Conservative Uncertainty Estimation By Fitting  Prior Networks

1 INTRODUCTION . Deep learning has achieved huge success in many applications . In particular , increasingly often , it is used as a component in decision-making systems . In order to have confidence in decisions made by such systems , it is necessary to obtain good uncertainty estimates , which quantify how certain the network is about a given output . In particular , if the cost of failure is large , for example where the automated system has the capability to accidentally hurt humans , the availability and quality of uncertainty estimates can determine whether the system is safe to deploy at all ( Carvalho , 2016 ; Leibig et al. , 2017 ; Michelmore et al. , 2018 ) . Moreover , when decisions are made sequentially , good uncertainty estimates are crucial for achieving good performance quickly ( Bellemare et al. , 2016 ; Houthooft et al. , 2016 ; Ostrovski et al. , 2017 ; Burda et al. , 2018 ) . Because any non-Bayesian inference process is potentially sub-optimal ( De Finetti , 1937 ) , these uncertainty estimates should ideally be relatable to Bayesian inference with a useful prior . Deep ensembles ( Lakshminarayanan et al. , 2017 ) , one of the most popular methods available for uncertainty estimation in deep networks today , struggle with this requirement . While deep ensembles can be related ( Rubin , 1981 ) to Bayesian inference in settings where the individual models are trained on subsets of the data , this is not how they are used in practice . In order to improve data efficiency , all ensembles are typically trained using the same data ( Lakshminarayanan et al. , 2017 ) , resulting in a method which does not have a theoretical justification . Moreover , deep ensembles can give overconfident uncertainty estimates in practice . On the other hand , Monte-Carlo dropout can be viewed ( Gal & Ghahramani , 2016 ) as a certain form of Bayesian inference . However , doing so requires requires either a limit to be taken or a generalization of variational inference to a quasi-KL divergence ( Hron et al. , 2018 ) . In practice , MC dropout can give arbitrarily overconfident estimates ( Foong et al. , 2019 ) . More broadly , a category of approaches , known as Bayesian Neural Networks ( Blundell et al. , 2015 ; Welling & Teh , 2011 ; Neal , 1996 ) , maintains a distribution over the weights of the neural network . These methods have a sound Bayesian justification , but training them is both difficult and carries an accuracy penalty , particularly for networks with convolutional architectures ( Osawa et al. , 2019 ) . Moreover , tuning BNNs is hard and achieving a good approximation to the posterior is difficult ( Brosse et al. , 2018 ) . We use another way of obtaining uncertainties for deep networks , based on fitting random priors ( Osband et al. , 2018 ; 2019 ) . Random priors are easy to train and were found to work very well in practice ( Burda et al. , 2018 ) . To obtain the uncertainty estimates , Affiliations : 1 . Microsoft Research Cambridge ; 2 . ETH Zurich ; 3 . University of Cambridge . The second author was an intern at Microsoft when contributing to this work . we first train a predictor network to fit a prior . Two examples of prior-predictor pairs are shown in the top two plots of Figure 1.Faced with a novel input point , we obtain an uncertainty ( Figure 1 , bottom plot ) by measuring the error of the predictor network against this pattern . Intuitively , these errors will be small close to the training points , but large far from them . The patterns themselves are drawn from randomly initialized ( and therefore untrained ) neural networks . While this way of estimating uncertainties was known before ( Osband et al. , 2019 ) , it did not have a theoretical justification beyond Bayesian linear regression , which is too limiting for modern applications . Contributions We provide a sound theoretical framework for obtaining uncertainty estimates by fitting random priors , a method previously lacking a principled justification . Specifically , we justify estimates in the uncertainty of the output of neural networks with any architecture . In particular , we show in Lemma 1 and Proposition 1 that these uncertainty estimates are conservative , meaning they are never more certain than a Bayesian algorithm would be . Moreover , in Proposition 2 we show concentration , i.e . that the uncertainties become zero with infinite data . Empirically , we evaluate the calibration and out-of-distribution performance of our uncertainty estimates on typical computer vision tasks , showing a practical benefit over deep ensembles and MC dropout . 2 PRELIMINARIES . We are going to reason about uncertainty within the formal framework of stochastic processes . We now introduce the required notations . A stochastic process is a collection of random variables { f ( x ) } . We consider processes where x ∈ RK and the random-variable f ( x ) takes values in RM . A stochastic process has exchangeable outputs if the distribution does not change when permuting theM entries in the output vector . Allowing a slight abuse of notation , we denote the finite-dimensional distribution of the process { f ( x ) } for the set X = { xi } i=1 , ... , N as f ( x1 , . . . , xN ) = f ( X ) . In practice , the finite-dimensional distribution reflects the idea of restricting the process to points x1 , . . . , xN and marginalizing over all the other points . Inference can be performed on stochastic processes similarly to probability distributions . In particular , we can start with some prior process { f ( x ) } , observe a set of N training points X = { xi } i=1 , ... , N and labels y = { yi } i=1 , ... , N and then consider the posterior process { fXy ( x ) } , whose finite-dimensional distributions are given by fXy ( x ? 