A Statistical Framework for Efficient Out of Distribution Detection in Deep Neural Networks

1 INTRODUCTION . Deep Neural Networks ’ ( DNNs ) predictions were shown to be overconfident and unreliable when the test samples are drawn from an unexpected distribution ( e.g. , Biggio et al . ( 2013 ) ; Szegedy et al . ( 2013 ) ; Goodfellow et al . ( 2014 ) ; Eykholt et al . ( 2017 ) ) . Therefore , numerous Out Of Distribution ( OOD ) detection methods were suggested to address this issue by informing an operator when not to trust the DNN ’ s prediction . In the context of DNNs , methods for OOD detection can be roughly divided into two classes . `` Mutators '' methods modify the network structure or loss , and depend on training its parameters to provide a confidence measure ( Hendrycks et al. , 2018 ; Malinin and Gales , 2018 ; Winkens et al. , 2020 ; Zhang et al. , 2020 ; Hsu et al. , 2020 ) . `` Observer '' methods model the prediction uncertainty of a pre-trained network , without modifying its architecture or parameters ( Hendrycks and Gimpel , 2016 ; Liang et al. , 2017 ; Lee et al. , 2018 ; Zisselman and Tamar , 2020 ; Raghuram et al. , 2020 ) . Mutators are potentially costly , as they require substituting or retraining the network and subsequently repeating an exhaustive hyperparameter tuning process . Also , the resulting model quality on the original task may vary . Alternatively , observer methods commonly utilize auxiliary models to monitor features produced by the reference network . Since this approach is decoupled from the network optimization process , the reference accuracy is retained at the cost of additional test time resources , such as compute and memory . Another differentiating factor between methods , regardless of their class , stems from fundamental assumptions regarding prior knowledge on the test distribution . Specifically , some methods ( e.g. , Hendrycks et al . ( 2018 ) ; Lee et al . ( 2018 ) ; Zisselman and Tamar ( 2020 ) ) rely on prior knowledge , such as access to the OOD data or its proxy . However , this prior can lead to undesirable bias and poor detection performance on other test distributions . Notably , OOD detection can be naturally cast as a statistical hypothesis testing problem ( Grubbs , 1969 ; Vovk et al. , 2005 ; Papadopoulos , 2008 ; Grosse et al. , 2017 ) . Nevertheless , authors of contemporary observer methods largely overlook the statistical aspects of their proposed algorithms . For example , they do not provide meaningful error guarantees nor explicitly define what hypothesis is being tested . For further details regarding related work , see Section 5 . In addition , these methods also tend to be computationally expensive ( see Section 4.4 ) . In this work , we aim to tackle these issues . Contributions . We first propose methodological innovations • We present a novel framework for OOD detection in DNNs , based on statistical hypothesis testing and phrase two relevant null hypotheses . The first is whether a sample is drawn from the same distribution of the classes in the training dataset ( any-class ) . The second is whether a sample is drawn from the distribution corresponding to the predicted class ( class-conditional ) . We construct valid tests for both hypotheses and further provide empirical evidence for these theoretical results ( Section 3 ) . • Our framework can be viewed as an extension of the inductive conformal predictor framework ( Papadopoulos , 2008 ) , which is a non-parametric method for conducting hypothesis testing with an arbitrary test statistic ( Vovk et al. , 2005 ) . However , instead of focusing solely on the last layer of the DNN predictor , we show how to normalize and aggregate multiple non-conformity ( i.e. , abnormality ) measures based on the entire network . Empirically , we demonstrate that this generalization is crucial for OOD detection . Namely , we find that a multi-layer detector improves accuracy by 50 % on average compared to single-layer variants ( Section 4.2 ) . These innovations lead to considerable practical advantages • Importantly , unlike current methods for OOD detection in DNNs , our framework returns a valid p-value for each tested sample . This guarantees maintaining a Type I Error ( detecting OOD as in-distribution ) on test data ( see Section 3.2 , Propositions 1 and 2 ) , and has additional advantages1 which do not exist in standard threshold-based OOD classification . • Based on this framework , we design a detection scheme denoted as MaSF ( Max-Simes-Fisher , Section 4.2 ) . MaSF achieves similar or better results compared to SOTA on the benchmark proposed by Lee et al . ( 2018 ) , while in our experiments , we show its test statistic inference is 35 times faster . Making MaSF appealing for real-world applications with a limited compute budget ( Section 4.4 ) . • We demonstrate potential avenues to further reduce the computational overhead via structured sparsity ( i.e. , channel sampling ) — further reducing the computational cost by a factor of 10 while degrading accuracy by no more than ∼ 1 % ( Section 4.5 ) . 