Deep Evidential Uncertainty

1 INTRODUCTION Recent advances in deep supervised learning have yielded super human level performance and precision . While these models empirically generalize well when placed into new test enviornments , they are often easily fooled by adversarial perturbations ( Goodfellow et al. , 2014 ) , and have difficulty understanding when their predictions should not be trusted . Today , regression based neural networks ( NNs ) are being deployed in safety critical domains of computer vision ( Godard et al. , 2017 ) as well as in robotics and control ( Bojarski et al. , 2016 ) where the ability to infer model uncertainty is crucial for eventual wide-scale adoption . Furthermore , precise uncertainty estimates are useful both for human interpretation of confidence and anomaly detection , and also for propagating these estimates to other autonomous components of a larger , connected system . Existing approaches to uncertainty estimation are roughly split into two categories : ( 1 ) learning aleatoric uncertainty ( uncertainty in the data ) and ( 2 ) epistemic uncertainty ( uncertainty in the prediction ) . While representations for aleatoric uncertainty can be learned directly from data , approaches for estimating epistemic uncertainty focus on placing probabilistic priors over the weights and sampling to obtain a measure of variance . In practice , many challenges arise with this approach , such as the computational expense of sampling during inference , how to pick an appropriate weight prior , or even how to learn such a representation given your prior . Instead , we formulate learning as an evidence acquisition process , where the model can acquire evidence during training in support of its prediction ( Sensoy et al. , 2018 ; Malinin & Gales , 2018 ) . Every training example adds support to a learned higher-order , evidential distribution . Sampling from this distribution yields instances of lower-order , likelihood functions from which the data was drawn ( cf . Fig . 1 ) . We demonstrate that , by placing priors over our likelihood function , we can learn a grounded representation of epistemic and aleatoric uncertainty without sampling during inference . In summary , this work makes the following contributions : 1 . A novel and scalable method for learning representations of epistemic and aleatoric uncertainty , specifically on regression problems , by placing evidential priors over the likelihood ; 2 . Formulation of a novel evidential regularizer for continuous regression problems , which we show is necessary for expressing lack of a evidence on out-of-distribution examples ; 3 . Evaluation of learned epistemic uncertainty on benchmark regression tasks and comparison against other state-of-the-art uncertainty estimation techniques for neural networks ; and 4 . Robustness evaluation against out of distribution and adversarially perturbed test data . 2 MODELLING UNCERTAINTIES FROM DATA . 2.1 PRELIMINARIES . Consider the following supervised optimization problem : given a dataset , D , of N paired training examples , ( x1 , y1 ) , . . . , ( xN , yN ) , we aim to learn a function f , parameterized by a set of weights , w , which approximately solves the following optimization problem : min w J ( w ) ; J ( w ) = 1 N N∑ i=1 Li ( w ) , ( 1 ) where Li ( · ) describes a loss function . In this work , we consider deterministic regression problems , which commonly optimize the sum of squared errors , Li ( w ) = 12 ‖yi − f ( xi ; w ) ‖ 2 . In doing so , the model is encouraged to learn the average correct answer for a given input , but does not explicitly model any underlying noise or uncertainty in the data when making its estimation . 2.2 MAXIMUM LIKELIHOOD ESTIMATION . We can also approach our optimization problem from a maximum likelihood perspective , where we learn model parameters that maximize the likelihood of observing a particular set of training data . In the context of deterministic regression , we assume our targets , yi , were drawn i.i.d . from a Gaussian distribution with mean and variance parameters θ = ( µ , σ2 ) . In maximum likelihood estimation , we aim to learn a model to infer θ = ( µ , σ2 ) that maximize the likelihood of observing our targets , y , given by p ( yi|θ ) . In practice , we minimize the negative log likelihood by setting : Li ( w ) = − log p ( yi|µ , σ2︸ ︷︷ ︸ θ ) = 1 2 log ( 2πσ2 ) + ( yi − µ ) 2 2σ2 . ( 2 ) In learning the parameters θ , this likelihood function allows us to successfully model the uncertainty of our data , also known as the aleatoric uncertainty . However , our model remains oblivious to the predictive model or epistemic uncertainty ( Kendall & Gal , 2017 ) . In this paper , we present a novel approach for estimating the evidence in support of network predictions by directly learning both the inferred aleatoric uncertainty as well as the underlying epistemic uncertainty over its predictions . We achieve this by placing higher-order prior distributions over the learned parameters governing the distribution from which our observations are drawn . 