatial correlations in the input layer of an FCNN (Fig. A. in the appendix), but no evidence for correlations between the channels of convolutional layer (Fig. A.5 in the appendix). Note though that this methodology cannot find structured correlations between channels, except at the input and output. This is because NN functions invariant to permutations of channels (Sussmann 1992]
MacKay 1992
Bishop et al. 1995 Aitchison 2020a Aitchison et al. 2020).
These findings suggest that better priors could be designed by explicitly taking this correlation structure into account. We hypothesize that multivariate distributions with non-diagonal covariance matrices could be good candidates for convolutional layer priors, especially when the covariances large for neighboring pixels within the convolutional filters (see Sec.
Additional evidence for the usefulness of correlated weights comes from the theory of infinitely wide CNNs and ResNets. Novak et al. (2019) noticed that the effect of weight-sharing disappears when infinite filters are used with isotropic priors. More recently, Garriga-Alonse (202T) showed that this effect can be avoided by using spatially correlated priors, leading to improved performance. Our experiments investigate whether this prior is also useful in the finite-width case.
van der Wilk
4 EMPIRICAL STUDY OF BAYESIAN NEURAL NETWORK PRIORS
We performed experiments MNIST and CIFAR-10. We compare Bayesian FCNNs, CNNs. and ResNets on these tasks. For the BNN inference, we used Stochastic Gradient Markov Chain Monte Carlo (CG-MCMC), in order scale to large training datasets. To obtain posterior samples that are close to the true posterior, we used an inference method that builds on the inference approach used in Wenzel et al. (2020a). which has been shown produce high-quality samples. particular, we combined the gradient-guided Monte Carlo (GG-MC) scheme from Garriga-Alonss & Fortuin (2021) with the cyclical learning rate schedule from Zhang et al. (019) and the preconditioning and convergence diagnostics from Wenzel et al. (2020a). We ran each chain for 60 cycles of 45 epochs each, taking one sample at the end of each of the last five epochs of each cycle, thus yielding 300 samples after 2,700 epochs, out of which temperature setting, dataset. model, and prior, we ran five such chains as replicates. Additional experimental results be found in Appendix A details about the evaluation metrics in Appendix
discarded the first 50 samples
burn-in. Per
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about the priors in Appendix and about the implementation in Appendix D In the figures, generally include an SGD baseline for the predictive error, where it is often competitive with some of the priors. For the likelihood, calibration, and OOD detection, the SGD baselines
out of the plotting range and are therefore not shown. For completeness, we show them in Appendix [A.4]
We show results for higher temperatures (7 > 1) in Appendix [A.6] for different prior variances in Appendix AA.7] and for different network architectures in Appendix AA.8 Moreover, while we focus on image classification tasks in this section, we provide results on UCI regression tasks in Appendix A.9. We also show inference diagnostics highlighting the accuracy of our MCMC sampling in [o.pendi|AA.10) Finally, we replicate our experiments on ResNets and CIFAR-10 for mean-field variational inference (Blundell et al. 2015) in Appendix A.11
4.1 PRIORS UNDER CONSIDERATION
We contrast the widely used isotropic Gaussian priors with heavy-tailed distributions, including the Laplace and Student-t distributions, and with correlated Gaussian priors. We chose these distributions based on our observations of the empirical weight distributions of SGD-trained networks (see Sec. 3 and for their ease of implementation and optimization. Further details on the distributions and their density functions can be found in Appendix C
The isotropic Gaussian distribution (Gauss) 1809) the de-facto standard for BNN priors in recent
work (e.g., Heeeandee--ooaaae & Adams| 2015 Louizos & Welling, 2017
Dusenberry et al. 2020
Wenzel et al. |000aa] Neal| 1992] Zhang et al. 2019) Osawa et al.. 2019: Immer et al. 2021b Lee
et al. 2017; Garriga-Alonso et al. 2019). However, its tails are re relatively light compared to some of the other distributions that we will consider and compared to the empirical weight distributions described above. The Laplace distribution (Laplace) 1774), for instance, has heavier tails than the Gaussian. It is often used in the context of (frequentist) lasso regression (Tibshirani 1996). Similarly, the Student-t distribution is also heavy-tailed. Moreover, it can be seen as Gaussian scale-mixture, where the scales are inverse-Gamma distributed (Helmert) 1875
Lüroth 1876).
For correlated Bayesian CNN priors, use multivariate Gaussian priors and define the covariance Σ to be block-diagonal, such that the covariance between weights in different filters is and between weights in the same filter is given by a Matérn kernel (u 1/2) on the pixel distances. Formally, for the weights wi,j and wy in filters and i' and for pixels j and j', the covariance
0o exp
d(j5') X
if =
cov(ur.j, wy
(4)
0
otherwise
where d(.--) the Euclidean distance between pixel positions and we set o X 1. This kernel was chosen to capture the decay with distance of spatial correlations (Fig.
4.2 BAYESIAN FCNN PERFORMANCE WITH DIFFERENT PRIORS
Following our observations from the empirical weight distributions (Sec. 3.J). D. we hypothesized that heavy-tailed priors should work better than Gaussian priors for Bayesian FCNNs. We tested this hypothesis by performing BNN inference with the same network architecture as in Sec. 3 using different priors. We report the predictive error and log likelihood on the MNIST test set. We follow Ovadia et al. (2019) in reporting the calibration of the uncertainty estimates on rotated MNIST digits and the out-of-distribution (OOD) detection accuracy on FashionMNIST (iiao et al. 2017 For more details about our evaluation metrics, see Appendix B
We observe that the heavy-tailed priors indeed outperform the Gaussian prior in terms of test error and test NLL in all cases, except for the Student-t distribution on MNIST at low temperatures (Fig. That said, calibration and OOD metrics are less clear, with heavy-tailed priors giving worse calibration and roughly similar OOD detection on MNIST and better calibration but worse OOD detection on FashionMNIST. Despite the unclear results on calibration and OOD detection, the error and NLL performance improvement for heavy-tailed priors at T = 1 is considerable, and suggests that Gaussian priors over the weights of FCNNs induce poor priors in the function space and inhibit the posterior from assigning probability mass to nigh-likelihood solutions, such as the SGD solutions analyzed above (Sec. 3. Finally, the cold posterior effect is removed- or even inverted- when using heavy-tailed priors, which supports the hypothesis that it is caused by prior misspecification in

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Figure 5: Performances of convolutional BNNs with different priors on MNIST, FNsHi and CIFAR-10 (see Sec. (4.)). The MNISS experiments used CNNs, while the CIFAR- 10 experiments used ResNet20. The correlated prior generally performs better than the isotropic ones, but still exhibits cold posterior effect, while the heavy-tailed priors reduce the cold posterior effect, but yield worse performance. Note the reversed y-axis for OOD detection on the right to ensure that lower values are better in all plots. Shaded regions represent one standard error.
FCNNs. Note that the cold posterior effect is typically observed in terms of performance metrics like error and NLL, and not calibration and OOD detection performance (Wenzel et al.) 2020a).. As such, even with Gaussian priors, we do not necessarily expect calibration and OOD detection to exhibit a cold posterior effect. Indeed, only calibration for FashionMNIST exhibits a cold posterior effect, with calibration for MNIST and all OOD detection results exhibiting an inverted cold posterior effect. Notably, we see in Appendix A.5 and Appendix [A.7 that these observations generalize to different activation functions and prior variances and in Appendix A.6] that warm posteriors (T > 1) deteriorate the performance for all considered priors, such that for the heavy-tailed priors, T N is indeed ideal.
