Uncertainty Sets for Image Classifiers using Conformal Prediction

1 INTRODUCTION . Imagine you are a doctor making a high-stakes medical decision based on diagnostic information from a computer vision classifier . What would you want the classifier to output in order to make the best decision ? This is not a casual hypothetical ; such classifiers are already used in medical settings ( e.g. , Razzak et al. , 2018 ; Lundervold & Lundervold , 2019 ; Li et al. , 2014 ) . A maximumlikelihood diagnosis with an accompanying probability may not be the most essential piece of information . To ensure the health of the patient , you must also rule in or rule out harmful diagnoses . In other words , even if the most likely diagnosis is a stomach ache , it is equally or more important to rule out stomach cancer . Therefore , you would want the classifier to give you—in addition to an estimate of the most likely outcome—actionable uncertainty quantification , such as a set of predictions that provably covers the true diagnosis with a high probability ( e.g. , 90 % ) . This is called a prediction set ( see Figure 1 ) . Our paper describes a method for constructing prediction sets from any pre-trained image classifier that are formally guaranteed to contain the true class with the desired probability , relatively small , and practical to implement . Our method modifies a conformal predictor ( Vovk et al. , 2005 ) given in Romano et al . ( 2020 ) for the purpose of modern image classification in order to make it more stable in the presence of noisy small probability estimates . Just as importantly , we provide extensive evaluations and code for conformal prediction in computer vision . Formally , for a discrete response Y ∈ Y = { 1 , . . . , K } and a feature vector X ∈ Rd , we desire an uncertainty set function , C ( X ) , mapping a feature vector to a subset of { 1 , . . . , K } such that P ( Y ∈ C ( X ) ) ≥ 1− α , ( 1 ) for a pre-specified confidence level α such as 10 % . Conformal predictors like our method can modify any black-box classifier to output predictive sets that are rigorously guaranteed to satisfy the desired coverage property shown in Eq . ( 1 ) . For evaluations , we focus on Imagenet classification ∗Equal contribution . Blog : https : //people.eecs.berkeley.edu/˜angelopoulos/blog/ posts/conformal-classification using convolutional neural networks ( CNNs ) as the base classifiers , since this is a particularly challenging testbed . In this setting , X would be the image and Y would be the class label . Note that the guarantee in Eq . ( 1 ) is marginal over X and Y—it holds on average , not for a particular image X . A first approach toward this goal might be to assemble the set by including classes from highest to lowest probability ( e.g. , after Platt scaling and a softmax function ; see Platt et al. , 1999 ; Guo et al. , 2017 ) until their sum just exceeds the threshold 1 − α . We call this strategy naive and formulate it precisely in Algorithm 1 . There are two problems with naive : first , the probabilities output by CNNs are known to be incorrect ( Nixon et al. , 2019 ) , so the sets from naive do not achieve coverage . Second , image classification models ’ tail probabilities are often badly miscalibrated , leading to large sets that do not faithfully articulate the uncertainty of the model ; see Section 2.3 . Moreover , smaller sets that achieve the same coverage level can be generated with other methods . The coverage problem can be solved by picking a new threshold using holdout samples . For example , with α =10 % , if choosing sets that contain 93 % estimated probability achieves 90 % coverage on the holdout set , we use the 93 % cutoff instead . We refer to this algorithm , introduced in Romano et al . ( 2020 ) , as Adaptive Prediction Sets ( APS ) . The APS procedure provides coverage but still produces large sets . To fix this , we introduce a regularization technique that tempers the influence of these noisy estimates , leading to smaller , more stable sets . We describe our proposed algorithm , Regularized Adaptive Prediction Sets ( RAPS ) , in Algorithms 2 and 3 ( with APS as a special case ) . As we will see in Section 2 , both APS and RAPS are always guaranteed to satisfy Eq . ( 1 ) —regardless of model and dataset . Furthermore , we show that RAPS is guaranteed to have better performance than choosing a fixed-size set . Both methods impose negligible computational requirements in both training and evaluation , and output useful estimates of the model ’ s uncertainty on a new image given , say , 1000 held-out examples . In Section 3 we conduct the most extensive evaluation of conformal prediction in deep learning to date on Imagenet and Imagenet-V2 . We find that RAPS sets always have smaller average size than naive and APSsets . For example , using a ResNeXt-101 , naive does not achieve coverage , while APS and RAPS achieve it almost exactly . However , APS sets have an average size of 19 , while RAPS sets have an average size of 2 at α = 10 % ( Figure 2 and Table 1 ) . We will provide an accompanying codebase that implements our method as a wrapper for any PyTorch classifier , along with code to exactly reproduce all of our experiments . 