E XPLORING THE U NCERTAINTY P ROPERTIES OF N EURAL N ETWORKS ’ I MPLICIT P RIORS IN THE I NFINITEW IDTH
L IMIT
Anonymous authors
Paper under double-blind review
A BSTRACT
Modern deep learning models have achieved great success in predictive accuracy for many data modalities.
However, their application to many real-world tasks is restricted by poor uncertainty estimates, such as overconﬁdence on out-ofdistribution (OOD) data and ungraceful failing under distributional shift.
Previous benchmarks have found that ensembles of neural networks (NNs) are typically the best calibrated models on OOD data.
Inspired by this, we leverage recent theoretical advances that characterize the function-space prior of an inﬁnitely-wide NN as a Gaussian process, termed the neural network Gaussian process (NNGP).
We use the NNGP with a softmax link function to build a probabilistic model for multi-class classiﬁcation and marginalize over the latent Gaussian outputs to sample from the posterior.
This gives us a better understanding of the implicit prior NNs place on function space and allows a direct comparison of the calibration of the NNGP and its ﬁnite-width analogue.
We also examine the calibration of previous approaches to classiﬁcation with the NNGP, which treat classiﬁcation problems as regression to the one-hot labels.
In this case the Bayesian posterior is exact, and we compare several heuristics to generate a categorical distribution over classes.
We ﬁnd these methods are well calibrated under distributional shift.
Finally, we consider an inﬁnite-width ﬁnal layer in conjunction with a pre-trained embedding.
This replicates the important practical use case of transfer learning and allows scaling to signiﬁcantly larger datasets.
As well as achieving competitive predictive accuracy, this approach is better calibrated than its ﬁnite width analogue.
I NTRODUCTION
Large, neural network (NN) based models have demonstrated remarkable predictive performance on test data drawn from the same distribution as their training data, but the demands of real world applications often require stringent levels of robustness to novel, changing, or shifted distributions.
Speciﬁcally, we might ask that our models are calibrated.
Aggregated over many predictions, well calibrated models report conﬁdences that are consistent with measured performance.
The Brier score (BS), expected calibration error (ECE), and negative log-likelihood (NLL) are common measurements of calibration (Brier, 1950; Naeini et al., 2015; Gneiting & Raftery, 2007).
Empirically, there are many concerning ﬁndings about the calibration of deep learning techniques, particularly on out-of-distribution (OOD) data whose distribution differs from that of the training data.
For example, MacKay (1992) showed that non-Bayesian NNs are overconﬁdent away from the training data and Hein et al.
(2019) conﬁrmed this theoretically and empirically for deep NNs that use ReLU.
For in-distribution data, post-hoc calibration techniques such as temperature scaling tuned on a validation set (Platt et al., 1999; Guo et al., 2017) often give excellent results; however, such methods have not been found to be robust on shifted data and indeed sometimes even reduce calibration on such data (Ovadia et al., 2019).
Thus ﬁnding ways to detect or build models that produce reliable probabilities when making predictions on OOD data is a key challenge.
Sometimes data can be only slightly OOD or can shift from the training distribution gradually over time.
This is called dataset shift (Quionero-Candela et al., 2009) and is important in practice for models dealing with seasonality effects, for example.
While perfect calibration under arbitrary distributional shift is impossible, simulating plausible kinds of distributional shift that may occur in practice at different intensities can be a useful tool for evaluating the calibration of existing models.
A recently proposed benchmark takes this approach (Ovadia et al., 2019).
For example, using several kinds of common image corruptions applied at various intensities, the authors observed the degradation in accuracy expected of models trained only on clean images (Hendrycks & Dietterich, 2019; Mu & Gilmer, 2019), but also saw very different levels of calibration, with deep ensembles (Lakshminarayanan et al., 2017) proving the best. Calibration metrics. There are many common metrics used to evaluate the calibration of a model.
Probably most common is the negative log-likelihood (NLL), which is a proper scoring rule and whose units, nats, are a coherent unit for information entropy. The Brier score (BS) from Brier (1950) is also a proper scoring rule, but its scale is harder to interpret and its use of the squared loss makes it sensitive to in/frequent events. We also consider the expected calibration error (ECE) of Naeini et al. (2015), which is not a proper scoring rule.
In particular, uniformly random predictions on a balanced, classiﬁcation dataset will produce an ECE of 0.
However, since it measures the absolute difference between predicted and observed probabilities, the scale of ECE is easier to interpret.
Finally, since the previous metrics require labels, we consider the average conﬁdence and entropy on OOD data, where the conﬁdence is the probability assigned to the most likely class and the entropy is the entropy of the predicted distribution over labels.
See (Ovadia et al., 2019) for more discussion of calibration metrics.
Bridging Bayesian Learning and Neural Networks. In principle, Bayesian methods provide a promising way to tackle calibration, allowing us to deﬁne models with and infer under speciﬁc aleatory and epistemic uncertainty.
Typically, the datasets on which deep learning has proven successful have high SNR (Signal-to-Noise Ratio), meaning epistemic uncertainty is dominant and model averaging is crucial because our overparameterized models are not determined by the training data.
Indeed Wilson (2020) argues that ensembles are a kind of Bayesian model average.