2 BACKGROUND . In the following section , we provide a summary of Null Hypothesis Statistical Testing ( NHST ) . We begin with a single hypothesis and then further expand on Global-Null tests . Null Hypothesis Statistical Testing . NHST is the process of choosing between two hypotheses - H0 ( null ) and H1 ( alternative ) regarding observation ( sample ) X . It is formalized as H0 : X ∼ P vs H1 : X ̸∼ P , where P is some distribution . The decision to reject H0 is based on the critical value γ and the test statistic T ( X ) . Throughout this section , we will focus on right-tailed tests . We call a test right-tailed when higher values of T ( X ) are more likely under H1 than under H0 . In those kinds of tests , H0 is rejected if T ( X ) ≥ γ . The critical value γ quantifies the required evidence to determine if H0 is false . Erroneously rejecting H0 , is called a Type I Error ( T1E ) . The significance level ( i.e. , probability of T1E ) is given by α = PH0 ( T ( X ) ≥ γ ) . Usually , α is predetermined , and γ is found accordingly . For a given significance level , the goal is to maximize the power of the test , PH1 ( T ( X ) ≥ γ ) , which is the probability of correctly rejecting H0 . Commonly , a p-value is used to summarize NHST . The p-value indicates the probability of obtaining a test statistic equal or more extreme than the observed test statistic , given that H0 is true . The p-value ( denoted by q ) is given by , q ( tobs ) = PH0 ( T ( X ) ≥ tobs ) , where tobs is the observed value of the test statistic . Assuming the null is correct , and that P is continuous , then the p-values are uniformly distributed ( Abramovich and Ritov , 2013 ) . This allows to simply reject the null if q ( tobs ) ≤ α . It is clear that 1For example , p-values enable adjusting the decision rule when more than one hypothesis is tested at a time while maintaining a certain error criterion ( Chapter 2 in Bretz et al . ( 2016 ) ) . For instance , one can potentially employ several OOD detectors on a single sample ( e.g. , each specialized on a specific type of outlier distribution ) . the significance level will be maintained if the p-value distribution is stochastically larger2 than or equal to the uniform distribution3 . Global Null Tests . Given a set of hypotheses , combination tests are used to determine if any hypothesis is false ( global null ) . Let q denote a vector of m p-values , q = ( q1 , . . . , qm ) , that is used to test the m null-hypotheses , H0,1 , . . . , H0 , m . The global null hypothesis is defined as H0 , . = ⋂m i=1 H0 , i . The test statistic is T ( q ) . The test power depends on the unknown distribution of the test statistic under H1 . Two ubiquitous tests of the global-null are the Fisher and Simes tests ( Fisher , 1992 ; Simes , 1986 ) .The Fisher test statistic is −2 ∑m i=1 log ( qi ) . It is suitable for detecting dense signals due to the summation of p-values . The Simes test statistic is min i∈ { 1 , ... , m } q ( i ) m i , where ( i ) is the index of the p-values after sorting . It is adept for detection of a sparse signal , as it focus on a single p-value . These complementing attributes are appealing for OOD detection , as we will demonstrate in the following sections . For a more detailed introduction of these tests see Section B.3 . An important benefit of using p-values is that they allow maintaining a meaningful error rate . There are many methods employed to maintain such error rates , which are applied on p-values ( Holm , 1979 ; Hochberg , 1988 ; Benjamini and Hochberg , 1995 ) . 3 HYPOTHESIS TESTING IN NEURAL NETWORKS . For the sake of simplicity , we focus the next section solely on convolutional neural networks ( CNNs ) for image classification tasks . The method presented , however , is applicable to any DNN . 3.1 PRELIMINARIES . Define [ k ] = { i ∈ N : i ≤ k } and let χc = { ( Xi , yi ) : i ∈ [ nc ] , yi = c } , where |χc| = nc , Xi is an image and yi , ŷi are its true and predicted class ( out of k classes ) . We further split the observations of each class to the CNN training set χctrain , and the validation set χ c val The training set is denoted by χtrain = ⋃k c=1 χ c train , and its cardinality is Ntrain = ∑k c=1 nc . We similarly define χval and Nval . Pc is the class distribution of Xi where yi = c , c ∈ [ k ] . The distribution of any test statistic under H0 is required to obtain p-values . For that reason we apply the empirical cumulative distribution function ( eCDF ) to estimate the null distribution of class c , function T at x using an observations set χc ∈ { χctrain , χcval } , P̂ ( x ; T , χc ) = ∑ ( Xi , yi ) ∈χc I ( T ( Xi ) ≤ x ) + 1 nc + 1 , ( 1 ) where I is the indicator function . The addition of 1 is necessary to ensure that the p-values are not equal to 0 , as it implies they are rejected at all significance levels . The problem of determining if an image Xtest is OOD , can be formalized as H∗0 : ∃c : Xtest ∼ Pc , H∗1 : Xtest ̸∼ Pc ∀c ∈ [ k ] ( 2 ) By rejecting H∗0 , we conclude that the image is not drawn from any of the class distributions . One can also test if an image is sampled from a specific class distribution , Hc0 : Xtest ∼ Pc , Hc1 : Xtest ̸∼ Pc . ( 3 ) Since the true label is unknown at test time , we are interested in the case where the class is assigned according to the CNN prediction , Hŷtest0 . Therefore , rejecting it indicates the image is either misclassified or OOD .

Summary: This paper constructs OOD detection as a hypothesis testing problem and extracts p-values from each layer and channel in CNN from the empirical cumulative distribution. Max, Simes, and Fisher test statistics are adopted as spatial, channel, and layer reduction methods (MaSF). Empirical results show that MaSF achieves comparable or better results than GRAM (Sastry and Oore, 2019), ResFlow (Zisselman and Tamar, 2020), and Mahalanobis (Lee et al., 2018) while significantly reducing the computational cost.