1.1 RELATED WORK . Reliably estimating predictive uncertainty for neural networks is an unsolved problem . Historically , the standard approach has been to train a Bayesian neural network to learn a distribution over network weights ( Quinonero-Candela et al. , 2005 ; MacKay , 1992 ; Neal , 2012 ; Kuleshov et al. , 2018 ; Gal , 2016 ) . This approach requires computational and algorithmic modifications ; other approaches avoid these via ensembles ( Lakshminarayanan et al. , 2017 ; Jiang et al. , 2018 ) or approximations of Bayesian inference ( Riquelme et al. , 2018 ; Sensoy et al. , 2018 ) . These methods also have major practical limitations ; for example , ensembling requires training many copies of a neural network adversarially . Therefore , the most widely used strategy is ad-hoc traditional calibration of the softmax scores with Platt scaling ( Platt et al. , 1999 ; Guo et al. , 2017 ; Nixon et al. , 2019 ) . This work develops a method for uncertainty quantification based on conformal prediction . Originating in the online learning literature , conformal prediction is an approach for generating predictive sets that satisfy the coverage property in Eq . ( 1 ) ( Vovk et al. , 1999 ; 2005 ) . We use a convenient data-splitting version known as split conformal prediction that enables conformal prediction meth- ods to be deployed for essentially any predictor ( Papadopoulos et al. , 2002 ; Lei et al. , 2018 ) . While mechanically very different from traditional calibration as discussed above , we will refer to our approach as conformal calibration to highlight that the two methodologies have overlapping but different goals . Conformal prediction is a general framework , not a specific algorithm—important design decisions must be made to achieve the best performance for each context . To this end , Romano et al . ( 2020 ) and Cauchois et al . ( 2020 ) introduce techniques aimed at achieving coverage that is similar across regions of feature space , whereas Vovk et al . ( 2003 ) ; Hechtlinger et al . ( 2018 ) and Guan & Tibshirani ( 2019 ) introduce techniques aimed at achieving equal coverage for each class . While these methods have conceptual appeal , thus far there has been limited empirical evaluation of this general approach for state-of-the-art CNNs . Concretely , the only works that we are aware of that include some evaluation of conformal methods on ImageNet—the gold standard for benchmarking computer vision methods—are Hechtlinger et al . ( 2018 ) , Park et al . ( 2019 ) , Cauchois et al . ( 2020 ) , and Messoudi et al . ( 2020 ) , although in all four cases further experiments are needed to more fully evaluate their operating characteristics for practical deployment . At the heart of conformal prediction is the conformal score - a measure of similarity between labeled examples which is used to compare a new point to among those in a hold out set . Our theoretical contribution can be summarized as a modification of the conformal score from Romano et al . ( 2020 ) to have smaller , more stable sets . Lastly , there are alternative approaches to returning prediction sets not based on conformal prediction ( Pearce et al. , 2018 ; Zhang et al. , 2018 ) . These methods can be used as input to a conformal procedure to potentially improve performance , but they do not have finite-sample coverage guarantees when used alone . 2 METHODS . In developing uncertainty set methods to improve upon naive , we are guided by three desiderata . First and most importantly , the coverage desideratum says the sets must provide 1−α coverage , as discussed above . Secondly , the size desideratum says we want sets of small size , since these convey more detailed information and may be more useful in practice . Lastly , the adaptiveness desideratum says we want the sets to communicate instance-wise uncertainty : they should be smaller for easy test-time examples than for hard ones ; see Figure 1 for an illustration . Coverage and size are obviously competing objectives , but size and adaptiveness are also often in tension . The size desideratum seeks small sets , while the adaptiveness desideratum seeks larger sets when the classi- Algorithm 1 Naive Prediction Sets Input : α , sorted scores s , associated permutation of classes I , boolean rand 1 : procedure NAIVE ( α , s , I , rand ) 2 : L← 1 3 : while ∑L i=1 si < 1− α do . Stop if 1− α probability exceeded 4 : L← L+ 1 5 : if rand then . Break ties randomly ( explained in Appendix B ) 6 : U ← Unif ( 0 , 1 ) 7 : V ← ( ∑L i=1 si − ( 1− α ) ) /sL 8 : if U ≤ V then 9 : L← L− 1 10 : return { I1 , ... , IL } Output : The 1− α prediction set , { I1 , ... , IL } fier is uncertain . For example , always predicting a set of size five could achieve coverage , but it is not adaptive . As noted above , both APSand RAPS achieve correct coverage , and we will show that RAPS improves upon APS according to the other two desiderata . We now turn to the specifics of our proposed method . We begin in Subsection 2.1 by describing an abstract data-splitting procedure called conformal calibration that enables the near-automatic construction of valid predictive sets ( that is , sets satisfying Eq . ( 1 ) ) . Subsequently , in Subsection 2.2 , we provide a detailed presentation of our procedure , with commentary in Section 2.3 . In Subsection 2.4 we discuss the optimality of our procedure , proving that it is at least as good as the procedure that returns sets of a fixed size , unlike alternative approaches .

Summary: In this paper, the authors propose a regularized conformal score for use in a conformal prediction framework. This regularizer is motivated by the instabilities of top-p variations on conformal scores (cf. Romano et. al., 2020) and the resulting high-variance in output conformal prediction set sizes. The proposed regularizer smooths top-p scores with top-k scores, which empirically results in more robust predictive sets. The authors also perform a large-scale evaluation on ImageNet with modern architectures, which serves as a helpful benchmark for conformal prediction algorithms.