Scale-invariant Bayesian Neural Networks with Connectivity Tangent Kernel

Setup and Definitions We consider a Neural Network (NN), $f(\cdot,\cdot):\mathbb{R}^{D}\times\mathbb{R}^{P}\rightarrow\mathbb{R}^{K}$, given input $x\in\mathbb{R}^{D}$ and network parameter $\theta\in\mathbb{R}^{P}$. Hereafter, we consider one dimensional vector a one-dimensional vector as a single column matrix unless otherwise stated. We use the output of NN $f(x,\theta)$ as a prediction for input $x$. Let $\mathcal{S}:=\{(x_{n},y_{n})\}_{n=1}^{N}$ denote the independently and identically distributed (i.i.d.) training data drawn from true data distribution $\mathcal{D}$, where $x_{n}\in\mathbb{R}^{D}$ and $y_{n}\in\mathbb{R}^{K}$ are input and output representation of $n$-th training instance, respectively. For simplicity, we denote concatenated input and output of all instances as $\mathcal{X}:=\{x:(x,y)\in\mathcal{S}\}$ and $\mathcal{Y}:=\{y:(x,y)\in\mathcal{S}\}$, respectively and $f(\mathcal{X},\theta)\in\mathbb{R}^{NK}$ as a concatenation of $\{f(x_{n},\theta)\}_{n=1}^{N}$. Given a prior distribution of network parameter $p(\theta)$ and a likelihood function $p(\mathcal{S}|\theta):=\prod_{n=1}^{N}p(y_{n}|x_{n},\theta):=\prod_{n=1}^{N}p(y_{n}|f(x_{n},\theta))$, Bayesian inference defines posterior distribution of network parameter $\theta$ as $p(\theta|\mathcal{S})=\frac{1}{Z(\mathcal{S})}\exp(-\mathcal{L}(\mathcal{S},\theta)):=\frac{1}{Z(\mathcal{S})}p(\theta)p(\mathcal{S}|\theta),Z(\mathcal{S}):=\int p(\theta)p(\mathcal{S}|\theta)d\theta$ where $\mathcal{L}(\mathcal{S},\theta):=-\log p(\theta)-\sum_{n=1}^{N}\log p(y_{n}|x_{n},\theta)$ is training loss and $Z(\mathcal{S})$ is normalizing factor. For example, the likelihood function for regression task will be Gaussian: $p(y|x,\theta)=\mathcal{N}(y|f(x,\theta),\sigma^{2}\mathbf{I}_{k})$ where $\sigma$ is (homoscedastic) observation noise scale. For classification task, we treat it as a one-hot regression task following Lee et al. [21] and He et al. [22]. While we applied this modification for theoretical tractability, Lee et al. [23], Hui and Belkin [24] showed this modification offers reasonable performance competitive to the cross-entropy loss. Details on this treatment is given in Appendix B.

Laplace Approximation In general, the exact computation for the Bayesian posterior of a network parameter is intractable. The Laplace Approximation (LA; [8]) proposes to approximate the posterior distribution with a Gaussian distribution defined as $p_{\mathrm{LA}}(\psi|\mathcal{S})\sim\mathcal{N}(\psi|\theta^{*},(\nabla^{2}_{\theta}\mathcal{L}(\mathcal{S},\theta^{*}))^{-1})$ where $\theta^{*}\in\mathbb{R}^{P}$ is a pre-trained parameter with training loss and $\nabla^{2}_{\theta}\mathcal{L}(\mathcal{S},\theta^{*})\in\mathbb{R}^{P\times P}$ is Hessian of loss function w.r.t. parameter at $\theta^{*}$.

Recent works on LA replace the Hessian matrix with (Generalized) Gauss-Newton matrix to make computation tractable [25, 26]. With this approximation, the LA posterior of regression problem can be represented as:

where $\alpha,\sigma>0$ and $\mathbf{I}_{P}\in\mathbb{R}^{P\times P}$ is a identity matrix and $\mathbf{J}_{\theta}\in\mathbb{R}^{NK\times P}$ is a concatenation of $\mathbf{J}_{\theta}(x,\theta^{*})\in\mathbb{R}^{K\times P}$ (Jacobian of $f$ w.r.t. $\theta$ at input $x$ and parameter $\theta^{*}$) along training input $\mathcal{X}$. Since covariance of equation 1 is inverse of $P\times P$ matrix, further sub-curvature approximation was considered including diagonal, Kronecker-factored approximate curvature (KFAC), last-layer, and sub-network [27, 28, 29]. Furthermore, they found that proper selection of prior scale $\alpha$ is needed to balance the dilemma between overconfidence and underfitting in LA.