Ongoing theoretical work has built a bridge between NNs and Bayesian methods (Neal, 1994a; Lee et al., 2018; Matthews et al., 2018b), by identifying NNs as converging to Gaussian processes in the limit of very large width.
Speciﬁcally, the neural network Gaussian process (NNGP) describes the prior on function space that is realized by an i.i.d.
prior over the parameters.
The function space prior is a GP with a speciﬁc kernel that is deﬁned recursively with respect to the layers.
While the many heuristics used in training NNs may obfuscate the issue, little is known theoretically about the uncertainty properties implied by even basic architectures and initializations of NNs.
Indeed theoretically understanding overparameterized NNs is a major open problem.
With the NNGP prior in hand, it is possible to disambiguate between the uncertainty properties of the NN prior and those due to the speciﬁc optimization decisions by performing Bayesian inference.
Moreover, it is only in this inﬁnite-width limit that the posterior of a Bayesian NN can be computed exactly.
1. S UMMARY OF CONTRIBUTIONS
This work is the ﬁrst extensive evaluation of the uncertainty properties of inﬁnite-width NNs.
Unlike previous work, we construct a valid probabilistic model for classiﬁcation tasks using the NNGP, i.e. a label’s prediction is always a categorical distribution.
We perform neural network Gaussian process classiﬁcation (NNGP-C) using a softmax link function to exactly mirror NNs used in practice.
Note that prior work on the NNGP has also used a softmax link-function, but considered approximate inference using inducing points on MNIST (Garriga-Alonso et al., 2019).
We perform a detailed comparison of NNGP-C against its corresponding NN on clean, OOD, and shifted test data and ﬁnd NNGP-C to be signiﬁcantly better calibrated and more accurate than the NN.
Next, we evaluate the calibration of neural network Gaussian process regression (NNGP-R) on both UCI regression problems and classiﬁcation on CIFAR10.
As the posterior of NNGP-R is a multivariate normal and so not a categorical distribution, a heuristic must be used to calculate conﬁdences for classiﬁcation problems.
On the full benchmark of Ovadia et al.
(2019), we compare several such heuristics, and against the standard RBF kernel and ensemble methods.
We ﬁnd the calibration of NNGP-R to be competitive with the best results reported in (Ovadia et al., 2019).
Finally, we consider NNs whose last layer only is inﬁnite by taking a pre-trained embedding and using an inﬁnite-width ﬁnal layer (abbreviated as NNGP-LL).
This mirrors an important practical use case for practitioners unable to retrain large models from scratch, but looking to adapt one to their particular data, which could have relatively few samples.
We compare the calibration of NNGP-LL with a multi-layer FC network, ﬁne-tuning of the whole network, and the gold standard ensemble method, and again we ﬁnd NNGP-LL to have competitive calibration.
While NNGP-LL is not a principled Bayesian method, it allows scaling to much larger datasets and potential synergy with recent advances in un- and semi-supervised learning due to the pre-training of the embedding, making this method applicable to many real-world uncertainty critical applications like medical imaging.
B ACKGROUND
Neal (1994b) identiﬁed the connection between inﬁnite width NNs and Gaussian processes, showing that the outputs of a randomly initialized one-hidden layer NN converges to a Gaussian process as the number of units in the layer approaches inﬁnity.
Let z li ( x ) describe the i th pre-activation following a linear transformation in the l th layer of a NN.
At initialization, the parameters of the NN are independent and random, so the central-limit theorem can be used to show that the pre-activations become Gaussian with zero mean and a covariance matrix K ( x, x (cid:48) )
= E [ z li ( x ) z li ( x (cid:48) )] .
Knowing the distributions of the outputs, one can apply Bayes theorem to compute the posterior distribution for new observations, which we detail in Sec. 3 for classiﬁcation and Sec. 4 for regression.
Moreover, when the parameters evolve under gradient ﬂow to minimize a squared loss, the output distribution 2 remains a GP but with a different kernel called the neural tangent kernel (NTK)
(Jacot et al., 2018; Lee et al., 2019).
This allows us to derive the exact formula for ensemble training of inﬁnite width networks; see Sec A for more details.
While we analyzed uncertainty properties of predictive output variance of NTK in conjunction, we observed that results and trends follow that of the NNGP.
These observations have recently been signiﬁcantly generalized to include NNs with more than one hidden layer (Lee et al., 2018; Matthews et al., 2018b) and with a variety of architectures, including weight-sharing (CNNs) (Xiao et al., 2018;
Novak et al., 2019b; Garriga-Alonso et al., 2019), skipconnections, dropout (Schoenholz et al., 2017; Pretorius et al., 2019), batch norm (Yang et al., 2019), weight-tying (RNNs)(Yang, 2019), self-attention (Hron et al., 2020), graphical networks (Du et al., 2019), and others.
In this work, we focus on FC and CNN-Vec NNGPs, whose kernels are derived from fully-connected networks and convolution networks without pooling respectively (see Sec A for precise deﬁnitions).
When it is required, we prepend FC- or C- to the NNGP to distinguish between these two variants.
We use the Neural Tangents library of Novak et al. (2019a) to automate the transformation of ﬁnite-width NNs to their corresponding Gaussian processes.
Another line of work has connected NNs and GPs by combining them in a single model.
