Robust uncertainty estimates with out-of-distribution pseudo-inputs training

Probabilistic models often use neural networks to control their predictive uncer-1 tainty . However , when making out-of-distribution ( OOD ) predictions , the often-2 uncontrollable extrapolation properties of neural networks yield poor uncertainty3 predictions . Such models then don ’ t know what they don ’ t know , which directly lim-4 its their robustness w.r.t unexpected inputs . To counter this , we propose to explicitly5 train the uncertainty predictor where we are not given data to make it reliable . As6 one can not train without data , we provide mechanisms for generating pseudo-inputs7 in informative low-density regions of the input space , and show how to leverage8 these in a practical Bayesian framework that casts a prior distribution over the9 model uncertainty . With a holistic evaluation , we demonstrate that this yields10 robust and interpretable predictions of uncertainty while retaining state-of-the-art11 performance on diverse tasks such as regression and generative modeling.12 1 Introduction13 Neural networks generally extrapolate arbitrarily [ Xu et al. , 2020 ] , and high quality predictions are14 limited to regions of the input space where the networks have been trained . This is to be expected and15 is only problematic if the associated predictions are not accompanied with a well-calibrated measure16 of uncertainty . If a neural network is used for estimating such a measure of uncertainty , we , however,17 quickly run into trouble , as the reported uncertainty then exhibits arbitrary behaviour in regions with18 no training data . Alarmingly , these are exactly the regions where evaluating the uncertainty is most19 important to the safe deployment of machine learning models in real world applications [ Amodei20 et al. , 2016 ] . One potential solution is to avoid using directly the output of neural networks for21 predicting uncertainty , and let it emerge from another mechanism , e.g . an ensemble [ Hansen and22 Salamon , 1990 , Lakshminarayanan et al. , 2017 ] or some notion of Monte Carlo [ MacKay , 1992 , Gal23 and Ghahramani , 2016 ] . Here we explore the alternative view that the networks should simply be24 trained where there is no data.25 Figure 1 : Pseudo-inputs are generated out of distribution , and there we train towards a prior ( grey density ) . But can we train without data ? The Bayesian formalism often26 does so implicitly : most conjugate priors can be seen as ad-27 ditional training data [ Bishop , 2006 ] , e.g . in Gaussian models,28 a mean prior N ( µ0 , 20 ) can be realised by additional training29 data of µ0 with 20 setting the amount of observations . Placing30 a prior over the output of a neural network can , thus , be inter-31 preted as additional training data . Unfortunately , this view is32 not practical as it implies additional data for all possible inputs33 to a neural network , resulting in infinite data . Our approach34 is simple : we locate regions of low data density in input space35 and implicitly place observations here in output space by min-36 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . imizing an appropriate KL divergence towards a prior ( see Fig . 1 ) . The result is a remarkably simple37 algorithm that drastically improves uncertainty estimates in both regression and generative modeling.38 1.1 Background and related work39 The predictive performance of machine learning models has drastically increased in the past decade,40 but the quality of the accompanying uncertainties have not followed . Uncertainties are reported as be-41 ing miscalibrated [ Guo et al. , 2017 ] and overconfident [ Lakshminarayanan et al. , 2017 , Hendrycks and42 Gimpel , 2016 ] . Some models even see higher likelihoods of out-of-distribution than in-distribution43 data [ Nalisnick et al. , 2019 , Nguyen et al. , 2015 , Louizos and Welling , 2017 ] .44 Neural networks commonly output distributions which gives a notion of predictive uncertainty . Clas-45 sifiers trained with soft-max is an ever-present example of such . These predictions are generally ob-46 served to be overconfident [ Lakshminarayanan et al. , 2017 , Hendrycks and Gimpel , 2016 ] and to carry47 little meaning outside the support of the training data [ Skafte et al. , 2019 , Lee et al. , 2017 ] . The latter48 is an artifact of the hard-to-control extrapolation