4.3 BAYESIAN CNN AND RESNET PERFORMANCE WITH DIFFERENT PRIORS
We repeated the same experiment for Bayesian CNNs on MNIST and FNSH (Fig. 5 first two rows). Given our observations about -GD-trained weights (Sec. B.I), might again expect heavy-tailed priors to outperform Gaussian priors. However, this is not the case: the Gaussian and correlated Gaussian priors perform better in almost all cases, with the exception of calibration for
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FaSHionMNIST.. Interestingly, the performance of different methods tends to be very similar at T = 1, and to diverge for lower temperatures, with performance improving for Gaussian and correlated Gaussian priors (indicating cold posterior effect), and worsening for heavy-tailed priors, indicating no cold posterior effect.
Our analysis of KGD-trained weights (Sec. (.2) also suggested that introducing spatial correlations in the prior (Sec. [411 might help. We observe that introducing correlations indeed improves performance compared the isotropic Gaussian prior (Fig. 5). Notably, the performance improvement is small for CNNs trained on MNIST and FNShionMNIST. and for ResNets trained on CIFAR-10 10 at higher temperatures, but more considerable for ResNets at lower temperatures. As such, correlated priors actually increase the magnitude of the cold posterior effect in ResNets trained on CIFAR-10 10. This might be because ResNets trained at very low temperatures on CIFAR-10 10 have a tendency to overfit, and imposing the prior helps to mitigate this overfitting. To support this hypothesis, we indeed see that correlated priors considerably improve over all other methods in terms of calibration and OOD) detection at low temperatures for ResNets trained on CIFAR-10.
To reiterate a point raised in Sec. 4.2 the original cold posterior paper (Wenzel et al. 2020a) considered only predictive performance (error and likelihood), and not other measures of uncertainty such as calibration and OOD detection. Indeed, we see different effects of temperature on these measures, with calibration improving for FashionMNIST at lower temperatures, but worsening for MNIST and C1 At the same time, we see performance at OOD detection worsen at lower temperatures in the smaller CNN model trained on MNIST and FashionMNIST, but increase lower temperatures in the ResNet trained on CIFAR-10. These results are consistent with other observations that measures of uncertainty do not necessarily correlate with predictive performance (Ovadia et al.) 2019, Izmailov et al., (00II. and indicate that the cold posterior effect is a complex phenomenon that demands careful future investigation. Again, we see in Appendix [A.5 and Appendix [A.7 that these observations generalize to different activation functions and prior variances and in Appendix th6t warm posteriors (T 1) deteriorate the performance for all considered priors.
In practice, models on this dataset are often trained using data augmentation (as is our model in Fig. 5) While this does indeed improve the performance (Fig. [1.1 in the appendix), it also strengthens the cold posterior effect. When we do not use data augmentation, the cold posterior effect (at least between 7 = and lower temperatures) is almost entirely eliminated (see Fig. 1 in the appendix and Wenzel et al. 2020a; Izmailov al. 2021). This observation raises the question of why data augmentation drives the cold posterior effect. Given that data augmentation adds terms to the likelihood while leaving the prior unchanged, we could expect that the problem is in the likelihood, as was recently argued by (Aitchison (2020b). On the other hand, van der Wilk et al. (2018) argued that treating synthetic augmented data extra datapoints for the purposes of the likelihood is incorrect from a Bayesian point of view. Instead, they express data augmentation in the prior, by constraining the classification functions to be invariant certain transformations. More investigation is hence needed into how data augmentation and the cold posterior effect relate.
RELATED WORK
Empirical analysis of weight distributions. There is some history in neuroscience of analysing
the statistics of data to inform inductive priors for learning algorithms, especially when it comes to
vision (Simoncelli 2009). For instance, it has been noted that correlations help in modeling natural
images Sriiasttva al. 2003), as well as sparsity in the parameters (Smyth et al.
20033] (ahani &
Linden 2003). In the context of machine learning, the empirical weight distributions of standard
neural networks have also been studied before (Bellido & Fiesler, 1993, Go & Lee 1999.. including
the insight that SGD can produce heavy-tailed weights (Gurbuzbalabas & Simsekli, 2020). but these
works have not systematically compared different architectures and did not use their insights to inform Bayesian prior choices.
BNNs in practice. Since the inception of Bayesian neural networks, scholars have thought about choosing good priors for them, including hierarchical (MacKay| 1992) and heavy-tailed ones (Neal 1996). In the context of infinite-width limits of such networks (Lee et al. 2017 Matthews et al. 2018 Garriga-Alonso et al. 2019: Yang, 2019 Tsuchida et al.) 2019) it has also been shown that networks with very heavy-tailed (i.c., infinite variance) priors have different properties from Emite-variance
priors Neal 1996: Peluchetti al. 2020). However, most modern applications of BNNs still relied
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on simple Gaussian priors. Although few different priors have been proposed for BNNs, these
were mostly designed for specific tasks (Atanov et al. [0188 Ghosh & Doshi- Velez, 2017] Overweg
jet al. 2019 Nalismick| 20188] Cui et al. 2020 Hafner et al. 2020) or relied heavily on non-standard
inference methods (Sun et al. 2019 Ma et al. 2019 Karaletsos & Bui 2020
Pearce et al. 2020)
Moreover, while many interesting distributions have been proposed as variational posteriors for BNNs
[ooizoos & Welling 2017: Swiatkowski et al., 2020 Dusenberry et al., 2020 Ober & Aitchison
2020 Aitchison al. 2020). these approaches have still used Gaussian priors. Others use non-
Gaussian prior, but approximate the posterior with a diagonal Gaussian (Blundell et al. 2015; Ghosh & Doshi Velez 2017 Nalisnick et al., 2015). somewhat limiting the prior's s effect. Another BNN posterior approximation is dropout (Gal & Ghahramani 20166; Kingma et al.) 2015), which is often poorly calibrated (Foong et al. 2019), but can also be seen to induce a scale-mixture prior, similar to our heavy-tailed priors (Molchanov et al. 2017).
BNN priors. Finally, previous work has investigated the performance of neural network priors chosen without reference to the empirical distributions of SGD-trained networks (Blundell et al. [0155] Ghosh
& Doshi-Velees 2017 Wu et al. 2018:
Atanov et al., 2018 Nalisnick] 2018: Overrwe et al., 20199]
Farquhar et al., 2019 Cui al., 2020 Rothfuss al. [2020] Hafner et all]|202)) Matsubara et al.
2020: Tran et al., 2020: Ober & Aitchison [200]] Sarriga-Alonaa & van der Wilk] 2001 Fortuin
2021 Immer al. [202 While these priors might in certain cireumstances offer performance
improvements, they did not offer a recipe for finding potentially valuable features to incorporate into
the weight priors. In contrast, we offer such recipe by examining the distribution of weights trained under a uniform prior with SGD, Importantly, unlike prior work, we use SG-MCMC with carefully evaluated convergence metries and systematically address the cold posterior effect.
Contemporaneous work? [Izmailov et al. 2021> compared gold-standard HMC inference with the more practical cyclical SG-MCMC used our work. They confirmed that cyclical SG-MCMC methods indeed have high-fidelity to the true posterior, and interestingly show that heavy-tailed priors offer slight performance improvements for language modeling tasks (though they do not assess the interaction of the cold posterior effect with these priors).