Hinton &
Salakhutdinov (2008) trained a deep generative model and fed its output in to a GP for discriminative tasks.
Wilson et al.
(2016b) consider a similar approach, creating a GP whose kernel is given as the composition of a nonlinear mapping ( e.g. a NN) and a spectral kernel.
They scale this approach to larger datasets in (Wilson et al., 2016a) using variational inference, but (Tran et al., 2018) found this method to be poorly calibrated.
F ULL B AYESIAN T REATMENT OF C LASSIFICATION WITH N EURAL K ERNELS
It is common to interpret the logits of a NN once mapped through a softmax as a categorical distribution over the labels for each point.
Indeed cross entropy loss is sometimes motivated as the KL divergence between the predicted distribution and the observed label.
Similarly, while the initialization scheme used for a NN’s parameters is often chosen for optimization reasons, it can also be thought of as a prior.
This implicit prior over functions and over the distribution of labels has effects, despite the decision of most common training algorithms in deep learning to forgo explicitly trying to ﬁnd its posterior.
In this section, we take seriously this implicit prior and utilize the simple characterization it has over logits in the inﬁnite width limit to deﬁne a probabilistic model for mutli-class classiﬁcation as
[Equation]
where K is the NNGP kernel.
In words, equation 1 states that the latent space is a Gaussian process with mean 0 and kernel function K and the softmax function is used to transform the latent space into a distribution over classes for each point x .
If this prior is a correct model for the data generation process, then the posterior is optimal for inference.
Thus, by avoiding heuristic approaches to inference, we are able to directly evaluate the statistical properties of the prior.
Then by comparing this to more standard gradient-based training methods, we can understand their effect on calibration.
For training data ( X , Y ) , the posterior on a test point x can be found by marginalizing out the latent space.
Denote the latent vector for the whole training data with F := f ( X ) and the test point with f := f ( x ) , and then let F denote their concatenation.
Then the posterior for the test label can be calculated by integrating over the posterior on the latent space of the test point:
[Equation]
Note that p ( f | x, X , Y ) is a marginal of the posterior distribution for the latent vectors on all points, i.e.
[Equation]
which, after neglecting the normalizing denominator, is proportional to the prior in the latent space and the probabilities under the softmax link-function, N ( F ; 0 , K ) (cid:81) i softmax( F i )
Y i . See Williams & Rasmussen (2006) for more details.
We generate samples from the joint posterior distribution of f and F using elliptical slice sampling (ESS) (Murray et al., 2010).
Note that the latent space dimension in the datasets we consider is substantial, which makes inference with ESS computationally intensive.
Speciﬁcally, each step of ESS requires sampling from the prior and thus a matrix multiplication of the Cholesky decomposition of the kernel matrix (whose running time is cubic in the dataset size but can be precomputed once), and the convergence times are longer for larger systems.
Moreover, while ESS is hyperparameter free, we found its convergence is affected by the kernel hyperparameters. Given these computational constraints on both tuning hyperparameters and running ESS, we focus our attention on FC and simple CNN-Vec kernels, which are much faster to compute that pooling based kernels.
3 Since we wish to make like-for-like comparisons between the Gaussian process and the corresponding NN, this restricts us to simplistic NN architectures that are far from state-of-the-art. We ﬁnd this acceptble as the goal of this section is relative comparison and not to obtain optimal absolute performance. To tune the kernel hyperparameters (weight and bias standard deviations, the activation function, the depth, and the kernel jitter)
we used the Google Vizier service (Golovin et al., 2017b) with a budget of 250 trials and selected the setting with the best log-likelihood on a validation set.
We use the same hyperparamters for the NN to make a direct comparison of the prior.
The additional hyperparameters required for the NN, like width and training time, were also tuned using Vizier.
We also compared against the NN performance when all its hyperparameters are tuned, and found the accuracy of the NN improved but the calibration results were broadly similar.
See the supplement for additional details.
Note that while optimizing hyperparamters based on a validation set likelihood is nonstandard for Bayesian methods, we do so here to make the fairest comparison to NNs, whose hyperparamters are normally optimized in this way.
In our experiments, we compare NNGP-C and a ﬁnite NN of the same architecture on in-distribution, shifted, and OOD test data when both models are trained using the standard CIFAR10 training set.
We consider CIFAR100, which contains classes not present in CIFAR10, and SVHN, which consists of images of house numbers, as two examples of OOD data.
The ﬁve levels of fog corruptions from Hendrycks & Dietterich (2019), which distorts the CIFAR10 test set with a fog-like effect, are used as increasing amounts of distributional shift as in Ovadia et al.
Our main ﬁndings for NNGP-C, summarized in Fig.
1 and Table 1, show that NNGP-C is well calibrated and outperforms [Figure] = y than the CNN, which shifts toward increasing overconﬁdence.
We see particularly signiﬁcant improvements in ECE and NLL, implying that NNGP-C is considerably better calibrated.
For complete results on other intensities of Fog corruption see Table S
[Table] the corresponding NN. This indicates that, rather than the prior induced by the random initialization scheme, the MAP-based training of the NN is partly responsible for its poor calibration.
R EGRESSION WITH THE NNGP
As observed in Sec. 3, inference with NNGP-C is challenging as the posterior is intractable.