3 EVIDENTIAL UNCERTAINTY FOR REGRESSION . 3.1 PROBLEM SETUP . We consider the problem where our observed targets , yi , are drawn i.i.d . from a Gaussian distribution now with unknown mean and variance ( µ , σ2 ) , which we seek to probabilistically estimate . We model this by placing a conjugate prior distribution on ( µ , σ2 ) . If we assume our observations are drawn from a Gaussian , this leads to placing a Gaussian prior on our unknown mean and an Inverse-Gamma prior on our unknown variance : ( y1 , . . . , yN ) ∼ N ( µ , σ2 ) µ ∼ N ( γ , σ2λ−1 ) σ2 ∼ Γ−1 ( α , β ) . where Γ ( · ) is the gamma function , m = ( γ , λ , α , β ) , and γ ∈ R , λ > 0 , α > 0 , β > 0 . Our aim is to estimate a posterior distribution q ( µ , σ2 ) = p ( µ , σ2|y1 , . . . , yN ) . To obtain an approximation for the true posterior , we assume that the estimated distribution can be factorized ( Parisi , 1988 ) such that q ( µ , σ2 ) = q ( µ ) q ( σ2 ) . Thus , our approximation takes the form of the Gaussian conjugate prior , the Normal Inverse-Gamma ( N.I.G . ) distribution : p ( µ , σ2︸ ︷︷ ︸ θ | γ , λ , α , β︸ ︷︷ ︸ m ) = βα √ λ Γ ( α ) √ 2πσ2 ( 1 σ2 ) α+1 exp { −2β + λ ( γ − µ ) 2 2σ2 } . ( 3 ) A popular interpretation of the parameters of the conjugate prior distribution is in terms of “ virtualobservations ” in support of a given property ( Jordan , 2009 ) . For example , the mean of a N.I.G . distribution can be interpreted as being estimated from λ virtual-observations with sample mean γ while its variance was estimated from 2α virtual-observations with sample mean γ and sum of squared deviations 2β . Following from this interpretation , we define the total evidence , Φ , of our evidential distributions as the sum of all inferred virtual-observations counts : ( Φ = λ+ 2α ) . Drawing a sample θj from the N.I.G . distribution yields a single instance of our likelihood function , namely N ( µj , σ2j ) . Thus , the N.I.G . hyperparameters , ( γ , λ , α , β ) , determine not only the location but also the dispersion concentrations , or uncertainty , associated with our inferred likelihood function . Therefore , we can interpret the N.I.G . distribution as higher-order , evidential , distribution on top of the unknown lower-order likelihood distribution from which observations are drawn . For example , in Fig . 2A we visualize different evidential N.I.G . distributions with varying model parameters . We illustrate that by increasing the evidential parameters ( i.e . λ , α ) of this distribution , the p.d.f . becomes tightly concentrated about its inferred likelihood function . Considering a single parameter realization of this higher-order distribution , cf . Fig . 2B , we can subsequently sample many lower-order realizations of our likelihood function , as shown in Fig . 2C . In this work , we use neural networks to infer the hyperparameters of this higher-order , evidential distribution , given an input . This approach presents several distinct advantages compared to prior work . First , our method enables simultaneous learning of the desired regression task , along with aleatoric and epistemic uncertainty estimation , built in , by enforcing evidential priors . Second , since the evidential prior is a higher-order N.I.G . distribution , the maximum likelihood Gaussian can be computed analytically from the expected values of the ( µ , σ2 ) parameters , without the need for sampling . Third , we can effectively estimate the epistemic or model uncertainty associated with the network ’ s prediction by simply evaluating the variance of our inferred evidential distribution . 3.2 LEARNING THE EVIDENTIAL DISTRIBUTION . Having formalized the use of an evidential distribution to capture both aleatoric and epistemic uncertainty , we next describe our approach for learning a model ( c.f . Fig . 2D ) to output the hyperparameters of this distribution . For clarity , we will structure the learning objective into two distinct parts : ( 1 ) acquiring or maximizing model evidence in support of our observations and ( 2 ) minimizing evidence or inflating uncertainty when the prediction is wrong . At a high level , we can think of ( 1 ) as a way of fitting our data to the evidential model while ( 2 ) enforces a prior to inflate our uncertainty estimates . ( 1 ) Maximizing the model fit . From Bayesian probability theory , the “ model evidence ” , or marginal likelihood , is defined as the likelihood of an observation , yi , given the evidential distribution parameters m and is computed by marginalizing over the likelihood parameters θ : p ( yi|m ) = p ( yi|θ , m ) p ( θ|m ) p ( θ|yi , m ) = ∫ θ p ( yi|θ , m ) p ( θ|m ) dθ . ( 4 ) The model evidence is not , in general , straightforward to evaluate since computing it involves integrating out the dependence on latent model parameters : p ( yi|m ) = ∫ ∞ σ2=0 ∫ ∞ µ=−∞ p ( yi|µ , σ2 ) p ( µ , σ2|m ) dµdσ2 ( 5 ) However , by placing a N.I.G . evidential prior on our Gaussian likelihood function an analytical solution for the model evidence does exist . For computational reasons , we minimize the negative logarithm of the model evidence ( LNLLi ( w ) ) . For a complete derivation please refer to Sec . 