CONCLUSION
We consider empirical weight distributions in non- -on-Baysii networks trained using SGD, finding that FCNNs displayed heavy-tailed weight distributions, and CNNs and ResNets displayed spatial correlations the convolutional filters. We therefore tested the performance of these priors and their interaction with the cold posterior effect. Indeed, we found that these priors improved performance, but their impact on the cold posterior effeet was more complex, with heavy-tailed priors in FCNNs climinating the cold posterior effect, correlated priors in CNNs trained on MNIST and FashionMNIST leaving the cold posterior largely unchanged, and correlated priors in ResNets trained on CIFAR-10 actually increasing the cold posterior effect, as they yield much larger performance improvements at lower temperatures. Importantly though, we do not expect there to be one "universal" prior that improves performance in all architectures and all tasks. The best prior is almost certain be highly task- and eechitectre-ddepedeeee and indeed we found that heavy-tailed priors offer little or no benefits for regression on UCI datasets (Sec. A.9
Thus, we can conclude that isotropic Gaussian priors are often non-optimal, and that it is worth exploring other priors more generally (as always though, the correct prior will heavily depend on the architecture and dataset). However, it is difficult to come to any strong conclusions regarding the origin of the cold posterior effect. At least in FCNNs, it does indeed appear that a misspecified prior can cause the cold posterior effect. However, in perhaps more relevant large-scale image models, we found that better (correlated) priors actually increase the cold posterior effect, which is consistent with other hypotheses, such as a misspecified likelihood (Aitchison 2020b), though of course we cannot rule out that there is a better prior that eliminates the cold posterior effect that we did not consider. We hope that our PoTchc library for BNN inference with different priors will catalyze future research efforts in this area and will also be useful on real-world tasks.
released arXiv two months after ours
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ACKNOWLEDGMENTS
VF was supported by a PhD fellowship from the Swiss Data Science Center. AGA was supported by UK Engineering and Physical Sciences Research Council studentship [1950008]. We thank Alexander Immer, Andrew Foong, David Burt, Seth Nabarro, and Kevin Roth for helpful discussions and the anonymous reviewers for valuable feedback. We also thank Edwin Thompson Jaynes for constant inspiration.
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A
ADDITIONAL EXPERIMENTAL RESULTS
A.1 COVARIANCE MATRICES OF FCNN, CNN AND RESNET
Here we report the full covariance matrices for the layers that were analyzed above (Sec. 3.2). We display the covariances for the FCNN for layer Frg..A.I), layer (Fig. A.2)) (Fig.(A.2) and layer 3 (Fi.. A.3).
The only discernable structure is in the first layer, presumably because the weights from neighboring pixels will be correlated. The other plots are less smooth than an empirical covariance matrix from an isotropic Gaussian (left image of every pair), but tend to have discernible structure.
Next, we give the covariances of CNN weights in layer (Fig. A.4) and layer (Fig, A.5). We have omitted layer 3 of the CNN because is just fully connected layer and also showed interesting structure.
Finally, Fig. AA.6 (left) measures the amount of covariance of every layer in the ResNet. We fit the lengthscale of a Gaussian distribution with squared exponential kernel, on the spatial correlations of the convolutional filters. The right-hand figure is the same Fig. 2h
A.2 EMPIRICAL OFF-DIAGONAL COVARIANCES
We report results for the distributions of off--diaonna covariances for the respective second layers of our FCNN and CNN in Figure A.7 The empirical distribution of off-diagonal elements in the covariance matrices is shown as a histogram, overlayed with a kernel density estimate of the expected distribution if the weights were samples from an isotropic Gaussian. We see that the empirical covariance distributions are generally more heavy-tailed than the ideal ones, that is, the empirical weights generally have larger covariances than would be expected from isotropic Gaussian weights. Note that, as observed above, the strongest covariances by far are found spatially in the CNN weights, that is, between weights within the same CNN filter. We report the same results for the other layers in the following. The FCNN results are shown in Figures A.8 and A.9 and the CNN results in Figure A.
A.3 THE INFLUENCE OF DATA AUGMENTATION THE COLD POSTERIOR EFFECT
When running the CIFAR- 10 experiments with Bayesian ResNets with and without data augmentation, we find that data augmentation seems to significantly increase the cold posterior effect (Fig. (A.II). Moreover, data augmentation seems to increase the performance of the models lot at colder temperatures, but not at the true Bayes posterior = 1. This suggests that data augmentation can also be one of the reasons for the cold posterior effect, as already hypothesized by Wenzel al. (2020a) and Aitchison (2020b).
A.4 SGD BASELINES
In terms of likelihood, calibration, and OOD detection, almost all our BNN models consistently outperformed the SGD baselines. The results including SGD are shown for FCNNs in Figure A.12 for CNNs in Figure A.13 and for ResNets in Figure A.14
A.5 ALTERNATIVE ACTIVATION FUNCTIONS
We repeated the experiments on MNIST with Bayesian FCNNs and CNNs and replaced the ReLU activation functions from Figure and Figure 5 with sigmoid (see Fig. A.155 and tanh (see Fig, A. activations respectively. We observe that while the performances overall worse than with ReLU activations (as generally expected), the effects the different priors are qualitatively very similar.
A.6 HIGHER TEMPERATURES
In the main body of the paper, we followed Wenzel et al. (2020a) in showing only posteriors with temperatures < 1, because we were interested in studying cold posteriors. Here, we also show results for warm posteriors, that is, T 1. We in Figure A.17 and Figure A.18 that these
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(a) Input weight covariance
(b) Output weight covariance
Figure A.I: FCNN layer empirical covariances of the weights, trained with SGD on MNIST. We can see correlations in the spatial direction in the weights of the input layer (left). In the other directions, the covariance matrix less smooth than we would expect from an isotropic Gaussian draw of the same size (left matrix of every pair), but otherwise has no discernible structure. This suggests that the weights are not isotropic Gaussian.
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(a) Input weight covariance
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Figure A.2: FCNN layer empirical covariances of the weights, trained with SGD on MNIST. The covariance matrix is less smooth than we would expect from isotropic Gaussian draw, but has no discernible structure.
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(a) Input weight covariance
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Figure A.3: FCNN layer 3 empirical covariances of the weights, trained with SGD on MNIST. The covariance matrix is less smooth than we would expect from isotropic Gaussian draw, but has no discernible structure.
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(a) Input weight covariance
(b) Output weight covariance
Figure A.4: CNN layer I empirical covariance of the weights, trained with SGD on MNIST. The input (also spatial) direction has correlations, also shown in Figure The output direction has no discernible structure.
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(a) Input covariance
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Figure A.5: CNN layer 2 empirical covariance of the weights, trained with SGD on MNIST. The input direction less smooth than the isotropic Gaussian, and some low-rank structures can be observed. should display the spatial correlation of Figure Pb. The output direction has discemnbbe structure.
warm posteriors generally do not improve the performance and that hence some of the priors (e.g.. heavy-tailed priors in FCNNs) do indeed achieve their optimal performance for x 1.
A.7 DIFFERENT PRIOR VARIANCES
In the main text, we use models where the prior variance is chosen according to the He initialization (He et al., 2016), which is motivated by the conservation of the activation norm across the depth of the networks. Here, we see in Figures 19 A.20 A.21 A.22 A.23 that our main observations regarding the ordering of the different priors and the cold posterior effect still hold, even for different prior variances (in this case, four times larger and smaller than the He variance).
A.8 DIFFERENT FCNN ARCHITECTURES
In the main we use FCNN models with three layers. Here, see in Figure A.24 that our main observations regarding the ordering of the different priors and the cold posterior effect still hold, even for different architectures (in this case, between and layers).
A.9 UCI REGRESSION
While the experiments the main paper focus on image classification, we also performed BNN experiments UCI regression tasks. The architecture is 3-layer FCNN, the hidden layers 64 units wide. We run GGMC for 30,000 epochs without minibatching on "boston". "energy", "yacch", and "wine", discarding runs where the potential diverges. For the other datasets, which are larger, run 3000 epochs, also without minibatching. The learning rate is flat 5 10-5 and we do not use a cosine schedule.