In this section, we consider Gaussian process regression using the NNGP (abbreviated as NNGP-R), which is deﬁned by the model
[Equation]
where [Equation] where K is the NNGP kernel and ε ∼ N (0 , σ 2 (cid:15) ) is an independent noise term.
One major advantage of NNGP-R is that the posterior is analytically tractable.
The posterior at a test point x has a Gaussian distribution with mean and variance given by
[Equation]
where ( X , Y ) is the training set of inputs and targets respectively and K (cid:15) ≡ K + σ 2 (cid:15) I .
Since K has a Kronecker factorization, the complexity of inference is O ( |X| 3 ) rather than O ( c 3 |X| 3 ) .
For regression problems where y ∈ R d , the variance describes the model’s uncertainty about the test point.
Note that while this avoids the difﬁculties of approximate inference methods like MCMC, computation of the kernel inverse means running time scales cubically with the dataset size.
4. B ENCHMARK ON UCI D ATASETS
We perform non-linear regression experiments proposed by Hernández-Lobato & Adams (2015), which is a standard benchmark for evaluating uncertainty of Bayesian NNs.
We use all datasets except for Protein and Year .
Each dataset is split into 20 train/test folds.
We report our result in Table 2, comparing against the following strong baselines: Probabilistic BackPropagation with the Matrix Variate Gaussian distribution (PBP-MV) (Sun et al., 2017), Monte-Carlo Dropout (Gal & Ghahramani, 2016) evaluated with hyperparameter tuning as done in Mukhoti et al.
and
Deep Ensembles (Lakshminarayanan et al., 2017).
We also evaluated a standard GP with RBF kernel
K RBF ( x, x (cid:48) )
=
β exp (cid:0) − γ || x − x (cid:48)
|| 2 (cid:1) for comparison.
Instead of maximizing train NLL for model selection, we performed hyperparameter search on a validation set (we further split the training set so that overall train/valid/test split is 80/10/10), as commonly done in NN model selection and in the BNN context applied in (Mukhoti et al., 2018).
For NNGP-R, the following hyperparameters were considered.
The activation function is chosen from (ReLU, Erf), number of hidden layers among [1, 4, 16], σ 2 w from [1, 2, 4], σ 2 b from [0., 0.09, 1.0], readout layer weight and bias variance are chosen either same as body or ( σ 2 w , σ 2 b )
=
For the GP with RBF kernel we evaluated γ ∈ { 10 k : k ∈ [ − 5 , − 4 , ..., 3] } and β ∈ { 10 k : k ∈ [ − 3 , − 4 , ..., 3] } .
For both experiments, σ ε ∈ np.logspace(-6, 4, 20) .
We found that NNGP-R can outperform and remain competitive with existing methods in terms of both root-mean-squared-error (RMSE) and negative-log-likelihood (NLL).
In Table 2, we observe that NNGP-R achieves the lowest RMSE on the majority (5/8) of the datasets and competitive NLL.
When using the NTK as the GP’s kernel instead, we saw broadly similar results. [Table] Average Negative [Table]
-Likelihood Test Performance
[Table]
4. C LASSIFICATION AS R EGRESSION
Formulating classiﬁcation as regression often leads to good results, despite being less principled (Rifkin et al., 2003; Rifkin & Klautau, 2004).
By doing so, we can compare exact inference for GPs to trained NNs on well-studied image classiﬁcation tasks.
Recently, various studies of inﬁnite NNs have considered classiﬁcation as regression tasks, treating the one-hot labels as independent regression targets (e.g. Lee et al.
; Novak et al. (2019b); Garriga-Alonso et al.
Predictions are then obtained as the argmax of the mean in Eq.
5, i.e. arg max k µ ( x ) k .
However, this approach does not provide conﬁdences corresponding to the predictions.
Note that the posterior gives support to all of R d , including points that are known to be impossible.
Thus, a heuristic is required to extract meaningful uncertainty estimates from the posterior Eq.
5, even though these conﬁdences will not correspond to the Bayesian posterior of any model.
Following (Albert & Chib, 1993; Girolami & Rogers, 2006), we produce a categorical distribution for each test point x , denoted p x , by deﬁning
[Equation]
where ( y 1 , . . .
, y d ) is sampled from the posterior for ( x, y ) and we used the independence of the posterior for each class.
6 Note that we also treat the predictions on different test points independently.
In general, Eq. 6 does not have an analytic expression, and we resort to Monte-Carlo estimation.
We refer readers to the supplement for comparison to other heuristics (e.g. passing the mean predictor through a softmax function and pairwise comparison).
While this is heuristic, we ﬁnd it is well calibrated (see Fig. 2).
This is perhaps because the posterior Eq.
5 still represents substantial model averaging, and most uncertainty in high SNR cases is epistemic rather than aleatory.
4. B ENCHMARK ON CIFAR
We examine the calibration of NNGP-R on increasingly corrupted images of CIFAR10-C (Hendrycks & Dietterich, 2019) using the benchmark of (Ovadia et al., 2019).
The results are displayed in Fig.
While FC-NNGP is similar to the standard RBF kernel, the C-NNGP outperforms both in terms of calibration and accuracy.