that comes with neural networks [ Xu et al. , 2021 ] .49 In general , since extrapolation is difficult to control , uncertainties predicted by neural networks will50 exhibit seemingly arbitrary behavior outside the support of the data , yielding untrustworthy results.51 Mean-variance networks for regression [ Nix and Weigend , 1994 ] model the conditional target52 density as a normal p ( y|x ) = N y|µ ( x ) , ( x ) 2 with mean and variance predicted by neural53 networks . The reported predictive uncertainty is generally accurate in regions near training data,54 but otherwise unreliable [ Hauberg , 2019 ] . To counter this , Arvanitidis et al . [ 2017 ] and Skafte et al.55 [ 2019 ] proposed variance network architectures to enforce a specified extrapolation value , but these56 heuristics tend to be difficult to tune , and lack principle . Mean-variance networks have seen a recent57 uptake within generative modeling , where they are applied as an encoder distribution in variational58 autoencders ( VAEs ) [ Kingma and Welling , 2013 , Rezende et al. , 2014 ] .59 Which uncertainty ? A commonly called-upon dichotomy [ Der Kiureghian and Ditlevsen , 2009 ] 60 is that the uncertainty of a model ’ s prediction can be decomposed into the uncertainty of the model61 ( epistemic ) and of the data ( aleatoric ) . The epistemic uncertainty can be lowered by increasing62 the amount of data , simplifying the model or otherwise reducing the complexity of the learning63 problem . The aleatoric uncertainty , on the other hand , is a property of the world , and can not be64 changed ; no prediction should ever be more certain than the uncertainty displayed by the associated65 data . Depending on the task at hand , we may be interested in different types of uncertainty : In active66 learning [ Settles , 2012 ] and Bayesian optimization [ Močkus , 1975 ] we request data for which we67 have high epistemic , but low aleatoric uncertainty to ensure maximal information gain ; while for68 classification and regression we often just want to minimize the overall predictive uncertainty.69 Bayesian methods are often used to quantify uncertainty due to their explicit formulation of70 uncertainty . Gaussian processes ( GPs ) [ Rasmussen and Williams , 2005 ] provide an elegant71 framework that provide state-of-the-art uncertainty estimates , but , alas , the corresponding mean72 predictions are often not up to the standards of neural networks . GPs are tightly linked to Bayesian73 neural networks ( BNNs ) [ MacKay , 1992 ] that place a prior over the network weights and seek the74 corresponding posterior . Despite advances in variational approximations [ Graves , 2011 , Kingma75 and Welling , 2013 , Blundell et al. , 2015 ] , expectation propagation [ Hernández-Lobato and Adams,76 2015 , Hasenclever et al. , 2017 ] , or Monte Carlo methods [ Welling and Teh , 2011 , Springenberg et al.,77 2016 ] , training BNNs remains difficult . Furthermore , the predictive uncertainty seems dependent78 on the degree of approximation and is thus controlled by the available compute power.79 Ensemble methods have long been used to produce aggregated predictions with uncertainty estimates80 [ Hansen and Salamon , 1990 , Breiman , 1996 ] . Deep ensembles [ Lakshminarayanan et al. , 2017 ] ,81 a collection of differently initialized networks trained on the same data , are generally reported as82 state-of-the-art for uncertainty quantification in deep models [ Thagaard et al. , 2020 , Ovadia et al.,83 2019 ] . As the models in the ensemble are trained on overlapping data , they are correlated , which84 influence the ensemble uncertainty in ways that remains unclear [ Breiman , 2001 ] . Monte-Carlo85 dropout [ Gal and Ghahramani , 2016 ] casts dropout training [ Srivastava et al. , 2014 ] as an ensemble86 model . It is computationally cheap , but experiments [ Ovadia et al. , 2019 , Skafte et al. , 2019 ] show87 that the increased correlation of ensemble elements amplifies the method ’ s overconfidence.88 Robustness to distribution shift is paramount to a well-behaved uncertainty predictor [ Ovadia89 et al. , 2019 ] and must be evaluated accordingly . For out-of-distribution detection , Liang et al.90 [ 2017 ] proposes a pre-processing perturbation step inspired by adversarial attacks [ Goodfellow et al.,91 2014a ] that helps the model distinguish in-distribution and out-of-distribution inputs . Hendrycks92 