Even with full batch MCMC, it is clear that the dynamics for regression networks are much less stable, especially at lower temperatures (which have "sharper" potential landscape). Figure A.25
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Lengthscale
Degrees of freedom
2
10
10
'L2
*L8
*L14
L19
L19
Layer index (input LO)
Figure A.6: Left: fitted lengthscale of a multivariate Gaussian with a squared exponential kernel (see eq eq.44 4) to the data of Figure All the entries of the SE covariance are positive, so this cannot capture all the features of the data, which has negative empirical covariance. Right: fitted degrees of freedom of a multivariate t-distribution, to same data. The empirical covariance was used in this case. The fitting criterion is the log-likelihood of the data. This is the same plot Figure]
Histogram counts
ind. Gaussian cov. entries
-5
0
5
-5
0
-0.5
0
0.5
-0.25
0
0.25
Entries of weight covariance matrix (x10-9 33
FCNN, input
b) FCNN, output
CNN, input
CNN, output
CNN, spatial
Figure A.7: Distributions off-diagonal elements in the empirical covariances the layer 2 weights of FCNNs and CNNs trained with SGD MNIST. The empirical distributions are plotted as histograms, while the idealized random Gaussian weights are overlaid in orange. see that the covariances of the empirical weights more heavy-tailed than for the Gaussian weights.
00:0- DOGS
D.005
0.02
0.
omrpttcal
ompliical
Layer I, columns
(b) Layer 1,
(c) Layer 3, columns
(d) Layer rows
Figure A.8: Distributions of off-diagonal elements in the empirical covariances of the weights of the FCNN in layers and 3. The empirical distributions are plotted as histograms, while the idealized random Gaussian weights overlaid in orange. We that the covariances of the empirical weights are more heavy-tailed than for the Gaussian weights.
03
1.0
0.20
0.15
04
0.10
0.1
8.8
6.0
0.00
35
10.0
(a) Layer 1
(b) Layer
Figure A.9: Distributions of singular values of the weight matrices of the FCNN in layers and 3. see that the spectra of the empirical weights decay faster than the ones of the Gaussian weights.
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shows that T = 1 is best for all datasets, in terms of the median mean squared enror (MSE) well as the quantiles and outliers. The priors are generally reasonably close in performance, such that is harder in this case to strongly prescribe certain prior choice. Of course, it is absolutely expect that different priors will be appropriate for different problems, especially when those problems are quite so distinct regression and image classification.
The split-Ai R diagnostics are considerably higher for regression (Table A.1 than for the classification setting (Sec. [A.10 10.2| for which the diagmostics look very good). Thus, the results here should be taken with grain of salt. They are representative of how GGMC-trained BNNs behave at each of these temperatures and priors, but the results may be different for other (more accurate) ways of approximating the posterior. However, one thing clear: UCI regression datasets exhibit no cold posterior effect, and for lower temperatures, the GGMC chains are less stable.
for every prior and UCI dataset. The split- R is generally high, which shows the chain has not fully explored the posterior. The best prior (in terms of median mean squared error, MSE) for each dataset is bolded, no prior is better overall. Additionally, each prior's performance is very similar for each dataset, which implies that the choice of prior does not matter much here (at least among these three).
Table A.1: Diagnostics and median performance temperature
split- ft diagnostic gaussian
Median MSE gaussian
laplace
student-t
laplace
student-t
boston
1.984856
1.664200
2.164047
0.051899
0,061199
0.056180
concrete
1.923906
1.960722
1.654736
0.962948
0.802203
0.728850
energy
2.026471
1.939554
2.309070
0.000759
0.000705
0.001500
kin8nm
1,523657
1.795404
1.609185
0,976947
1.520133
0.958203
naval
1,506857
1.700794
1,583324
1.241157
1.146679
1.679176
power
1.666368
1.738743
2,471113
0.156302
0,286625
0.148974
protein
1.574833
2.037037
1.510434
1.151049
1,296588
1.259395
wine
2.133220
1.936430
1.844044
0.674713
0.617795
0.604185
yacht 2.034536
1.880282
2.414279
0.000612
0.000559
0.000599
A. INFERENCE DIAGNOSTICS
One of the main goals of our work is make statements about the true BNN posteriors that are as accurate as possible. To this end, we closely monitored the accuracy of our inference algorithm. In order check the correctness of our SG-MCMC inference, we estimated the temperature of the sampler using the two diagnostics from Wenzel et al. (2020a), namely the kinetic temperature and the configurational temperature.
The kinetic temperature is derived from the sampler's momentum m Rd The inner product m M -mm. for the (in this case diagonal) mass matrix M, an estimate of the scaled variance of the momenta. If the sampler is correct it should, in expectation, be equal to the desired temperature. The configurational temperature is slightly more involved and is discussed in Appendix A. 10.
m,
As an example, we show the estimated kinetic temperatures for our ResNet experiment on CIFAR-10 in Figure The desired temperature is shown as dotted horizontal line. The kinetic temperatures for the other experiments look qualitatively similar and are shown in Appendix I. We see that the kinetic temperatures generally agree well with the true temperatures, so the sampler works as expected there. In contrast, the configurational temperature estimates can be somewhat larger than T. especially when T is small (see Appendix AA. This suggests that there could be small inference inaccuracies at low temperatures. However, these inaccuracies are small, and the configurational temperature certainly decreases as decreases, so there should be no impact on the overall trends.
We also computed the rank mmn-normalized split-R diagnostic Vehtari et al. 2202)) which measures how well a collection of independent Markov chains have mixed. The split- R is related to the ratio of between-chain and within-chain variances, and should be as close to 1 as possible. Given the complexity of neural network weight posteriors, we report the R for the quantities we are interested in estimating (the y-values in Figs. 4 aldd 5) For every considered model and function, Table A.2
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contains the worst (highest) R estimate obtained across all priors. Appendix A.10.2 contains more detailed explanation and empirical R estimates for different priors.
We can see that, for most experiments, the chains have mixed sufficiently. Only for the larger models (CIFARIO Resdets)-and, to lesser extent, Student-t FCe chains have mixed less well. Interestingly, for all convolutional networks, the correlated prior mixes best. This further supports its suitability as a prior for image data and CNNs.
A.10.1 KINETIC AND CONFIGRAATINNAL TEMPERATURE ESTIMATES
As described above, we use two temperature diagnostics (inspired by Wenzel et al. (020aa)) the kinetic temperature and the configurational temperature. The kinetic temperature derived from the sampler's momentum matrix M, is an estimate of the scaled variance of the momenta. is always positive and should, in expectation, be equal to the desired temperature. In contrast, the configurational temperature is M m + const is the Hamiltonian. In
R. The inner product 3 M Mm'm, for the (in this case diagonal) mass
e H m), where H (0, m) = log p(O () + m 3 expectation, this should also equal T. Unlike the kinetic temperature estimator, the configurational temperature estimator is not guaranteed to be always positive, even though the temperature is always positive. Using subsets of a parameter or momentum also yields estimators of the temperature.
In both cases, we estimate the mean and its standard error from a weighted average of parameters or momenta. That is. for each separate NN weight matrix or bias vector, we estimate its kinetic and configurational temperature using the expressions above. Then, we take their average and condaard-dviationn weighted by the number of elements in that parameter matrix or vector.
A.30 A.3II and A.32 as mean one standard error. The desired temperature is shown as dotted horizontal line. The kinetic temperatures generally agree well with the true temperatures, so our sampler works as expected there.
We show the estimated temperatures of all our BNN experiments in Figures | A.28 A.29
The configurational temperature estimates have higher variance than the kinetic ones. Especially in the regime of small true temperatures, they often tend to slightly over- or underestimate the temperature. This is not surprising, since at low temperatures the noise in the gradients is dominated by the minibatching as opposed to the temperature noise. Correctly estimating the temperature from the gradients thus becomes harder.
Note that while the relative deviations can seem large in this regime, the absolute deviations are still quite small. Note also that while the conditioned momenta are strictly positive, the inner products between gradients and parameters can become negative in principle, which is why at low temperatures (close to 0) the configurational temperature estimates might sometimes be bit below 0. Overall, the sampler is still within the tolerance levels of working correctly here, but there could be some small inaccuracies low temperatures. However, judging from the shape of the actual tempering curves (see Sec. (4), the measures usually change more in the higher temperature regimes than in the lower ones, so there no strong reason believe that the inference at low temperatures was too inaccurate to support the results.