Moreover, we ﬁnd that at severe corruption levels, the C-NNGP actually outperforms all methods in (Ovadia et al., 2019) (compare against their Table G1) in BS and ECE. [Figure]
B AYESIAN OR I NFINITE -W IDTH L AST L AYER
As we have seen NNGP-C and NNGP-R are remarkably well-calibrated.
However, obtaining high performing models can be computationally intensive, especially for large datasets.
NNGP-C
and
NNGP-R have running times that are cubic in the dataset size, due to computation of the kernel’s
Cholesky decomposition, with NNGP-C suffering additionally from potentially slow convergence of
MCMC.
Moreover, performant NNGP kernels require substantial compute to obtain (Novak et al., 2019b; Arora et al., 2019;
Novak et al., 2019a; Li et al., 2019) in contrast to training a NN to similar accuracies.
Moreover, even though the most performant NNGP kernels are SotA for a kernel method (Shankar et al., 2020), they still under-perform SotA NNs by a large margin.
To combine the beneﬁts of the NNGP and NNs, obtaining models that are both performant and well calibrated, we propose stacking an inﬁnite-width sub-network on top of a pre-trained NN.
More precisely, we use features obtained from a pre-trained model as inputs for the NNGP.
As such, the outputs of the combined network are drawn from a GP and we may use Eqs.
5 and 6 for inference.
We refer to this as neural network Gaussian process last-layer (NNGP-LL).
Mathematically, the model is
[Equation]
where g is a pre-trained embedding, K is the NNGP kernel, and ε ∼ N (0 , σ
[Equation]
) is a noise term. For the implementation of this method, we imagine that g is an input to the algorithm.
The computation of the kernel requires precomputing the embeddings of all data points, which are then considered as inputs to the NNGP-R algorithm, resulting in a running time that is cubic in the training set size (see Sec. E). The main advantage is that the embedding g allows for great expressive power which can be coupled with computationally cheaper kernels.
This method draws inspiration from (Hinton & Salakhutdinov, 2008; Wilson et al., 2016b; Bradshaw et al., 2017), but with a multi-layer NNGP kernel.
Note we are speciﬁcally interested in the calibration properties and so the innovations in computational efﬁciency in (Wilson et al., 2016b) are complementary to our work.
This setup mirrors an important use case in practice, e.g. for customers of cloud ML services, who may use embeddings trained on vast amounts of non-domain-speciﬁc data, and then ﬁne-tune this model to their speciﬁc use case.
This ﬁne tuning consists of either ﬁtting a logisitic regression layer or deeper NNs using the embeddings obtained from their data, or perhaps training the whole NN generating the embedding by simply initializing with the pre-trained weights (Kornblith et al., 2019).
These strategies allow practitioners to obtain highly accurate models without substantial data or computation.
However, little is understood about the calibration of these transfer learning approaches.
We consider the EfﬁcientNet-B3 (Tan & Le, 2019) embedding from TF-Hub 7 and TF Keras Applictions 8 that is trained on ImageNet (Deng et al., 2009), and perform our evaluations on CIFAR and its corruptions.
For our experiments, we mainly use a multi-layer FC-NNGP as the top subnetwork since its kernel is very fast to compute and the ﬁnal FC layer of EfﬁcientNet-B3 removes any spatial structure that might be exploited by convolutions.
However, we also explore other kernel types (linear, 9 CNNs with pooling and self-attention layers) in Fig.
3 by using earlier layers of EfﬁcientNet-B3.
We compare our method with other popular last-layer methods for generating uncertainty estimates (Vanilla logisitic regression, using a deep NN for the last layers, temperature scaling Platt et al.
; Guo et al.
, MC-Dropout (Gal & Ghahramani, 2016), ensembles of several last-layer deep NNs).
In the supplement we further investigate these results with a WideResNet (Zagoruyko & Komodakis, 2016) that we can train from scratch, using the initialization method in (Dauphin &
Schoenholz, 2019), which achieves good test performance on CIFAR-10 without BatchNorm (Ioffe & Szegedy, 2015).
This allows us to compare against the gold standard ensemble method.
We ﬁnd that in the transfer learning case, ensembles are quite ineffective alone.
The best previous method is given by combining MC-Dropout, temperature scaling, and ensembles.
However, this is still bested by NNGP-LL (see Table 3).
We also examine the effect of the ﬁne-tuning dataset size on calibration and performance.
Remarkably, NNGP-LL is able to achieve accuracy and calibration comparable to ensembles with as few as 1000 training points.
Valid Test    5 Corruption Intensity 0.
A cc u r a cy
Valid Test    5 Corruption Intensity 0.
E C E
Method LINEAR-LLFC-NNGP-LL
C-NNGP-LLSELF-ATTN-NNGP-LL
[Figure]
Ens/Drp/T RBF-LL 1K 5K 10K NNGP-LL
[Table]
D ISCUSSION
In this work, we explored several methods that exploit neural networks’ implicit priors over functions in order to generate uncertainty estimates, using the corresponding Neural Network Gaussian Process (NNGP) as a means to harness the full distribution over function-space in the inﬁnite-width limit.
Using the NNGP, we performed fully Bayesian classiﬁcation (NNGP-C) and regression (NNGP-R) and also examined heuristics for generating conﬁdence estimates when classifying via regression.