et al . [ 2018 ] used a Generative Adversarial Network ( GAN ) [ Goodfellow et al. , 2014b ] to generate93 out-of-distribution pseudo-inputs whose inclusion in the training under an additional regularizing94 term in the loss function , called outlier exposure , enhances the predictor ’ s ability to discriminate95 out-of-distribution inputs [ Lee et al. , 2017 , Dai et al. , 2017 ] .96 1.2 Robust uncertainty estimates97 Our work is strongly inspired by the critical assessment of the issues that undermine variance estima-98 tion ran by Skafte et al . [ 2019 ] and by the proposal of Stirn and Knowles [ 2020 ] which we detail here.99 Notation . Let the observed variable x 2 X follow the data generating distribution pdata ( x ) , only100 known through the training dataset of N i.i.d samples Dtrain = { xn } Nn=1 . In the case of supervised101 learning , the observed variables x = ( x , y ) , with x 2 Rd being the input and y 2 Rd0 the target102 for the model , follow the joint decomposition pdata ( x , y ) = pdata ( y|x ) pdata ( x ) . The proposed103 probabilistic model p✓ ( x ) , whose weights are indicated by ✓ , aims to accurately emulate pdata ( x ) .104 Practical problems in variance estimation . Gaussian likelihoods in the form of p✓ ( x ) =105 N x|µ✓ ( x ) , ✓ ( x ) 2 are widely adopted to model continuous covariates . Real world data can not be106 expected to be homoscedastic , i.e constant throughout input space , and thus the predictive uncertainty,107 ✓ ( x ) , most often uses neural networks to map continuously the observed x onto the parameter space.108 Beyond the well-known unreliable extrapolation properties of neural networks , this parametrisation109 of predictive uncertainty is hamstringed by serious defects . Firstly , the predictive variance scales the110 learning rates of the mean and variance updates by 1/2 ✓ ( x ) 2 , resulting in a bias for data regions with111 low uncertainty [ Nix and Weigend , 1994 ] . Secondly , the maximisation of the modeled likelihood is112 particularly sensitive to scarce data , as local gradient updates for the variance point towards the then113 undefined maximum likelihood estimate ( MLE ) [ Skafte et al. , 2019 ] . Lastly , and more worryingly , such114 model ’ s likelihood is ill-defined [ Mattei and Frellsen , 2018a ] , as it can arbitrarily and without bound115 increase when the variance estimates collapse towards a detrimental 0 . Overall , the naive maximisa-116 tion of model likelihood seems insufficient to generate robust and well-behaved uncertainty estimates.117 Student-t likelihood . The Bayesian formalism , by imposing to learn a parametrised distribution118 over the predictive uncertainty , offers an attractive view to approaching the problem of uncertainty119 estimation . Skafte et al . [ 2019 ] notably adopts a Gamma distributed precision , 1/ 2 = ⇠ ( ↵ , ) ,120 as the conjugate of an unknown precision for a Gaussian likelihood , to yield a non-standard Student-t121 distributed marginal likelihood1 . It is known to offer a more robust likelihood , especially in the scarce122 data regime [ Gelman et al. , 2013 ] ,123 p✓ ( x ) = Z N ( x|µ , ) ( |↵ , ) d = T ⇣ x|⌫ = 2↵ , µ̂ = µ , ̂ = p /↵ ⌘ . ( 1 ) Interestingly , its variance Var [ x ] = / ( ↵ 1 ) = ( /↵ ) · ( ↵/ ( ↵ 1 ) ) can be explicitly decomposed124 to an aleatoric term /↵ and an epistemic term1 ↵/ ( ↵ 1 ) [ Jørgensen , 2020 , p16 ] , and offers a125 direct verification of whether a model knows what it knows.126 Variational variance . Stirn and Knowles [ 2020 ] assumes a latent model precision . This is127 generated by a prior p ( ) and its posterior is approximated variationally by the family of Gamma128 distributions , conditioned on the inputs to reflect heteroscedasticity . Through amortized variational129 inference ( AVI ) [ Kingma and Welling , 2013 ] neural networks f map to the posterior parameters from130 data , q ( z|f ( x ) ) . As such , variational variance preserves the modelling capacity and robustness of131 the non-standard Student-t marginal likelihood , without modifying its parameter architecture , while132 the definition of a prior over the latent precision induces a more robust training objective . Assuming133 the likelihood precision is the unique latent code , the evidence lower bound ( ELBO ) ,134 L ( q ; x ) = Eq ( ) [ log p ( x| ) ] DKL ( q ( |x ) || p ( ) ) = 1 2 ✓ ( ↵ ) log log ( 2⇡ ) ↵ ( x µ ) 2 ◆ DKL ( q ( |x ) || p ( ) ) , ( 2 ) takes the form of a regularised log-likelihood , exposing the benefits of the prior regularisation . It135 penalises predicted variances that would unrealistically get arbitrarily close to either the detrimental136 1See Section I. of the supplementary materials . limits of 0 or 1 , reducing the concerns regarding the ill-definition of the objective . Additionally,137 the scaling effect of the learning rates of the likelihood parameters is reduced . Naturally , the effect138 of the regularisation will be highly dependent on the prior selected . Here , because we are mostly139 interested in enforcing a constant desired uncertainty extrapolation , we adopt an homoscedastic140 Gamma distributed prior , p ( ) = ( |a , b ) , that matches the level of uncertainty observed in data,141 and leave it for future practitioners to adopt the most adequate prior for the task at hand.142 2 Out-of-distribution pseudo-inputs training143 2.1 Dissipative loss144 In the variational variance formalism , due to AVI , the predictive uncertainty is controlled by ↵ and ,145 the independent neural networks parametrising the posterior distribution , Var [ x ] = ( x ) / ( ↵ ( x ) 1 ) .146 The unreliable extrapolation properties of neural networks therefore directly challenge the robustness147 of the method ’ s uncertainty estimates outside of its training support , limiting the applicability of the148 method . We consider that this flawed extrapolation is not inevitable.149 Inspired by outlier exposure [ Hendrycks et al. , 2018 ] , we propose to include deliberately generated150 out-of-distribution pseudo-inputs , { x̂k } Kk=1 where x̂k ⇠ pout ( x ) , in the training of our variational151 objective to constrain the extrapolation of the posterior parametrisation . The optimal variational152 objective q⇤ is chosen such that it minimises our proposed dissipative loss over the consolidated153 dataset D = Dtrain [ Dout , where Dout = { x̂k } Kk=1,154 Loss ( q ; D ) = h Lin ( q ; Dtrain ) + Lout ( q ; Dout ) i . ( 3 ) The in-distribution component of the loss function Lin ( q ; D ) naturally arises as the standard ELBO155 over the training set . The out-of-distribution component Lout ( q ; D ) operates on a fundamentally156 different source of data . As the only information available regarding the pseudo-inputs is that they157 are out-of-distribution , we assert for them a constant , non-informative likelihood p ( x̂| ) = c , that158 has thus no influence on optimisation . This is similar to the strategy of censoring [ Lee and Wang,159 2003 ] where different likelihoods are used for observations with different properties . As a result , the160 dissipative loss becomes,161 Loss ( q ; D ) = h X x2Dtrain Eq ( |x ) [ p✓ ( x| ) ] DKL ( q ( |x ) || p ( ) ) X x̂2Dout DKL ( q ( |x̂ ) || p ( ) ) i . ( 4 ) It share the same motivating intuition as the confidence loss of Lee et al . [ 2017 ] and completes the162 variational variance formalism with a principled mechanism to learn robust variance estimates with163 the desired extrapolation properties . It indeed explicitly forces the predictor to match our high-entropy164 prior expectations on out-of-distribution samples while learning the low-entropy covariate dependent165 distribution , hence the name of dissipative . The reliance of the model ’ s predictive uncertainty166 on its mean predictions implies that it is primordial here to safeguard its generative performance.167 We guarantee it with the implementation of a split training procedure [ Skafte et al. , 2019 ] ; the168 out-of-distribution regularisation is only applied after the model ’ s mean has been trained.169 2.2 Pseudo-input generators ( PIGs ) 170 Minimising the posterior KL divergence out-of-distribution requires an efficient sampling procedure171 of pseudo-inputs . As exposed in Fig . 