A.10.2 BETWEEN- BETWEEN-CHAIN AND WITHIN-CHAIN VARIANCES
The split-R Rt estimator measures the difference between posterior variance estimate in each chain, and between chains. It is roughly the square root of the between-chain variance divided by the within-chain variance (Vehtari et al., 2021 eq. 1-3). Its value is usually not smaller than 1, and
a chain that has mixed well should have a value no larger than R < 1.01 (Vehtari et al. 2021 (Previously, a threshold of 1.1 was considered enough (Gelman et al.) 2013] Section 11.5).)
Neural network functional forms have a large number of parameter symmetries (for example. per- mutation invariance). Accordingly, the true BNN posterior should sample from all these modified parameters with probability proportional to their prior. However, for prediction purposes, it does not matter the parameters are stuck in single "permutation" and do not mix.
Therefore, for the purposes of this paper, we calculate the π diagnostic not directly on the parameters, but on symmetry-invarial functions of the parameters, In practice, this amounts to evaluating the
NN test set, and calculating the R diagnostic for functions of the logits and the prior probability.
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Table A.3: Estimated R values for the different models and priors with respect to the loss.
Gaussian
Laplace
Student-t
Correlated
MNIST FCNN
1,000
1,001
1.006
FashionMNIST FCNN
1.000
1.000
1.007
MNIST CNN
1,000
1.000
1.002
1.000
FashionMNIST CNN
1.001
1.003
1.009
1.001
CIFARIO ResNet
1.115
1.115
1.125
1.109
CIFARIO ResNet (augmented)
1.057
1.054
1.047
1.066
Table A.4: Estimated R values for the different models and priors with respect the potential.
Gaussian
Laplace
Student-t
Correlated
MNIST FCNN
1.000
1.002
1.023
FashionMNIST FCNN
1,000
1,000
1.013
MNIST CNN
1,000
1.000
1.001
1.000
PashionMNIST CNN
1,000
1.002
1.007
1.000
CIFARIO ResNet
1.166
1.147
1.171
1.139
CIFARIO ResNet (augmented)
1.085
1.083
1.073
1.090
Tables A.3 A.4 and A.5 display the value of the diagnostic Å for different such functions: the log-likelihood, the unnormalized log-posterior (potential), and the log-prior, respectively.
We employ the rank-normalized i estimator (Vehtari al. 2021 eq. 14) implemented in the ArviZ library (Kumar et al. 2019).
The diagnostics are generally favorable ( < 1.01, mostly) for smaller NNs (FCNNs and 2-layer CNNs) and for MNIST. Within the ResNets applied to CIFAR10, the prior distribution with the R closer is the correlated Gaussian. This provides evidence that inference is easier in the case of the correlated Gaussian, and therefore that the correlated Gaussian is better prior (Gelman et al. 2013) Yang et al. 2015). This because the prior is good, the data are plausible simulations from it; so the posterior is close to the prior and will be easy to approximate.
A.11 VARIATIONAL INFERENCE
In this paper, our experimental results have focused on inference with SG-MCMC, as we wished to obtain the most reliable posterior possible. However, can-sampling approaches such variational inference (VI; e.g., Graves 201 I. Blundell et al.) 2015 Dusenberry et al., 2020) and Laplace's method (c.g. Immer al.) 202 remain popular the literature. Therefore, might be valuable to understand the effect of the prior on the performance these methods. In this section, we focus variational inference /Wainwright et al. 2008), in particular the mean-field VI (MFVT) approach
(Graves 2011 Blundell et al. 2015).
Table A.5: Estimated & values for the different models and priors with respect to the log prior.
Gaussian
Laplace
Student-t
Correlated
MNIST FCNN
1.000
1.005
1.101
FashionMNIST FCNN
1,000
1.003
1.104
MNIST CNN
1,001
1,002
1.013
1.001
PashionMNIST CNN
1.002
1,006
1,013
1.001
CIFARIO ResNet
1.404
1.232
1.366
1.195
CIFAR1O ResNet (augmented)
1.274
1.346
1.264
1.198
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Variational inference attempts to approximate the true intractable posterior p(w]z,y) y) by tractable approximate posterior q(w) from an approximating family Q by maximizing the evidence lower bound (ELBO):
9º (w) arg max C(g; A) 9E
arg max Egglog w)] - XKL(g(w) Ip(w)).
(5)
TE@
For À = 1, the ELBO true lower bound to the marginal likelihood of the model, and the true posterior is recovered as the optimal solution when Q is the family of all distributions over w. For MFVI, we restrict the approximating distribution to be fully-factorized Gaussian, that is, ((w) = (V: [ui (i), so that there is no correlation structure in the approximate posterior. The variational parameters (Jis 0;) can then be optimized using the reparameterization trick (Kingma Welling 2014 Rezende et al. 2014).
As with (SG-)MCMC, we can temper the posterior by adjusting X, with 0 < À < resulting in a "cold posterior". However, we note that apart from the case = T' = 1, where we target the true posterior in both VI and MCMC, there is no straightforward, direct relationship between the cold posterior obtained in Eq. [ and that obtained from Eq. (for discussion see Wenzel et al. (2020a), particularly App. E).
A.11.1 EXPERIMENTAL DETAILS AND RESULTS
We replicate the experiment in Sec. 4.3 for the ResNet architecture using CIFAR-IO. 10. We train each model for 1,000 epochs on batches of 500 augmented datapoints, using Adam (Kingma & Ba 2015) with an initial learning rate of 0.01, which we reduce to 0.001 after 500 epochs. We are able to use these relatively high learning rates because we follow the parameterization introduced in Ober & Aitchison (2020); we also follow their step-wise tempering scheme for the first 100 epochs, which gradually increases the influence of the KL term. We
sample from the approximate posterior for training and 10 samples for testing. Finally, we again run 5 replicates for each model.
We plot the results of this experiment in Figure A.33 We immediately make a few observations. First, the performance of MFVI is far worse than that of SG-MCMC on all metrics with the exception of calibration. We note that the performance at documented behavior that tempering with À < is required for decent performance with MFVI (e.g., Wenzel et al. 2020a). Finally, it does not seem that the choice prior has much effect on the performance of MFVI, as all priors perform similarly. We hypothesize that this is largely due to the field-field assumption imposed on the approximate posterior. which severely restricts its expressiveness and can lead to pathological behavior (Foong et al. 2019 Trippe & Turner (2018). The mean-field assumption leads to a poor approximation to the true posterior, and therefore will not be influenced by the choice of prior as SG-MCMC. However, we leave full investigation of these effects to future work.
particularly bad, which reflects the well-
B EVALUATION METRICS
When using BNNs, practitioners might care about different outcomes. In some applications, the predictive accuracy might be the only metric of interest, while in other applications calibrated uncertainty estimates could be crucial. We therefore use range of different metrics in our experiments in order to highlight the respective strengths and weaknesses of different priors. Moreover, we compare the priors to the empirical weight distributions of conventionally trained networks.
B. EMPIRICAL TEST PERFORMANCE
Test error The test error is probably the most widely used metric in supervised learning. It intuitively measures the performance of the model on a held-out test set and is often seen as an empirical approximation to the true generalization error. While it is often used for model selection, it comes with the risk of overfitting to the used test set (Bishop, 2006) and in the case of BNNs also fails to account for the predictive variance of the posterior.
Test log-likelihood The predictive log- likelihod also requires a test set for its evaluation, but takes the predictive posterior variance into account. It can thus offer a built-in tradeoff between the
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mean fit and the quality of the uncertainty estimates. Moreover, it is proper scoring rule (Gneiting Raftery 2007).