Across the board, we found that the NNGP provides good uncertainty estimates and generally delivers well-calibrated models even on OOD data.
We found that NNGP-R is competitive with SOTA methods on the UCI regression task and remained calibrated even for severe levels of corruption.
Despite their good calibration properties, as pure kernel methods, NNGP-C and NNGP-R cannot always compete with modern NNs in terms of accuracy.
Adding an NNGP to the last-layer of a pre-trained model (NNGP-LL), allowed us to simultaneously obtain high accuracy and improved calibration.
Moreover, we found NNGP-LL to be a simple and efﬁcient way to generate uncertainty estimates with potentially very little data, and that it outperforms all other last-layer methods for generating uncertainties we studied.
Overall, we believe that the inﬁnite-width limit provides a promising direction to improve and better understand uncertainty estimates for NNs.
Supplementary Material A D ETAILED D ESCRIPTION OF THE NNGP AND NTK
In this section, we describe the FC-NNGP and the C-NNGP.
Most of the contents are adopted from
Lee et al.
; Novak et al.
(2019b)
; Lee et al.
(2019), which we refer readers to for more technical details.
NNGP:
Let D ⊆ R n 0 × R K denote the training set and X = { x : ( x, y ) ∈ D} and Y =
{ y : ( x, y ) ∈ D} denote the inputs and labels, respectively.
Consider a fully-connected feed-forward network with L hidden layers with widths n l , for l = 1 , ..., L and a readout layer with n L +1 = K .
For each x ∈ R n 0 , we use h l ( x ) , x l ( x ) ∈ R n l to represent the pre- and post-activation functions at layer l with input x .
The recurrence relation for a feed-forward network is deﬁned as
[Equation]
and
[Equation]
where φ is a point-wise activation function, W l +1 ∈ R n l × n l +1 and b l +1 ∈ R n l +1 are the weights and biases, ω lij and b lj are the trainable variables, drawn i.i.d.
from a standard Gaussian ω lij , β lj ∼ N (0 , 1) at initialization, and σ 2 ω and σ 2 b are weight and bias variances.
As the width of the hidden layers approaches inﬁnity, the Central Limit Theorem (CLT) implies that the outputs at initialization { f ( x ) } x ∈X converge to a multivariate Gaussian in distribution.
Informally, this occurs because the pre-activations at each layer are a sum of Gaussian random variables (the weights and bias), and thus become a Gaussian random variable themselves.
See Poole et al.
Schoenholz et al.
; Lee et al.
Xiao et al.
; Yang & Schoenholz (2017) for more details, and Matthews et al.
(2018a);
Novak et al.
(2019b)
for a formal treatment.
Therefore, randomly initialized neural networks are in correspondence with a certain class of GPs (hereinafter referred to as NNGPs), which facilitates a fully Bayesian treatment of neural networks (Lee et al., 2018; Matthews et al., 2018b).
More precisely, let f i denote the i -th output dimension and
K denote the sample-to-sample kernel function (of the pre-activation) of the outputs in the inﬁnite width setting,
[Equation]
then f ( X ) ∼ N (0 , K
( X , X )) , where K i,j ( x, x (cid:48) ) denotes the covariance between the i -th output of x and j
-th output of x (cid:48) , which can be computed recursively (see Lee et al.
For a test input x ∈ X T , the joint output distribution f
([ x, X ]) is also multivariate Gaussian.
Conditioning on the training samples, f ( X )
=
Y , the distribution of f ( x ) | X , Y is also a Gaussian N (cid:0) µ ( x ) , σ 2 ( x ) (cid:1) ,
[Equation]
and where K = K ( X , X ) .
This is the posterior predictive distribution resulting from exact Bayesian inference in an inﬁnitely-wide neural network.
C-NNGP:
The above arguments can be extended to convolutional architectures Novak et al.
(2019b).
By taking the number of channels in the hidden layers to inﬁnity simultaneously, the outputs of
CNNs also converge weakly to a Gaussian process (C-NNGP).
The kernel of the C-NNGP takes into account the correlation between pixels in different spatial locations and can also be computed exactly via a recursively formula; e.g. , see Eq.
(7) in (Novak et al., 2019b, §2.2).
Note that for convolutional architectures, there are two canonical ways of collapsing image-shaped data into logits.
One is to vectorlize the image to a one-dimensional vector (CNN-Vec) and the other is to apply a global average pooling to the spatial dimensions (CNN-GAP).
The kernels induced by these two approaches are very different and so are the C-NNGPs.
We refer the readers to Section 3.2 of (Novak et al., 2019b, §3) for more details.
In this paper, we have focused mostly on vectorization since it is more efﬁcient to compute.
ATTN-NNGP:
There is also a correspondence between self-attention mechanisms and GPs.
Indeed, for multi-head attention architectures, as the number of heads and the number of features tend to inﬁnity, the outputs of an attention model also converge to a GP (Hron et al., 2020).
We refer the readers to Hron et al.
(2020) for technical details.
NTK: When neural networks are optimized using continuous gradient descent with learning rate η on mean squared error (MSE) loss, the function evaluated on training points evolves as
[Equation]
where J t ( X ) is the Jacobian of the output f t evaluated at X and Θ t ( X , X )
=
J t ( X ) J t
( X ) T is the
Neural Tangent Kernel (NTK).