PAC-Bayes bound with data-dependent prior We consider a PAC-Bayes generalization error bound of classification task used in McAllester [30], Perez-Ortiz et al. [31] (especially, equation (7) of Perez-Ortiz et al. [31]). Let $\mathbb{P}$ be any PAC-Bayes prior distribution over $\mathbb{R}^{P}$ independent of training dataset $\mathcal{S}$ and $\mathrm{err}(\cdot,\cdot):\mathbb{R}^{K\times K}\rightarrow[0,1]$ be a error function which is defined separately from the loss function. For any constant $\delta\in(0,1]$ and $\lambda>0$, and any PAC-Bayes posterior distribution $\mathbb{Q}$ over $\mathbb{R}^{P}$, the following holds with probability at least $1-\delta$: $\mathrm{err}_{\mathcal{D}}(\mathbb{Q})\leq\mathrm{err}_{\mathcal{S}}(\mathbb{Q})+\sqrt{\frac{\mathrm{KL}[\mathbb{Q}\|\mathbb{P}]+\log(2\sqrt{N}/\delta)}{2N}}$ where $\mathrm{err}_{\mathcal{D}}(\mathbb{Q}):=\mathbb{E}_{(x,y)\sim\mathcal{D},\theta\sim\mathbb{Q}}[\mathrm{err}(f(x,\theta),y)]$, $\mathrm{err}_{\mathcal{S}}(\mathbb{Q}):=\mathbb{E}_{(x,y)\sim\mathcal{S},\theta\sim\mathbb{Q}}[\mathrm{err}(f(x,\theta),y)]$, and $N$ denotes the cardinality of $\mathcal{S}$. That is, $\mathrm{err}_{\mathcal{D}}(\mathbb{Q})$ and $\mathrm{err}_{\mathcal{S}}(\mathbb{Q})$ are generalization error and empirical error, respectively. The only restriction on $\mathbb{P}$ here is that it cannot depend on the dataset $S$.

Following the recent discussion in Perez-Ortiz et al. [31], one can construct data-dependent PAC-Bayes bounds by (i) randomly partitioning dataset $\mathcal{S}$ into $\mathcal{S}_{\mathbb{Q}}$ and $\mathcal{S}_{\mathbb{P}}$ so that they are independent, (ii) using a PAC-Bayes prior distribution $\mathbb{P}_{\mathcal{D}}$ only dependent of $\mathcal{S}_{\mathbb{P}}$ (i.e., independent of $\mathcal{S}_{\mathbb{Q}}$ so $\mathbb{P}_{\mathcal{D}}$ belongs to $\mathbb{P}$), (iii) using a PAC-Bayes posterior distribution $\mathbb{Q}$ dependent of entire dataset $\mathcal{S}$, and (iv) computing empirical error $\mathrm{err}_{\mathcal{S}_{\mathbb{Q}}}(\mathbb{Q})$ with target subset $\mathcal{S}_{\mathbb{Q}}$ (not entire dataset $\mathcal{S}$). In summary, one can modify the aforementioned original PAC-Bayes bound through our data-dependent prior $\mathbb{P}_{c}$ as

where $N_{\mathbb{Q}}$ is the cardinality of $\mathcal{S}_{\mathbb{Q}}$. We denote sets of input and output of splitted datasets ($\mathcal{S}_{\mathbb{P}},\mathcal{S}_{\mathbb{Q}}$) as $\mathcal{X}_{\mathbb{P}},\mathcal{Y}_{\mathbb{P}},\mathcal{X}_{\mathbb{Q}},\mathcal{Y}_{\mathbb{Q}}$ for simplicity.

Our goal is to construct scale-invariant $\mathbb{P}_{\mathcal{D}}$ and $\mathbb{Q}$. To this end, we first assume a pre-trained parameter $\theta^{*}\in\mathbb{R}^{P}$ of the negative log-likelihood function with $\mathcal{S}_{\mathbb{P}}$. This parameter can be attained with standard NN optimization procedures (e.g., stochastic gradient descent (SGD) with momentum). Then, we consider linearized NN at the pre-trained parameter with the auxiliary variable $c\in\mathbb{R}^{P}$ as