B.2 UNCERTAINTY ESTIMATES
Uncertainty calibration Bayesian methods are often chosen for their superior uncertainty esti- mates, so many users of BNNs will not be satisfied with only fitting the posterior mean well. The calibration measures how well the uncertainty estimates of the model correlate with predictive perfor- mance. Intuitively, when the model is for instance 70 % certain about a prediction, this prediction should be correct with 70 % probability. Many deep learning models are not well calibrated, because they are often overconfident and assign too low uncertainties to their predictions (Ovadia et al. 2019 Wenzel et al. 20205). When the models are supposed to be used in safety-critical scenarios, is often crucial to be able to tell when they encounter input that they are not certain about (Kendall &
Gal 2017). For these applications, metrics such as the expected calibration error (Nacini al. 2015 might be the most important criteria.
-uttof-distributtin detection The out-of-distribution (OOD) detection measures how well one tell in-distribution and out-of-distribution examples apart based the uncertainties. This is important when we believe that the model might be deployed under some degree of dataset shift. In this case, the model should be able detect these OOD examples and be able reject them, that is, refuse to make a prediction on them.
C DETAILS ABOUT THE CONSIDERED PRIORS
We contrast the widely used isotropic Gaussian priors with heavy-tailed distributions, including the Laplace and Student-t distributions, and with correlated Gaussian priors. We chose these distributions based on our observations of the empirical weight distributions SGD-trained networks (see Sec. 3 and for their of implementation and optimization. We give quick overview over these different distributions and their most salient properties.
Gaussian. The isotropic Gaussian distribution (Gauss 1809 the de-facto standard for BNN
priors in recent work (e.g.. -ormanndzzLLbbtto & Adams| [ & Welling [017 Dusenberry
al. 2020 Wenzel al. [2200a] Neal, 1992] Zhang et al. [219]] Osawa et al.) 2019: [mmer et al.
2021b
Lee et al. 2017: Garriga-Alonoo et al. 2019). Its probability density function (PDF) is
(x 11)²
p(:; ut, o
exp
2
20²
with mean i and standard deviation σ.. attractive, because it is the central limit of all finite- variance distributions (Billingsley [196]) and the maximum entropy distribution for given and scale (Bishop. 2006). However, its tails are relatively light compared to some of the other distributions that we will consider.
Laplace. The Laplace distribution discontinuous μ.. Its PDF is
(Laplace 1774) has heavier tails than the Gaussian and is
1
m b
p(z; b)
26 exp
with mean and scale b. It is often used in the context of (frequentist) lasso regression (Tibshirani 1996).
Student-t.
The Student-t distribution characterizes the mean of finite number of samples from Gaussian distribution (Student 1908). Its PDF
((
1 (x (60-143 11)2
p(z;
r(5)
where μ is the mean, I' is the gamma function, and y are the degrees of freedom. The Student-t also arises as the marginal distribution over Gaussians with inverse-Gamma prior over the variances
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(Helmert. 1875 Lüroth 1876). For oc, the Student-t distribution approaches the Gaussian. For any finite it has heavier tails than the Gaussian. Its k-th moment is only finite for > k. The parameter thus offers convenient way to adjust the heaviness of the tails. Note that it also controls the variance of the distribution, which is w/(v- 2) (or else undefined). Unless otherwise stated, we set = 3 in experiments, such that the distribution has rather heavy tails, while still having finite mean and variance.
Multivariate Gaussian with Matérn covariance. For our correlated Bayesian CNN priors, use multivariate Gaussian priors
p(:; na Σ) (( det E
exp
- x 11122-1
with || ||00-113-- (x 1-))==1- M- 11)
where d is the dimensionality.
In our experiments, we set μ and define the covariance E be block-diagonal, such that the covariance between weights in different filters is and between weights in the same filter is given by a Matérn kernel (u = 1/2) on the pixel distances, as applied by Garriia-AAlono van der Wilk (2021) in the infinite-width case. Formally, for the weights wi,j and wy. in filters and i' and for pixels and j', the covariance is
0² exp
(-dGis") 2
ifi= 1'
cov(wes we
(6)
else
where :(:,)) is the Euclidean distance in pixel space and we set a = A = 1.
D IMPLEMENTATION DETAILS
Training setup. For all the MNIST BNN experiments, perform 60 cycles of SG-MCMC (Zhang et al. 2019) with 45 epochs each. We draw one sample each at the end of the respective last five burn-in of the chain.
epochs of each cycle. From these 300 samples, discard the first 50 Moreover, in each cycle, we only add Langevin noise in the last 15 epochs (similar to Zhang et al. (2019)) We start each cycle with learning rate of 0.01 and decay using cosine schedule. We use a mini-batch size of 128.
For the SGD experiments yielding the empirical weight distributions, do not add any Langevin noise. We also do not use any cycles and just train the networks once to convergence, which in our case took 600 epochs.
the settings, but
We ran the experiments on GPUs of the type NVIDIA GeForce GTX 1080 Ti and NVIDIA GeForce RTX 2080 on our local cluster. The main experiments (see Fig. and Fig. took around 10,000 GPU hours to run.
FCNN architecture. For the FCNN experiments, we used feedforward neural network with three layers, hidden layer width of 100, and Rel.U activations.
CNN architecture. For the CNN experiments, we use convolutional network with two convolu- tional layers and one fully connected layer. The hidden convolutional layers have 64 channels each and use 3 x convolutions and ReLU activations. Each convolutional layer is followed by a 2 x max-pooling layer.
ResNet architecture and data augmentation. For the ResNet experiments on CIFAR-10 10, we use ResNet20 architecture (He et al. 2016). equal to the used in Wenzel al. (2020a). For data augmentation, we pad all the images with pixels on each border and then randomly crop out a 32x32 image out of that padded one and then randomly flip half of the images horizontally
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Software packages. We implemented the inference and models with the PyTorch library (Paszke al. 2019). To manage our experiments and schedule runs with several settings, we used Sacred (Greff et al. 2017) and Jug (Coelho 2017) respectively. For the diagnostics, we also use Arviz (Kumar et al.
2019
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wagtt
0.0
0.10
0.1
0.00
-0.10
-0.05
0.00
0.10
-2.10
-0.55
0.00
0.05
0.10
emptiical
emptticll
gaussann
(a) Layer columns
(b) Layer rows
(c) Layer singular values
Figure A.10: Distributions of off- aff-diagonal elements in the empirical covariances of the weights and singular values of the CNN in the other layer. The empirical distributions are plotted as histograms, while the idealized random Gaussian weights overlaid an orange line. We see that the covariances of the empirical weights are more heavy-tailed than for the Gaussian weights and that the singular value spectrum for the empirical weights decays faster than the Gaussian ones.
Error
Likelihood
Calibration
OOD detection
w/ o data aug.
0.50
Student
lafacee
0.315
COD AUROC
0.82
goussian
1 0.300
correlated
0.12
SCD
0.325
0.86
0.10
10-1 tomperature
10-1
10-1
10-1
10-1
10-1
10-1
10-4
10-1
temperature
temporature
tomperature
W/ data ang.
0.35
0.85
0.50
0.25
0.05
temperature
temperature
Figure A.11: Performances of Bayesian ResNets with different priors on CIFAR-10 with and without data augmentation in terms of different metrics. Data augmentation seems to increase the cold posterior effect.
Error
Likelihood
Callbration
0OD detection
0.20
MNIST
0.025
crea
0:100
0.30
0.020
0.10
10-2
10-1
Temperature
temperature
Temperatuure
temperatinn
FashionMNNSST
0.14
AUROC
3.7
0.050
0.0
8.17
0.4
10 tomperature
10
10
10
10:7
30
30
10:2
tomperature
Semporature
Cemporature
Figure A.12: Performances of fully connected BNNs with different priors on MNIST and Fashion- MNIST terms of different metrics, compared to SGD solutions. The heavy-tailed priors perform better for Fashion MNIST, and perform better for MNIST at least for Laplace for error and NLL., heavy-tailed priors also eliminate the cold posterior effect (they get worse as temperature falls).