In the inﬁnite-width limit, the NTK remains constant ( Θ t = Θ ) throughout training (Jacot et al., 2018).
Thus the above equation is reduced to a constant coefﬁcient
ODE
[Equation]
and the time-evolution of the outputs of unseen input X T can be solved in closed form as a Gaussian with mean and covariance
[Equation]
Note that the randomness of the solution is the consequence of the random initialization f ( X ) ∼
N (0 , K
( X , X )) .
Finally, the above arguments do not rely on the choice of architectures and we could likewisely deﬁne
CNTK and ATTN-NTK, the NTK for CNNs and attention models, respectively.
B A DDITIONAL F IGURES FOR NNGP-C
In this section, we show some additional plots and results comparing NNGP-C against standard NNs.
Mainly, we address the method of hyperparameter tuning considered in the main text, where we ﬁxed the hyperparameters that are common to both the NNGP and the NN, then only tuned the additional
NN hyperparameters.
Here, we show results for tuning all of the NN’s hyperparameter from scratch.
Additional Tuning details.
For any tuning of hyperparameters, we split the original training set of CIFAR10 into a 45K training set and a 5K validation set.
All models were trained using the 45K points, and we then selected the hyperparameters from the validation set performance.
We introduced a constant that multiples the whole NNGP kernel, or equivalently scales the whole latent space vector or the last layer bias and weight standard deviations—we called this constant the kernel scale.
For
FC-NNGP-C, the activation function was tuned over { ReLU , erf } , the weight standard deviation was tuned over [0 . 1 , 2 . 0] on a linear scale, the bias standard deviation was tuned over [0 . 1 , 0 . 5] on a linear scale, the kernel scale was tuned over [10 − 2 , 100] on a logarithmic scale, the depth was tuned over { 1 , 2 , 3 , 4 , 5 } , and the diagonal regularizer was tuned over [0 ., 0 .
01] on a linear scale.
For the FC-NN, there are additional hyperparameters: the learning rate, training steps, and width.
For the NN, we considered two types of tuning.
Either, as in the main text, the hyperparameters that are shared with the NNGP are ﬁxed and the additional hyperparameters are tuned, or, as we present in the supplement, all of the NN’s hyperparameters are tuned from scratch.
In either case, the activation, the weight standard deviation, the bias standard deviation, the kernel scale, and the depth were tuned as above.
The learning rate was tuned over [10 − 4 , 0 . 1] on a logarithmic scale, the total training steps was tuned over [10 5 , 10 7 ] on a logarithmic, and the width was tuned over { 64 , 128 , 256 , 512 } .
For the C-NNGP-C, all hypermaramters were treated as for the FC-NNGP-C case, except depth which was limited to { 1 , 2 } .
For the CNN, we again considered the two types of tuning: either ﬁxing common hyperparameters or retuning all hyperparameters from scratch.
The CNN’s learning rate was tuned over [10 − 4 , 0 . 1] on a logarithmic scale, the total training steps was tuned over [2 17 , 2 22 ] on a logarthimic scale, and the width was tuned over { 64 , 128 , 256 , 512 } .
[Figure]
[Figure]
[Table]
C C OMPARISON OF H EURISTICS FOR G ENERATING C ONFIDENCES FROM NNGP-R
In Secs.
4 and 5, we utilized a heuristic to generate conﬁdence from exact GPR posterior distribution.
Here we denote the heuristic described in Eq.
(6) as exact conﬁdence, which is the probability of a class probit being maximal under an independent multivariate Gaussian distribution.
We consider two more heuristics.
One is denoted pairwise , where we take conﬁdence to be proportional to the probability that the i -th class probit is larger than other probits in pairwise fashion, i.e.
[Equation]
where Φ( · ) is Gaussian cumulative distribution function.
In order to obtain conﬁdence, we normalize by the sum so that the heuristic conﬁdence sums up to 1 .
This is following the one-vs-one multiclass classiﬁcation strategy Hastie & Tibshirani (1998).
We note that, we introduce temperature scaling with temperature T by replacing posterior variances as
[Equation]
Another heuristic is denoted softmax , where we apply the softmax function to the posterior mean:
[Equation]
In this case, the posterior variance is not used to construct the heuristic conﬁdences.
A comparison for these three-different heuristics for C-NNGP-R is shown in Fig.
S4 with and without temperature scaling.
We note that exact and pairwise heuristics remain well calibrated without temperature scaling.
However with temperature scaling the softmax heuristic can be competitive to other heuristics.
In Sec. 4 and 5, we focused on the exact heuristic [Table] [Table] Valid
Test
Corruption Intensity 0.
[Figure]
cy
Method
RBF FC
[Figure]
NNGP [Figure] FC [Figure]
NTK
[Figure]
R
C
[Figure]
NNGP
[Figure]
NTK-R
Valid
Test
Corruption Intensity 0.
E C E
Valid
Test
Corruption Intensity 0.
[Figure]
[Figure]
[Figure]
r
[Figure]
Valid
Test
Corruption Intensity
[Figure]
[Figure]
hood [Figure] D A DDITIONAL F IGURES FOR NNGP-LL The results in the main text focused on a ﬁxed embedding (see Table 3) and we show additional results for this as a box-plot in Fig.