where $\mathrm{diag}$ is a vector-to-matrix diagonal operator. Note that equation 3 is the first-order Taylor approximation (i.e., linearization) of NN with perturbation $\theta^{*}\odot c$ given input $x$ and parameter $\theta^{*}$: $g^{\mathrm{pert}}_{\theta^{*}}(x,c):=f(x,\theta^{*}+\theta^{*}\odot c)=f(x,\theta^{*}+\mathrm{diag}(\theta^{*})c)\approx g^{\mathrm{lin}}_{\theta^{*}}(x,c)$, where $\odot$ denotes element-wise multiplication (Hadamard product) of two vectors. Here we write the perturbation in parameter space as $\theta^{*}\odot c$ instead of single variable such as $\delta\in\mathbb{R}^{P}$. This design of linearization matches the scale of perturbation (i.e., $\mathrm{diag}(\theta^{*})\odot c$) to the scale of $\theta^{*}$ in a component-wise manner. Similar idea was proposed in the context of pruning at initialization [32, 33] to measure the importance of each connection independently of its weight. In this context, our perturbation can be viewed as perturbation in connectivity space by decomposing the scale and connectivity of parameter.

Based on this, we define a data-dependent prior ($\mathbb{P}_{\mathcal{D}}$) over connectivity as

This distribution can be translated to a distribution over parameter by considering the distribution of perturbed parameter ($\psi:=\theta^{*}+\theta^{*}\odot c$): $\mathbb{P}_{\theta^{*}}(\psi):=\mathcal{N}(\psi\,|\,\theta^{*},\alpha^{2}\mathrm{diag}(\theta^{*})^{2})$. We now define the PAC-Bayes posterior over connectivity $\mathbb{Q}(c)$ as follows:

where $\mathbf{J}_{c}\in NK\times P$ is a concatenation of $\mathbf{J}_{c}(x,\mathbf{0}_{P}):=\mathbf{J}_{\theta}(x,\theta^{*})\mathrm{diag}(\theta^{*})\in\mathbb{R}^{K\times P}$ (i.e., Jacobian of perturbed NN $g^{\mathrm{pert}}_{\theta^{*}}(x,c)$ w.r.t. $c$ at input $x$ and connectivity $\mathbf{0}_{P}$) along training input $\mathcal{X}$. Our PAC-Bayes posterior $\mathbb{Q}_{\theta^{*}}$ is the posterior of Bayesian linear regression problem w.r.t. connectivity $c$ : $y_{i}=f(x_{i},\theta^{*})+\mathbf{J}_{\theta}(x_{i},\theta^{*})\mathrm{diag}(\theta^{*})c+\epsilon_{i}$ where $(x_{i},y_{i})\in\mathcal{S}$ and $\epsilon_{i}$ are i.i.d. samples of $\mathcal{N}(\epsilon_{i}|\mathbf{0}_{K},\sigma^{2}\mathbf{I}_{K})$. Again, it is equivalent to the posterior distribution over parameter $\mathbb{Q}_{\theta^{*}}(\psi)=\mathcal{N}\left(\psi|\theta^{*}+\theta^{*}\odot\mu_{\mathbb{Q}},(\mathrm{diag}(\theta^{*})^{-2}/\alpha^{2}+\mathbf{J}_{\theta}^{\top}\mathbf{J}_{\theta}/\sigma^{2}\right)^{-1})$ where $\mathrm{diag}(\theta^{*})^{-2}:=(\mathrm{diag}(\theta^{*})^{-1})^{2}$ by assuming that all components of $\theta^{*}$ are non-zero. Note that this assumption can be easily satisfied by considering the prior and posterior distribution of non-zero components of NNs only. Although we choose this restriction for theoretical tractability, future work can modify this choice to achieve diverse predictions by considering the distribution of zero coordinates. We refer to Appendix C for detailed derivations of $\mathbb{Q}_{\theta^{*}}(c)$ and $\mathbb{Q}_{\theta^{*}}(\psi)$.

Now we provide an invariance property of prior and posterior distributions w.r.t. function-preserving scale transformations as follows: The main intuition behind this proposition is that Jacobian w.r.t. connectivity is invariant to the function-preserving scaling transformation, i.e.,
$\mathbf{J}_{\theta}(x,\mathcal{T}(\theta^{*}))\mathrm{diag}(\mathcal{T}(\theta^{*}))=\mathbf{J}_{\theta}(x,\theta^{*})\mathrm{diag}(\theta^{*})$.

Now we plug in prior and posterior into the modified PAC-Bayes generalization error bound in equation 2. Accordingly, we obtain a novel generalization error bound, named PAC-Bayes-CTK, which is guaranteed to be invariant to scale transformations (hence without the contradiction of FM hypothesis mentioned in Sec. 1).