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Error
Likelihood
Calibration
detection
MNIST
0.0175
0.15
0.97
0.0100
0.98
0.0055
0.62
0.99
10-4
FashionMNIs
0.00
0.075
AUROC
0.70
0:75
NILL
goussion
g 0000
correlated
0.00
0.5
0.25
0.67
10
temperature
temperature
temperature
tempereture
Figure A.13:
Performances of convolutional BNNs with different priors on MNIST and Fashion- MNIST in terms of different metries, compared SGD solutions. The correlated prior generally performs better than the isotropic ones, but still exhibits a cold posterior effect, while the heavy-tailed priors reduce the cold posterior effect, but yield a worse performance.
Eeror
Likelibood
Calibeation
OOD detection
w//o data aug.
students
0.3
0C0 AUROC
1.0
laplace
0.14
gaussian
0.2
6.80
0.12
0.5
cor cornettted
500
0.10
10-"
10-1 tomperature
tomperature
temparature
tomperature
w/ data aug.
0.85
0.0
0.2
COD
0.00
0.1
10 temperature
Temperature
temperature
temperature
Figure A.14 Performances of Bayesian ResNets with different priors on CIFAR-10 with and without data augmentation in terms of different metrics, compared to SGD solutions. The correlated prior generally outperforms the other ones. Moreover, data augmentation seems to increase the cold posterior effect.
Error
Likelibood
Calibration
0OD detection
FCNN
0.02
gaussion
Isplace
0.15
0.30
0.32
020 AUROC
0.85
0.80
0.08
0.06
0.30
10 temperature
10
10a
10.2
10
20
30 7
15:7
10e
10
107
temperature
temperature
temperature
CNN
coner
0.04
Rg loc
Rudontt
0.000
0.02
gelssien
wot uted
0.10
0.000
10"
10-4
30
10
10-4 temperature
10-h
10
10t4
10-1
10
Comperature
temparature
temparature
Figure A.15: Performances of fully connected and convolutional BNNs with sigmoid activation functions on MNIST. The observed effects are qualitatively similar to the ones with ReI.U activations in the main body of the paper.
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Error
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Figure A. Performances of fully connected and convolutional BNNs with tanh activation functions on MNIST. The observed effects are qualitatively similar to the ones with ReLU activations in the main body of the paper.
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Figure A.17: Performances of Bayesian FCNNs with different priors on (Fashion-)MNIS including temperatures T > 1. The performances generally do not improve for warm posteriors, such that T ss 1 is indeed optimal for some priors.
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Figure A.18: Performances of Bayesian CNNs and Resnets with different priors on (Fashion-)MNNSTT and CIFAR, including temperatures T >> 1. The performances generally do not improve for warm posteriors, such that T indeed optimal for some priors. Note that here, we do not use data augmentation for CIFAR.
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Published as a conference paper at ICLR 2022
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Figure A919: Performances of Bayesian FCNNs with different priors and different prior variances on MNIST, The qualitative behavior is similar to the one for the He variance in the main text.
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Figure A.20: Performances of Bayesian CNNs with different priors and different prior variances on MNIST. The qualitative behavior is similar to the one for the He variance in the main text.
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Figure A.21: Performances of Bayesian FCNNs with different priors and different prior variances on Fashion- MNISS. The qualitative behavior is similar to the one for the He variance in the main text.
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Published as a conference paper at ICLR 2022
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Figure A.22: Performances of Bayesian CNNs with different priors and different prior variances on Fashion-MNIST The qualitative behavior is similar to the one for the He variance in the main text.
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Figure A.23: Performances of Bayesian Resnets with different priors and different prior variances on CIFAR-10. The qualitative behavior is similar to the one for the He variance in the main text. Note that here, we do not use data augmentation.
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Figure A.24: Performances of Bayesian FCNNs with different priors and different depths on MNIST. The qualitative behavior for the different numbers of layers is similar the one for three layers in the main text.
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Published as a conference paper at ICLR 2022
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We exclude runs where the potential diverges. Temperature is
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Published as a conference paper at ICLR 2022
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Figure A.27: Temperature diagnostics of the MNIST experiment with FCNNs.
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Figure A.29: Temperature diagnostics of the FashionMNIST experiment with FCNNs.
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Published as a conference paper at ICLR 2022
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Figure A.30: Temperature diagnostics of the FashionMNIST experiment with CNNs.
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Figure A.31: Temperature diagnostics of the CIFAR-10 experiment with ResNets without data augmentation.
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Figure A.32: Temperature diagnostics of the CIFAR-10 experiment with ResNets with data augmen- tation.
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Published as a conference paper at ICLR 2022
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Figure A.33: Performances mean-field variational inference ResNets with different priors on CIFAR-10. Note the reversed y-axis for OOD detection the right ensure that lower values better in all plots. Shaded regions represent one standard error.
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Reviews:
[{"review_id": "mvdK9DuJVv2", "reviewer": "Reviewer_sapj", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "*Update after rebuttal* I have read the authors’ response and other reviews. I would like to thank both the authors and other reviewers for providing an interesting discussion. Although the other reviews raise valid points on how to make the paper stronger, I believe it is worth publishing in its current form.\n\n======= \n\n\n\n\nStrong features:\n1. Very well written paper. I really enjoy reading this paper. One can rarely see this style nowadays with more textbook, discussion-like language. \n2. Very thorough experiments. Although it should be expected from a paper which main contribution is empirical evaluation of a phenomenon, but not always empirical papers hold this expectations. This submission does. Everything is considered from different angles, everything is tested - different activation functions, different variance, convergence, etc. \n3. The paper is well placed in the context of the current literature. \n4. The paper provides interesting insights into the default choice of the prior for BNNs as well as provides a recipe to explore this direction further.\n\nWeak features:\nI do not really see major weak points for this paper. Of course, there is always a room for improvement in anything, but in this case it would be \"nice to have\", rather that \"should/must have\". It would be interesting to see other types of priors, bigger datasets, other domains, but the current experiments have already taken 10,000 GPU hours as claimed by the authors. \nThe actual insight from the paper, i.e. that isotropic Gaussians are not the best choice of prior, may be not very exciting and surprising, but the paper provides a thorough analysis to show this and a recipe how to choose a better prior.\n\nEven with a non-surprising result, I like this paper and enjoy reading it. I believe it provides useful and interesting discussion to the community and therefore I recommend acceptance of this paper. There is no anything particular novel in terms of the algorithms and methods in the paper, but rather a very well done empirical analysis that addresses a relevant problem. \n\nSpecific suggestions/comments (of different significance for assessment, but mostly for further improvement of the paper):\n1. The acronym SGD in abstract is not introduced\n2. The second paragraph in Introduction. \"at lower temperatures\" - a temperature has not been introduced yet\n3. Eq. (1) - strictly speaking x and y should be defined separately\n4. In the empirical analysis of weight distributions (Section 3), it is claimed that the posterior distribution reached by SGD would be a good choice for a prior. However, the posterior distribution family = the prior distribution family only when a prior is conjugate to a likelihood. This argument is therefore should be made more carefully\n5. Figure 2a. It would be interesting to see some explanation/discussion of why the degree of freedom decreases between L14 and L19\n6. Details on data augmentation is missing\n7. Figure 3. It is claimed that the strength increases for later layers, but the max variance is less for them\n8. The acronym SGLD is not introduced\n9. Section 4. “To the best of our knowledge, this procedure constitutes the best SGLD-based inference approach” - before that it was stated that the authors used a combination of 3 different methods, therefore this claim looks unjustified\n10. Figures 4-… What do shaded areas represent?