7.1 , LNLLi ( w ) = − log p ( yi|m ) = − log ( 2 1 2+αβα √ λ 2π ( 1 + λ ) ( 2β + λ ( γ − yi ) 2 1 + λ ) − 12−α ) . ( 6 ) Instead of modeling this loss using empirical Bayes , where the objective is to maximize model evidence , we alternatively can minimize the sum-of-squared ( SOS ) errors , between the evidential prior and the data that would be sampled from the associated likelihood . Thus , we define LSOSi ( w ) as LSOSi ( w ) = Eθ′∼p ( θ|m ) [ Ey′∼p ( y|θ′ ) [ ||y′ − yi||22 ] ] ( 7 ) = ∫ ∞ σ2=0 ∫ ∞ µ=−∞ Ey′∼p ( y|µ , σ2 ) [ ||y′ − yi||22 ] p ( µ , σ2|m ) dµdσ2 ( 8 ) = ( Γ ( α− 12 ) 4 Γ ( α ) λ √ β ) ( 2β ( 1 + λ ) + ( 2α− 1 ) λ ( yi − γ ) 2 ) . ( 9 ) A step-by-step derivation is given in Sec . 7.1 . In our experiments , using LSOSi ( w ) resulted in greater training stability and increased performance , compared to the LNLLi ( w ) loss . Therefore , LSOSi ( w ) is used in all presented results . ( 2 ) Minimizing evidence on errors . In the first term of our objective above , we outlined a loss function for training a NN to output parameters of a N.I.G . distribution to fit our observations , either by maximizing the model evidence or minimizing the sum-of-squared errors . Now , we describe how to regularize training by applying a lack of evidence prior ( i.e. , maximum uncertainty ) . Therefore , during training we aim to minimize our evidence ( or maximize our uncertainty ) everywhere except where we have training data . This can be done by minimizing the KL-divergence between the inferred posterior , q ( θ ) , and a prior , p ( θ ) . This has been demonstrated with success in the categorical setting where the uncertainty prior can be set to a uniform Dirichlet ( Malinin & Gales , 2018 ; Sensoy et al. , 2018 ) . In the regression setting , the KL-divergence between our posterior and a N.I.G . zero evidence prior ( i.e. , { α , λ } = 0 ) is not well defined ( Soch & Allefeld , 2016 ) , please refer to Sec . 7.2 for a derivation . Furthermore , this prior needs to be enforced specifically where there is no support from the data . Past works in classification accomplish this by using the ground truth likelihoood classification ( i.e. , the one-hot encoded labels ) to remove the non-misleading evidence . However , in regression , labels are provided as point targets ( not ground truth Gaussian likelihoods ) . Unlike classification , it is not possible to penalize evidence everywhere except our single point estimate , as this space is infinite and unbounded . Thus , these previously explored approaches for evidential optimization are not directly applicable . To address both of these shortcomings of past works , now in the regression setting , we formulate a novel evidence regularizer , LRi , based on the error of the i-th prediction , LRi ( w ) = ‖yi − E [ µi ] ‖p · Φ = ‖yi − γ‖p · ( 2α+ λ ) , ( 10 ) where ‖x‖p represents the L-p norm of x . The value of p impacts the penalty imposed on the evidence when a wrong prediction is made . For example , p = 2 , heavily over-penalizes the evidence on larger errors , whereas p = 1 and p = 0.5 saturate the evidence penalty for larger errors . We found that p = 1 provided the optimal stability during training and use this value in all presented results . This regularization loss imposes a penalty whenever there is an error in the prediction that scales with the total evidence of our inferred posterior . Conversely , large amounts of predicted evidence will not be penalized as long as the prediction is close to the target observation . We provide an ablation analysis to quantitatively demonstrate the added value of this evidential regularizer in Sec 7.3.2 . The combined loss function employed during training consists of the two loss terms for maximizing model evidence and regularizing evidence , Li ( w ) = LSOSi ( w ) + LRi ( w ) . ( 11 )

Summary: This paper proposed deep evidential regression, a method for training neural networks to not only estimate the output but also the associated evidence in support of that output. The main idea follows the evidential deep learning work proposed in (Sensoy et al., 2018) extending it from the classification regime to the regression regime, by placing evidential priors over the Gaussian likelihood function and performing the type-II maximum likelihood estimation similar to the empirical Bayes method [1,2]. The authors demonstrated that the both the epistemic and aleatoric uncertainties could be estimated in one forward pass under the proposed framework without resorting to multiple passes and showed favorable uncertainty comparing to existing methods. Robustness against out of distribution and adversarially perturbed data is illustrated as well.