Note that recent works on solving FM contradiction only focused on the scale-invariance of sharpness metric: Indeed, their generalization bounds are not invariant to scale transformations due to the scale-dependent terms (equation (34) in Tsuzuku et al. [14] and equation (6) in Kwon et al. [15]). On the other hand, generalization bound in Petzka et al. [16] (Theorem 11 in their paper) only holds for single-layer NNs, whereas ours has no restrictions for network structure. Therefore, to the best of our knowledge, our PAC-Bayes bound is the first scale-invariant PAC-Bayes bound. To highlight our theoretical implications, we show the representative cases of $\mathcal{T}$ in Proposition 2.2 in Appendix D (e.g., weight decay for network with BN), where the generalization bounds of the other studies are variant, but ours is invariant, resolving the FM contradiction on bound level.

The following corollary explains why we name PAC-Bayes bound in Theorem 2.3 PAC-Bayes-CTK.

Note that Corollary 2.4 clarifies why $\sum_{i=1}^{P}h(\beta_{i})/4N_{\mathbb{Q}}$ in Theorem 2.3 is called the sharpness term of PAC-Bayes-CTK. As eigenvalues of CTK measures the sensitivity of output w.r.t. perturbation on connectivity, a sharp pre-trained parameter would have large CTK eigenvalues, so increasing the sharpness term and the generalization gap by according to Corollary 2.4.

Finally, Proposition 2.5 shows that empirical CTK is also scale-invariant.

To verify that PAC-Bayes bound in Theorem 2.3 is non-vacuous, we compute it for real-world problems. We use CIFAR-10 and 100 datasets [36], where the 50K training instances are randomly partitioned into $\mathcal{S}_{\mathbb{P}}$ of cardinality 45K and $\mathcal{S}_{\mathbb{Q}}$ of cardinality 5K. We pre-train ResNet-18 [37] with a mini-batch size of 1K on $\mathcal{S}_{\mathbb{P}}$ with SGD of initial learning rate 0.4 and momentum 0.9. We use cosine annealing for learning rate scheduling [38] with a warmup for the initial 10% training step. We fix $\delta=0.1$ and select $\alpha,\sigma$ based on the negative log-likelihood of $\mathcal{S}_{\mathbb{Q}}$.

To compute the equation 8, one need (i) $\mu_{\mathbb{Q}}$-based perturbation term, (ii) $\mathbf{C}^{\theta^{*}}_{\mathcal{X}}$-based sharpness term, and (iii) samples from PAC-Bayes posterior $\mathbb{Q}_{\theta^{*}}$. $\mu_{\mathbb{Q}}$ in equation 6 can be obtained by minimizing $\arg\min_{c\in\mathbb{R}^{P}}L(c)=\frac{1}{2N}\|\mathcal{Y}-f(\mathcal{X},\theta^{*})-\mathbf{J}_{c}c\|^{2}+\frac{\sigma^{2}}{2\alpha^{2}N}c^{\top}c$ by first-order optimality condition. Note that this problem is a convex optimization problem w.r.t. $c$, since $c$ is the parameter of the linear regression problem. We use Adam optimizer [39] with fixed learning rate 1e-4 to solve this. For sharpness term, we apply the Lanczos algorithm to approximate the eigenspectrum of $\mathbf{C}^{\theta^{*}}_{\mathcal{X}}$ following [40]. We use 100 Lanczos iterations based on the their setting. Lastly, we estimate empirical error and test error with 8 samples of CL/LL implementation of Randomize-Then-Optimize (RTO) framework [41, 42]. We refer to Appendix E for the RTO implementation of CL/LL.

Table 1 provides the bound and related term of the resulting model. First of all, we found that our estimated PAC-Bayes-CTK is non-vacuous [43]: estimated bound is better than guessing at random. Note that the non-vacuous bound is not trivial in PAC-Bayes analysis: only a few PAC-Bayes literatures [44, 43, 31] verified the non-vacuous property of their bound, and other PAC-Bayes literatures [45, 14] do not. To check the invariance property of our bound, we scale the scale-invariant parameters in ResNet-18 (i.e., parameters preceding BN layers) for fixed constants $\{0.5,1.0,2.0,4.0\}$. Note that this scaling does not change the function represented by NN due to BN layers, and the error bound should be preserved. Table 1 shows that our bound and related terms are invariant to these transformations. On the other hand, PAC-Bayes-NTK is very sensitive to parameter scale, as shown in Table 7 in Appendix J.