2 , their generation should leverage a-priori knowledge about172 pdata ( x ) to resolve the undefined nature of pout ( x ) . In this simple regression case , we show the173 predictive uncertainty of variational variance models trained on artificial heteroscedastic data . We174 use a prior uncertainty level that matches the maximum of the data uncertainty . As anticipated,175 without pseudo-inputs , the model extrapolates uncertainty to a constant , arbitrary level , and only the176 introduction of pseudo-inputs near the training data results in the desired uncertainty extrapolation.177 Reassuringly , this suggests that we do not need to regularise our model ’ s extrapolation in the entire178 out-of-distribution space . Instead , we can focus on the simpler task of generating pseudo-inputs179 in low-density regions of the input space that neighbours training data , as they can enforce correct180 extrapolation in the rest of the out-of-distribution space . Lee et al . [ 2017 ] gives supporting arguments.181 Recent contributions have relied on GANs for generating a useful representation of pout ( x ) [ Lee182 et al. , 2017 , Dai et al. , 2017 ] . Although conceptually intuitive , GANs incur a heavy computational183 burden and most likely induce serious practical challenges as a result of the instability of their training184 [ Shrivastava et al. , 2017 ] . Furthermore , as one need to understand what is in-distribution to model185 what it is not , we instead propose to directly leverage the information at hand about the data.186 Algorithm 1 : Pseudo-Input Generator ( PIG ) 8k 2 [ 1 , K ] , x̂k ⇠ pdata ( x ) . iterations = 0 . ✏ = 1 ; while ( iterations < max_iterations ) & ( ✏ > tolerance ) do compute 8k 2 [ 1 , K ] , rxp ( x ) ( x̂k ) ; ✏ = maxk2 [ 1 , K ] ( rxp ( x ) ( x̂k ) ) ; 8k 2 [ 1 , K ] , x̂k = x̂k rxp ( x ) ( x̂k ) ; iterations = iterations + 1 ; end Algorithm 1 gives a simple proce-187 dure for generating pseudo-inputs us-188 ing the data density . Pseudo-inputs189 are originally sampled from pdata ( x ) ,190 and their positions iteratively updated191 with gradient descent , with step size ,192 by following the directions that min-193 imise their likelihood under pdata ( x ) ,194 similarly to reversed adversarial steps195 [ Goodfellow et al. , 2014a ] .196 The procedure can run prior to training , in parallel for all x̂k with automatic differentiation , and197 thus results in limited additional complexity for the optimisation2 . It relies on the availability of198 a differentiable density estimate of the data , which is , depending on the use case , either directly199 available ( see Sec . 3.2 ) , or can be approximated through a variety of methods such as Bayesian200 Gaussian mixture models [ Bishop , 2006 ] , or various normalising flows [ Rezende and Mohamed,201 2015 ] based methods such as masked autoregressive flows [ Papamakarios et al. , 2017 ] ( see Sec . 3.1 ) .202 A caveat here is that depending on the PIG ’ s parameters , and on the quality of the density estimate203 available , pseudo-inputs might be generated in undesired regions of the input space , e.g uninformative204 density minima . In practice , we adopted conservative density estimates and parameters and did not205 observe any significant degradation of the predictive uncertainty due to the addition of pseudo-inputs.206 3 Experiments207 Holistic evaluation of uncertainty estimates . The ground truth for uncertainty estimates is usually208 unknown , making their evaluation non-trivial . Similarly as in Stirn and Knowles [ 2020 ] , we propose209 to assess them using a collection of metrics . Calibration , which evaluates probabilistic predictions210 w.r.t the long-run frequencies that actually occur [ Dawid , 1982 ] can be measured by proper scoring211 rules [ Lakshminarayanan et al. , 2017 ] such as the model log-likelihood log p✓ ( x| ) . Additionally,212 the root mean squared error ( RMSE ) between the predictive and empirical variance , Var [ x ] 213 ( Eq ( z|x ) [ p✓ ( x| ) ] x ) 2 , offers a quantification of the model ’ s awareness of its own uncertainty . It214 nevertheless requires an understanding of the model ’ s mean predictive performance , as commonly215 measured by the RMSE of the mean residuals , Eq ( |x ) [ p✓ ( x| ) ] x . We further propose to evaluate216 the cooperation of mean and uncertainty estimates for the generation of credible samples , which217 constitutes a consistency check for the learned precision distribution [ Gelman et al. , 2013 ] , by218 measuring the RMSE of sample residuals x⇤ x , with x⇤ ⇠ p✓ ( x ) . Finally , The ELBO , despite the219 absence of theoretical grounding for it [ Blei et al. , 2017 ] , is commonly reported as an approximation220 of the marginal likelihood , and thus of the overall model ’ s predictive performance.221 2Running times are reported in Sec . IV . of the supplementary materials . A complete assessment of a model ’ s uncertainty estimates further requires their evaluation under222 distributional shift [ Ovadia et al. , 2019 ] , which we either introduce voluntarily through deliberate223 splitting of the training and test sets , as in Sec . 