S5 here.
However, it is also common in practice to tune all of the embedding’s weights and simply initialize at their pre-triained values.
We explore this setting in the supplement in Table S3 and Fig.
S6 by considering the EfﬁcientNet-B3 embedding and ﬁne tuning it on CIFAR10.
We also show a results comparing the FC-NNGP-LL with using the standard RBF kernel for the same purpose (see Fig. S7)
[Figure]
D. A COMPARISON OF E NSEMBLE AND NNGP-LL ON W IDE R ESNET
To compare the NNGP-LL method against the gold standard ensemble method [Figure] we train a WideResnet 28-10 on CIFAR-10 from scratch with 5 different random initialization.
The model is trained using
MetaInit (Dauphin & Schoenholz, 2019), Delta-Orthogonal (Xiao et al., 2018), mixup (Zhang et al., 2017) and without BatchNorm (Ioffe & Szegedy, 2015).
The model achieves about 94% accuracy on the clean test set.
See Table S4 and Fig.
S8 for the comparison.
We ﬁnd the ECE for NNGP-LL is very competitive, even with small dataset size, compared to baseline methods including ensembles. E C OMPLEXITY OF NNGP-LL
There are two main steps in applying NNGP-LL.
One is computing the NNGP kernel K and the other is inference.
For NNGP-LL, we note that the computational of the kernel has the added cost of inference using the pretrained NN on the whole dataset, which adds a cost that is linear in dataset size but with a potentially large constant factor. For inference, we need to invert the whole kernel matrix of size c |X|× c |X| , the computational complexity is usually O ( c 3 |X| 3 ) where c is the number of classes and |X| is the training set size. However, since K has a Kronecker factorization, the complexity is reduced to O ( |X| 3 ) . [Table]
0.947 Accuracy (50th) 0.
0.896 Accuracy (25th) 0.
Valid
Test    5 Corruption Intensity 0. [Figure]
[Figure]
LL ENS
[Figure]
ENS DRP
[Figure]
NNGP
[Figure]
FT
[Figure]
NN
[Figure]
NNGP
[Figure]
The complexity of computing the NNGP kernels is architecture dependent. For notational simplicity, assume the cardinality of the training and test sets are m and n .
[Figure]
LL
[Figure]
NNGP
[Figure] [Figure]
E. FCN
The FCN kernel is relatively cheap to compute since it depends only on the dot product between inputs. It takes O ( L ( mn + n 2 )) time, where L is the number of layers of the FCN. In practice, the whole CIFAR-10 kernel could be otained in minutes. E. CNN-VEC
The CNN-VEC kernel is slightly more expensive to compute.
Each entry K ( x, x (cid:48) ) is a function of {(cid:104) x a , x (cid:48) a (cid:105) : a ∈ [ d ] } where d is the total number of pixels in one image (e.g. d = 32 ∗ 32 for
CIFAR-10).
Computing K ( x, x (cid:48) ) takes O ( Ld ) time and the whole kernel takes O (( mn + n 2 ) Ld ) time.
E. CNN-GAP AND A TTENTION
CNN-GAP and Attention kernels are very expensive to compute. Each entry K ( x, x (cid:48) ) is a function of {(cid:104) x a , x (cid:48) b (cid:105) : a, b ∈ [ d ] } . The complexity for one entry is O ( d 2 L ) and O (( mn + n 2 ) d 2 L for the whole kernels. For a 10 layers Myrtle kernel, it takes about 400 GPU hours (using NVIDIA Tesla V100) to compute the upper triangle part for the whole CIFAR10 (where d = 32 2 ).
However, for the last layer
[Table]
Valid Test
[Table] [Figure]
method, the images are down-sampled to smaller 8 × 8 image-patches. The whole kernel could be obtained in about 400 / (4 4 ) ∼ 1 . 6 GPU hours
[Figure]
While FCN or CNN-Vec NNGP performance is not as competitive as modern CNNs on CIFAR classiﬁcation tasks, recently Shankar et al.
(2020) has shown that the Myrtle architecture along with ZCA regularized preprocessing can obtain highly performant kernels.
Lee et al.
(2020) has shown that this is generally true for pooling kernels. Here we evaluate uncertainty properties of Myrtle-10 NNGP. Note that this kernel is extremely compute intensive (see SM E) and evaluation on the full CIFAR10 and CIFAR10-C requires 12,
GPU hours (using NVIDIA Tesla V100s).
We observe that NNGP-R with the Myrtle-10 kernel is the least robust under corruption when compared to other kernel architectures (Fig. S10).
However, comparing to the ﬁnite Myrtle network trained via gradient descent, we observe that NNGP-R method is as robust as the deepE ensemble method when average over 10 random seeds (Fig S11). While we consider these results preliminary and requiring further investigation, it suggests there may be a necessary tradeoff between the accuracy and calibration obtainable by NNGP-R.
[Figure]
-R compared with Myrtle-10 architecture (Shankar et al., 2020).
Here we compare NNGP-R in Fig.
2 to the pooling based Myrtle-10 kernel with ReLU activation. All methods except for the Myrtle architecture remain well calibrated for all intensities of shifts.
See Tab. S2 for detailed numbers.
[Figure] [Figure]