Now, we focus on the fact that the trace of CTK is also invariant to the parameter scale by Proposition 2.5. Unlike PAC-Bayes-CTK/NTK, a trace of CTK/NTK does not require onerous hyper-parameter selection of $\delta,\alpha,\sigma$. Therefore, we simply define $\textbf{CS}(\theta^{*}):=\mathrm{tr}(\mathbf{C}^{\theta^{*}}_{\mathcal{X}})$ as a practical sharpness measure at $\theta^{*}$, named Connectivity Sharpness (CS) to detour the complex computation of PAC-Bayes-CTK. This metric can be easily applied to find NNs with better generalization, similar to other sharpness metrics (e.g., trace of Hessian), as shown in [7]. We evaluate the detecting performance of CS in Sec. 4.1. The following corollary shows how CS can explain the generalization performance of NNs, conceptually.

As the trace of a matrix can be efficiently estimated by Hutchinson’s method [46], one can compute the CS without explicitly computing the entire CTK. We refer to Appendix F for detailed procedures of computing CS. As CS is invariant to function-preserving scale transformations by Theorem 2.5, it also does not contradict the FM hypothesis.

One can view parameter space version of $\mathbb{Q}_{\theta^{*}}$ as a modified version of $p_{\mathrm{LA}}(\psi|\mathcal{S})$ in equation 1 by (i) replacing isotropic damping term to the parameter scale-dependent damping term ($\mathrm{diag}(\theta^{*})^{-2}$) and (ii) adding perturbation $\theta^{*}\odot\mu_{\mathbb{Q}}$ to the mean of Gaussian distribution. In this section, we focus on the effect of replacing the damping term of LA in the presence of weight decay of batch normalized NNs. We refer to [47, 48] for the discussion on the effect of adding perturbation to the LA with linearized NNs.

The main difference between covariance term of LA equation 1 and equation 7 is the definition of Jacobian (i.e. parameter or connectivity) similar to the difference between empirical CTK and NTK in remark 2.6. Therefore, we name $p_{\mathrm{CL}}(\psi|\mathcal{S})\sim\mathcal{N}(\psi|\theta^{*},\left(\mathrm{diag}(\theta^{*})^{-2}/\alpha^{2}+\mathbf{J}_{\theta}^{\top}\mathbf{J}_{\theta}/\sigma^{2}\right)^{-1})$ as Connectivity Laplace (CL) approximated posterior.

To compare CL posterior and existing LA, we explain how weight decay regularization with BN produces unexpected side effects in existing LA. This side effect can be quantified if we consider linearized NN for LA, called Linearized Laplace (LL; Foong et al. [49]). Note that LL is well known to be better calibrated than non-linearized LA for estimating ’in-between’ uncertainty. By assuming $\sigma^{2}\ll\alpha^{2}$, the predictive distribution of linearized NN for equation 1 and CL are

for any input $x\in\mathbb{R}^{d}$ where $\mathcal{X}$ in subscript of CTK/NTK means concatenation. We refer to Appendix G for the detailed derivations. The following proposition shows how weight decay regularization on scale-invariant parameters introduced by BN can amplify the predictive uncertainty of equation 10. Note that the primal regularizing effect of weight decay originates through regularization on scale-invariant parameters [50, 13].

Recently, [47, 48] observed similar pitfalls in Proposition 3.1. However, their solution requires a more complicated hyper-parameter search: independent prior selection for each normalized parameter group. On the other hand, CL does not increase the hyper-parameter to be optimized compared to LL. We believe this difference will make CL more attractive to practitioners.

To verify that the CS actually has a better correlation with generalization performance compared to existing sharpness measures, we evaluate the three correlation metrics on the CIFAR-10 dataset: (a) Kendall’s rank-correlation coefficient ($\tau$) [51] (b) granulated Kendall’s coefficient and their average ($\Psi$) [7] (c) conditional independence test ($\mathcal{K}$) [7]. For all correlation metrics, a larger value means a stronger correlation between sharpness and generalization.