3.1 , or by using test data from a different dataset224 altogether , as in Sec . 3.2.225 3.1 Regression226 In a regression setting where the proposed model must capture the conditioning between targets and227 inputs y|x , the precision of a Gaussian likelihood is the only assumed latent code.228 Faithfully to variational variance [ Stirn and Knowles , 2020 ] we229 adopt a Gamma heteroscedastic variational posterior q ( |x ) =230 ( |↵ ( x ) , ( x ) ) parametrised by the independent ↵ and 231 networks , with weights , uniquely conditioned on the inputs ( see232 Fig . 3 ) . This approximate posterior , independent of the targets , gives233 up on the dependency of the true posterior on both covariates to234 guarantee heteroscedasticity3.235 For strictly more than 2 degrees of freedom , or equivalently,236 ↵ ( x ) > 1 , the marginal predictive probability p✓ , ( y|x ) =237 T ⇣ y|2↵ ( x ) , µ✓ ( x ) , p ( x ) /↵ ( x ) ⌘ , has its first two moments238 defined , E [ y|x ] = µ✓ ( x ) and Var [ y|x ] = ( x ) / ( ↵ ( x ) 1 ) , pro-239 viding explicit mean and uncertainty estimates with a single forward240 pass in the single layered , fully connected , ↵ , and µ✓ networks used here . To ensure definition of241 both the posterior distribution and of the marginal distribution ’ s variance , the parameter maps use a242 soft-plus activation on their last layer to ensure positivity , and the ↵ network is further shifted by 1.243 The unique dependence of the posterior on the inputs implies that the generation of pseudo-inputs244 should only rely on the input density . In a general regression setting , it is unknown , and we estimate245 it here prior to training with a Bayesian Gaussian mixture model [ Bishop , 2006 ] . We refer to it246 henceforth as dissipative variational variance ( d-VV ) . The specific implementation details are listed247 in Section II . of the supplementary materials.248 3.1.1 Toy regression249 The desiderata for our method are clear : capture of the data heteroscedasticity , extrapolation to a250 higher uncertainty level , no underestimation of the predictive uncertainty , and posterior extrapolation251 to the prior out-of-distribution . Skafte et al . [ 2019 ] first showed on the toy regression task , y =252 x sin ( x ) +0.3 ✏1+0.3x ✏2 , where ✏1 , ✏2 ⇠ N ( 0 , 1 ) , that amongst a collection of methods , only their253 proposed variance network architecture could realise our first three expectations . Fig . 4 demonstrates254 that our more principled approach also fulfills all of our requirements , without the need for arbitrarily255 enforcing the desired extrapolation in our architecture . The importance of out-of-distribution training256 is also revealed as the standard variational variance approach fails to produce uncertainty estimates257 that extrapolate correctly and are robust to distributional shift ( bottom row of Fig . 4 ) .258 3See Section II . of the supplementary materials for the expression of the true posterior . Table 1 : UCI benchmarks . Each square shows the performance of a given model ( rows ) on a given dataset ( columns ) . The intensity of the colouring represents the certitude that the associated model performed best on the given dataset . Grey rows mean impossible evaluation for a metric . Table 2 : Evaluation of the generative modeling . For each dataset , we report mean ± std over 5 trials . FashionMNIST SVHN CIFAR log p ( x ) VAE 2215.54± 68.81 4304.90± 58.45 2930.64± 14.82 d-V3AE 2349.71± 11.80 4133.41± 64.28 2668.85± 13.23 RMSE ( x , x̃ ) VAE 0.171± 0.003 0.097± 7e-4 0.154± 5e-4d-V3AE 0.158± 0.003 0.087± 0.002 0.129± 7e-4 Decomposition of the model and data uncertainty . Fig . 5259 presents the decomposition of the predictive uncertainty . The260 aleatoric component captures the heteroscedastic increase of261 uncertainty in the training data while the epistemic uncertainty,262 constant in distribution , extrapolates to higher values . The263 proposed method therefore demonstrates , to the best of our264 knowledge , a principled decomposition of uncertainty factors.265 3.1.2 UCI Benchmarks266 Real world regression datasets from the UCI repository4 are used to evaluate our model against267 curated baselines , analogically to the setup from Hernández-Lobato and Adams [ 2015 ] and Skafte268 et al . [ 2019 ] 5 . As revealed by the summarising Tab . 