1 . . . x ? N ′ ) = f ( x ? 1 . . . x ? N ′ |x1 , . . . , xN , y1 , . . . , yN ) for any set of testing points x ? 1 . . . x ? N ′ . We use subscripts to denote conditioning on the dataset throughout the paper . We denote the variance of fXy ( x ? ) with σ2Xf ( x ? ) . A stochastic process is called Gaussian if if all its finite-dimensional distributions are Gaussian . Given a test point x ? , we denote the posterior GP mean with µXy ( x ? ) and posterior GP variance with σ2X ( x ? ) . We provide more background on GPs in Appendix D . 3 ESTIMATING UNCERTAINTY FROM RANDOM PRIORS . Intuition Uncertainties obtained from random priors have an appealing intuitive justification . Consider the networks in the top part of Figure 1 . We start with a randomly initialized prior network , shown in red . Whenever we see a datapoint , we train the predictor network ( green ) to match this prior . Uncertainties can then be obtained by considering the squared error between the prior and the predictor at a given point . An example uncertainty estimate is shown as the shaded blue area in the bottom of Figure 1 . While it may at first seem that the squared error is a poor measure of uncertainty because it can become very small by random chance , we formally show in Section 4.1 that this is very improbable . In Section 4.2 , we show that this error goes down to zero as we observe more data . Similarly to GP inference , uncertainty estimation in our framework does not depend on the regression label . The prediction mean ( blue curve in the bottom part of Figure 1 ) is obtained by fitting a completely separate neural network . In section 6 , we discuss how this framework avoids the overconfidence characteristic of deep ensembles ( Lakshminarayanan et al. , 2017 ) . Prior The process of obtaining network uncertainties involves randomly initialized prior networks , which are never trained . While this may at first appear very different from they way deep learning is normally done , these random networks are a crucial component of our method . We show in Section 4.1 that the random process that corresponds to initializing these networks can be interpreted as a prior of a Bayesian inference procedure . A prior conveys the information about how the individual data points are related . The fact that we are using random networks has both practical and theoretical benefits . Practically , since the prior does not depend on the data , there is no way that it can overfit . The use of random priors also has strong empirical support – randomly initialized networks have been recently used as priors to obtain state-of-the-art performance on computer vision tasks ( Ulyanov et al. , 2018 ; Cheng et al. , 2019 ) . Theoretically , using random priors satisfies the likelihood principle ( Robert , 2007 ) . Moreover , random priors can be viewed as a safe choice since they make the minimum reasonable assumption that the network architecture is appropriate for the task . In fact , whenever deep learning is used , with or without uncertainty estimates , practitioners are already implicitly making that assumption . Algorithm 1 Training the predictors . function TRAIN-UNCERTAINTIES ( X ) for i = 1 . . . B do f i ∼ { f ( x ) } . random prior hXfi ← FIT ( X , f i ( X ) ) end for return fi , hXfi end function function FIT ( X , f i ( X ) ) L ( h ) , ∑ x∈X ‖f i ( x ) − h ( x ) ‖2 hXfi ← OPTIMIZE ( L ) . SGD or similar return hXfi . return trained predictor end function Algorithm The process of training the predictor networks is shown in Algorithm 1 . The function TRAIN-UNCERTAINTIES first generates random priors , i.e . neural networks with random weights . In our notation , it corresponds to sampling functions from the prior process { f ( x ) } . These priors , evaluated at points from the dataset X = { xi } i=1 , ... , N are then used as labels for supervised learning , performed by the function FIT . After training , when we want to obtain an uncertainty estimate φ at a given test point x ? , we use the formula σ̂2 ( x ? ) = max ( 0 , σ̂ 2 µ ( x ? ) + βv̂σ ( x ? ) − σ2A ) . ( 1 ) Here , the quantity σ̂2µ is the sample mean of the squared error . We will show in Section 4 that it is an unbiased estimator of a variable that models the uncertainty . On the other hand , v̂σ is the samplebased estimate of the standard deviation of squared error across bootstraps , needed to quantify our uncertainty about what the uncertainty is . The hyper-parameter β controls the degree to which this uncertainty is taken into account . Formally , the quantities are defined as σ̂2µ ( x ? ) , ∑B i=1 1 MB ‖f ( x ? ) − hXfi ( x ? ) ‖ 2 , ( 2 ) v̂σ ( x ? ) , √∑B i=1 1 B ( σ̂ 2 µ ( x ? ) − 1M ‖f ( x ? ) − hXfi ( x ? ) ‖2 ) 2 . ( 3 ) In the above equations , B is the number of prior functions and each prior and predictor network has M outputs . Because the predictors are trained independently , uncertainty estimates obtained from each of the B predictor-prior pairs are independent . We defer the discussion of details of network architecture to Section 5 . Our experiments ( Section 7 ) show that it is often sufficient to use B = 1 in practice .

Summary: This paper introduces a new method for uncertainty estimation which utilizes randomly initialized networks. Essentially, instead of training a single predictor that outputs means and uncertainty estimates together, authors propose to have two separate models: one that outputs means, and one that outputs uncertainties. The later one consists of two networks: a randomly initialized “prior” which is fixed and is not trained, and a “predictor”, which is then trained to predict the output of the randomly initialized “prior” applied to the training samples.