We compare CS to following baseline sharpness measures: trace of Hessian ($\mathrm{tr}(\mathbf{H})$; Keskar et al. [19]), trace of empirical Fisher ($\mathrm{tr}(\mathbf{F})$; Jastrzebski et al. [52]), trace of empirical NTK at $\theta^{*}$ ($\mathrm{tr}(\mathbf{\Theta^{\theta^{*}}})$), Fisher-Rao (FR ;Liang et al. [18]) metric, Adaptive Sharpness (AS; Kwon et al. [15]), and four PAC-Bayes bound based measures: Sharpness-Orig. (SO), Pacbayes-Orig. (PO), Sharpness-Mag. (SM), and Pacbayes-Mag. (PM), which are eq. (52), (49), (62), (61) in Jiang et al. [7]. For computing granulated Kendall’s correlation, we use 5 hyper-parameters (network depth, network width, learning rate, weight decay, and mini-batch size) and 3 options for each (thus we train models with $3^{5}=243$ different training configurations). We vary the depth and width of NN based on VGG-13 [53]. We refer to Appendix H for experimental details.

In Table 2, CS shows the best results for $\tau$, $\Psi$, and $\mathcal{K}$ compared to all other sharpness measures. Also, granulated Kendall of CS is higher than other sharpness measures for 3 out of 5 hyper-parameters and competitive to other sharpness measures with the leftover hyper-parameters. The main difference of CS with other sharpness measures is in the correlation with network width and weight decay: For network width, we found that sharpness measures except CS, $\mathrm{tr}(\mathbf{F})$, AS and FR fail to capture strong correlation. While SO/PO can capture the correlation with weight decay, we believe it is due to the weight norm term of SO/PO. However, this term would interfere in capturing the sharpness-generalization correlation related to the number of parameters (i.e., width/depth), while CS/AS does not suffer from such a problem. Also, it is notable that FR fails to capture this correlation despite its invariant property.

To assess the effectiveness of CL as a general-purpose Bayesian NN, we consider uncertainty calibration on UCI dataset and CIFAR-10/100.

UCI regression datasets We implement full-curvature versions of LL and CL and evaluate these to the 9 UCI regression datasets [54] and its GAP-variants [49] to compare calibration performance on in-between uncertainty. We train MLP with a single hidden layer. We fix $\sigma=1$ and choose $\alpha$ from {0.01, 0.1, 1, 10, 100} using log-likelihood of validation dataset. We use 8 random seeds to compute the average and standard error of the test negative log-likelihoods. The following two tables show test NLL for LL/CL and 2 baselines (Deep Ensemble [55] and Monte-Carlo DropOut (MCDO; Gal and Ghahramani [56])). Eight ensemble members are used in Deep Ensemble, and 32 MC samples are used in LL, CL, and MCDO. Table 3 show that CL performs better than LL for 6 out of 9 datasets. Although LL shows better calibration results for 3 datasets in both settings, we would like to point out that the performance gaps between LL and CL were not severe as in the other 6 datasets, where CL performs better.

Image Classification We evaluate the uncertainty calibration performance of CL on CIFAR-10/100. As baseline methods, we consider Deterministic network, Monte-Carlo Dropout (MCDO; [56]), Monte-Carlo Batch Normalization (MCBN; [57]), and Deep Ensemble [55], Batch Ensemble [58], LL [25, 9]. We use Randomize-Then-Optimize (RTO) implementation of LL/CL in Appendix E. We measure Expected Calibration Error (ECE; Guo et al. [59]), negative log-likelihood (NLL), and Brier score (Brier.) for ensemble predictions. We also measure the area under receiver operating curve (AUC) for OOD detection, where we set the SVHN [60] dataset as an OOD dataset. For more details on the experimental setting, please refer to Appendix I.

Table 4 shows uncertainty calibration results on CIFAR-100. We refer to Appendix I for results on other settings, including CIFAR-10 and VGGNet [53]. Our CL shows better results than baselines for all uncertainty calibration metrics (NLL, ECE, Brier., and AUC) except Deep Ensemble. This means scale-invariance of CTK improves Bayesian inference, which is consistent with the results in toy examples. Although the Deep Ensemble presents the best results in 3 out of 4 metrics, it requires full training from initialization for each ensemble member, while LL/CL requires only a post-hoc training upon the pre-trained NN for each member. Particularly noteworthy is that CL presents competitive results with Deep Ensemble, even with much smaller computations.

Robustness to the selection of prior scale Figure 1 shows the uncertainty calibration (i.e. NLL, ECE, Brier) results over various $\alpha$ values for LL, CL, and Deterministic (Det.) baseline. As mentioned in previous works [27, 28], the uncertainty calibration results of LL is extremely sensitive to the selection of $\alpha$. Especially, LL shows severe under-fitting for large $\alpha$ (i.e. small damping) regime. On the other hand, CL shows stable performance in the various ranges of $\alpha$.