16 , our method retains the mean predictive power269 of the variational variance method . The log-likelihood and RMSE of variance and sample residuals270 further show the improved calibration resulting from the imposition of a prior on the variance , as271 the VV methods generally outperform the MLE Student-t ( VV ( no prior ) ) that shares the same272 architecture . The table thus proves that holistically , the dissipative loss strengthens the variational273 variance model ’ s performance , which itself generally surpasses the chosen baselines.274 The robustness of the methods to distributional shift is further evaluated as in Foong et al . [ 2019 ] . For275 each input feature , a hole is created in the training data by assigning the middle third of observations276 to the test set , when sorted w.r.t that feature . Interestingly , we see that our method ’ s calibration277 slightly improves under the shift , highlighting the robustness benefits of the OOD prior regularisation.278 We note that both MC Dropout and the combined method of Skafte et al . [ 2019 ] generally perform279 well , confirming their interest for regression tasks requiring uncertainty quantification , but the280 former ’ s calibration is not robust to data shifts , as is also reported in Ovadia et al . [ 2019 ] , and the281 latter is in practice difficult to tune and lacks principle.282 4https : //archive.ics.uci.edu/ml/index.php 5See Sec . II . for details about the chosen baselines , datasets and implementations specifics . 6The full numbers are included in Sec . II . of the supplements . 3.2 Generative models283 We extend the evaluation of our proposal to the case of generative models through the lens of VAEs284 [ Kingma and Welling , 2013 , Rezende et al. , 2014 ] . Variational auto-encoders infer a low dimensional285 latent encoding of the data z 2 RD , on which is conditioned the generative process p✓ ( x|z ) . Its286 predictive uncertainty , which evaluates the confidence of the model in its ability to adequately287 reconstruct inputs is known to be untrustworthy.288 In the case of continuous or seemingly continuous in-289 puts , the adoption of a Gaussian decoder p✓ ( x|z ) =290 N x|µ✓ ( z ) , ✓ ( z ) 2 results in an ill-defined model likelihood291 [ Mattei and Frellsen , 2018a ] that encourages decoder variance292 collapse , making the training of the model notoriously harder293 [ Skafte et al. , 2019 ] . Most implementations therefore choose294 to fix the variance to a set level e.g ✓ ( z ) = 0.1 , or elude the295 challenge by adopting a Bernoulli likelihood.296 Motivated by our previous results , we now aim to demonstrate297 that VAEs , whose decoder is fitted with our method , are298 able to provide robust uncertainty estimates . Assuming a299 latent generative precision , the latent variables of the model300 are decomposed into z = { z , } , with z the latent input301 representations . The marginalisation of the Gamma distributed302 latent variance results in a Student-T decoder , as detailed in303 Eq . 1 . The overall architecture of the variational variance variational auto encoder ( V3AE ) [ Stirn304 and Knowles , 2020 ] is shown in Fig . 6 , and yields , with the addition of our out-of-distribution305 pseudo-inputs training , the dissipative loss function7,306 Loss ( q , ✓ ; Dtrain ) = h X x2Dtrain L ( q , ✓ ; x ) Eqout ( z ) [ DKL ( q ( |z ) || p ( ) ) ] i . ( 5 ) Because only the decoder is regularised , the pseudo-inputs lie in the space of latent representations,307 Dout = { ẑk } Kk=1 2 RD . The distribution of training inputs is therefore readily accessible as the308 aggregate variational posterior q ( z|Dtrain ) = q ( z|x1 ) · · · q ( z|xN ) . Here again , we rely on a split309 training procedure to leverage this perk ; the encoder parameter maps µ✓ and ✓ , as well as the310 decoder mean µ are first trained until convergence , allowing the generation of the out-of-distribution311 pseudo-inputs and subsequently , the training of the decoder variance.312 Image data . We evaluate the performance of our proposed dissipative313 V3AE ( d-V3AE ) against a fully Gaussian VAE on image data , coming from314 FashionMNIST , SVHN and CIFAR10 . For both models , all parameter315 maps share the same underlying architecture , with the addition of either316 a softplus and/or a shifting last layer to ensure definition of both the317 variational and the generative distribution ’ s moments8.318 Tab . 