\n11. For the correlated Gaussian, it is not discussed why the Matern kernel in particular has been chosen\n12. Sec A.1. “The covariances of the FCNN weights are shown in Figure A.5” – seems that it should be Figure A.1\n13. Figure A.10 is not referred to in the text\n14. Figure A.12 – not sure I can agree with the conclusions about heavy-tailed distributions made in the caption", "summary_of_the_paper": "The paper provides an analysis of priors other than standard isotropic Gaussians for Bayesian neural network. The authors first empirically estimate the posterior distributions of the weights of a BNN and then use these distributions as a prior for these BNNs. They show that those priors allow better performance of BNNs. The paper discusses this analysis in the context of the cold posterior effect.", "summary_of_the_review": "I vote for acceptance of this paper. I believe it provides a thorough empirical insights on the prior choice for BNNs and a recipe for future prior exploration.", "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": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "iuUeFPFMtgD", "reviewer": "Reviewer_yXaK", "summary": "", "questions": "", "limitations": "", "rating": 5, "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": "Strengths:\n1) The paper took an empirical approach by first analyzing the weight distributions of the SGD trained model, then identifying the characteristics of these distributions, and finally proposing priors with the corresponding characteristics to facilitate the Bayesian learning. An expectation-maximization procedure is provided to justify the above approaches and empirical results support the choice of the priors.\n\n2) The impact of these new priors on different type of networks are discussed, especially the cold-posterior effect which relates to the setting the the temperature in the posterior.\n\nWeakness:\n1) It would be great if the impact to the computational speed could be discussed further. Would the new priors drastically increase the computational demand, or the computation needed is similar to the isotropic gaussian prior?\n\n2) As discussed in section 3 equation (2), SGD is used to sample from the posterior on w. However, it is unclear to me why SGD could be used to sample from the posterior weights of the networks. According to my understanding, a proper tool to sample from neural network posteriors is stochastic gradient Langevin dynamics (https://icml.cc/2011/papers/398_icmlpaper.pdf) rather than SGD. The \"stochastic\" part of SGD is with respect to the batch sampling from the data distribution, not with respect to the weight parameters. More clarification is needed regarding this choice of the weight sampler.\n\n3) One motivation of the work from the introduction section is to impose proper prior distributions on the network weights, so that principled Bayesian inference can be carried out without temperature adjustment as in cold-posterior. However, according to my understanding, when the temperature is set to be 1, the results with the proposed priors do not always match or outperform the (non-Bayesian) SGD result (see Figure 4 and 5). This somewhat weakens the motivation and impact of the paper.", "summary_of_the_paper": "This paper proposed to use distributions different from isotropic gaussian for Bayesian neural network priors. For fully connected neural networks the heavy-tailed distributions such as laplacian and student-t are used, while for convolutional neural networks multivariate gaussian with spatial correlations are employed. Empirical results show that these alternative priors give better results than isotropic gaussian on several image classification tasks. Cold-posteriors versions of the isotropic gaussian prior and the proposed priors are also provided for comprehensive comparison. Overall this is an interesting paper with novel discoveries.", "summary_of_the_review": "This paper studied the important question of what would be a proper prior for Bayesian neural networks, and proposals were made based on empirical investigations of weight distributions of different neural networks. Experimental results on some image datasets validated the choice of the priors. In summary this is a nice paper with interesting findings. Still there are some technical details that need to be addressed (see my comments above).", "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": "i8C8HzTqYkv", "reviewer": "Reviewer_Ergj", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "# Weaknesses\n\n * There is no clear take-at-home message.\n * The authors generalize their results on *one* FCNN to *all* FCNNs and on *one* CNN to *all* CNNs, on one task. The consistency across models and datasets should have been checked.\n * There is no heuristics or explanation of the observed results (except for the influence of the data augmentation on the \"cold posterior effect\".\n\n# Strengths\n\nThe experimental report is interesting for researchers who want to have a better intuition of the influence of the prior.\n\n# EDIT\n\nI have taken note of the modification made by the authors in the Discussion section, which moderates a bit the main claims. But I do not change my recommendation, since the main claims (in the first sections) are still too affirmative regarding the experiments.\n\nAnyway, as an impressive empirical work, this paper deserves to be published. ", "summary_of_the_paper": "This paper presents numerous empirical facts about the distribution of the weights after training a neural network. The authors considered one FCNN, one CNN, and one ResNet, and computed several statistics on their weights.\n\nThe authors also propose, in a Bayesian setup, to compare the performance of trained BNNs with different priors.", "summary_of_the_review": "Weak accept.\n\nDespite the numerous and contradictory experiments (heavy tailed priors are better for FCNNs, but worse for CNNs), and the difficulty the authors have to interpret their results, the empirical results are significant enough for helping other researchers to guide their research. \n\nFor instance:\n * a heavy-tailed distribution of the weights is not necessarily a problem;\n * the distribution of weights after training a FCNN and a CNN may be structurally different;\n * the \"cold posterior effect\" can be partly explained by data augmentation;\n * the \"cold posterior effect\" can be removed when using a prior with heavier tail.\n\nA major issue remains: the authors should have checked that their results about \"FCNNs\" and \"CNNs\" are consistent for *several* FCNNs/CNNs and several datasets.", "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": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "Q4rS15mzBSK", "reviewer": "Reviewer_CSpQ", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "__Comments__:\n\n\n- The results are kind of unstable, on the many plots (Figure 5, Figure A.19, Figure A.20) Gaussian prior behaves similarly or better than heavy-tailed analogs. \n\n\n- The results on the cold-posterior effect are also unstable between models, which is reported in the paper.\n\n\n- SGD baselines are undertrained (Figure 5, Figure A.11), e.g., ResNet-20 (w/ data aug.) error should be approx 8%, while 10% is reported. That is a quite important issue that may significantly flip the results of the experiments.\n\n\n- The work studies only a single inference technique, while the results will not necessarily generalize between inference techniques. So, it is hard to make any general conclusions.", "summary_of_the_paper": "The work studies prior distributions for Bayesian CNNs. The work report that conventionally used priors e.g., Gaussian poorly fit empirical distributions of the trined weights. The empirical distributions appear to be heavy-tailed and correlated. Thus the paper proposes to use \"heavy-tailed priors\" for FCNNs and correlated Gaussian priors for CNNs that tend to improve classification performance.", "summary_of_the_review": "The work studies an interesting problem of prior selections for BNNs. At the same time, the results of the two main experiments are mixed and do not clearly demonstrate the importance of the heavy-tailed priors. The study also considers a single inference technique. Overall that makes me conclude that results are not generalizable and reliable. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "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": {}}]

Author rebuttal:

Meta-review: This paper provides some empirical investigation of the choice of the prior distribution for the weights in Bayesian neural networks. It shows empirically that, when trained via SGD, weights in feedforward neural networks exhibit heavy-tails, while weights in convolutional neural networks are spatially correlated. From this observation they show that the use of such priors leads to some improved performances compared to the iid Gaussian prior in some experimental settings.

Reviewers have conflicting views on this paper, that have not been reconcilied after the author's response and the discussion.
On the plus side, the paper is very well written, the experimental part is carefully conducted, and provides some insights on the choice of the prior in Bayesian neural networks, which could lead to further developments. 
On the negative side, the claims made in the introduction are not fully supported by the experiments (the claims have been slightly amended in the revised version), and the take-home message is not so clear. In particular, Bayesian approaches with the proposed priors still underperform compared to SGD without tempering. The authors could also have considered a broader sets of experiments.

Overall, I think the contributions outweight the limitations of this paper, and I would recommend acceptance.
Decision: Accept (Poster)