2 compares model performance on two metrics , the log-likelihood319 and the RMSE between the original inputs x and reconstructed samples x̃,320 where x̃ ⇠ p✓ ( x| , z ) , ( , z ) ⇠ q ( , z|x ) . Unlike most previous imple-321 mentations , we focus on actual samples , and not the mean , of the generative322 distributions . This comparison emphasize the cooperation between the323 decoder ’ s mean and variance , allowing evaluation of the models ’ uncer-324 tainty estimates . Our method both qualitatively ( Fig . 7 ) , and quantitatively325 improves on a Gaussian VAE ’ s sampling ability . The prior smoothens the326 uncertainty estimates , resulting in more realistic and less crisp samples.327 The log-likelihoods , evaluated at test time using truncation , i.e . ptrunc ( x ) =328 p✓ ( x ) / ( Fx ( 1 ) Fx ( 0 ) ) , to account for the finite support of data , reveal329 that our model can achieve a better fit , if the prior is selected correctly . In330 SVHN and CIFAR10 , the presence of color channels complicates the selection process and challenges331 our choice of a single homoscedastic prior for all pixels and channels . We note that the dissipative loss332 also applies to classic VAEs with Bernoulli-only decoders ; see Sec . III . of the supplements for details.333 7The derivation of the dissipative loss function is provided in Sec . III . of the supplementary materials . 8Implementation details are provided in Sec . III . of the supplementary materials . Applications of robust generative uncertainty . In Figs . 8 & 9 , the colouring of the 2D latent space334 represent the aggregated decoder variance Pdi=1 ( ✓ ( z ) 2 ) i . It is clear that our method displays more335 regular uncertainty estimates , and provides the extrapolation guarantees we strove for . Beyond in-336 creased robustness and better generative power , this unlocks meaningful out-of-distribution detection,337 beating previous state-of-the-art [ Havtorn et al. , 2021 ] . For Figs . 9 & 10 , as argued in Mattei and338 Frellsen [ 2018b ] , we refit at test time the encoder of models trained on FashionMNIST on MNIST.339 The regularity and structure of the decoder variance rewards the encoder for learning to place represen-340 tations of OOD data outside of the region of in-distribution latent encodings , resulting in a model that341 is aware of its own inability to reconstruct plausible data , as displayed by the row x̃refit of d-V3AE.342 4 Conclusion343 We have introduced a novel loss , the dissipative loss , that leverages artificial out-of-distribution344 pseudo-inputs for learning robust uncertainty estimates . We demonstrate through a Bayesian approach345 that casts a prior distribution over the model ’ s variance a principled mechanism for controlling the346 extrapolation properties of neural networks governing the predictive uncertainty . Our experimental347 results reflect the benefits of our principled and scalable approach , displaying better calibrated and348 more robust uncertainty estimates , while matching the predictive power of known baselines . Finally,349 and most interestingly , our approach can instill into probabilistic models a notion of their own350 ignorance , increasing their ability to know what they don ’ t know.351 Limitations . The largest limitation of our approach is that it depends on an estimate of the density of352 the input data . In our experience , even coarse-grained densities are sufficient to significantly improve353 upon current approaches . However , as one rarely have guaranteed good estimates of the input density,354 our method can not be approached as a black-box . One exception seems to be the application to VAEs,355 where the aggregated posterior , in our experience , always provide a suitable density estimate.356 Negative societal impact . 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Summary: This paper considers the idea of generating pseudo-inputs (i.e. extra augmented samples) to improve OOD uncertainty estimates by training with an uncertainty-increasing ("dissipative") loss at these pseudo-inputs. The pseudo-inputs are generated by minimizing estimated density under the data distribution via a simple gradient algorithm. The idea is evaluated in supervised learning (regression) and generative modeling (VAE for images).