Learning Optimal Conformal Classifiers

1 INTRODUCTION . In classification tasks , for input x , we approximate the posterior distribution over classes y ∈ [ K ] : = { 1 , . . . , K } , denoted πy ( x ) ≈ P ( Y = y|X = x ) . Following Bayes ’ decision rule , the single class with highest posterior probability is predicted for optimizing a 0-1 classification loss . This way , deep networks πθ , y ( x ) with parameters θ achieve impressive accuracy on held-out test sets . However , this does not guarantee safe deployment . Conformal prediction ( CP ) ( Vovk et al. , 2005 ) uses a posttraining calibration step to guarantee a user-specified coverage : by allowing to predict confidence sets C ( X ) ⊆ [ K ] , CP guarantees the true class Y to be included with confidence level α , i.e . P ( Y ∈ C ( X ) ) ≥ 1 − α when the calibration examples ( Xi , Yi ) , i ∈ Ical are drawn exchangeably from the test distribution . This is usually achieved in two steps : In the prediction step , so-called conformity scores ( w.r.t . to a class k ∈ [ K ] ) are computed to construct the confidence sets C ( X ) . During the calibration step , these conformity scores on the calibration set w.r.t . the true class Yi are ranked to determine a cut-off threshold τ for the predicted probabilities πθ ( x ) guaranteeing coverage 1−α . This is called marginal coverage as it holds only unconditionally , i.e. , the expectation is being taken not only w.r.t . ( X , Y ) but also over the distribution of all possible calibration sets , rather than w.r.t . the conditional distribution p ( Y |X ) . CP also outputs intuitive uncertainty estimates : larger confidence sets |C ( X ) | generally convey higher uncertainty . Although CP is agnostic to details of the underlying model πθ ( x ) , the obtained uncertainty estimates depend strongly on the model ’ s performance . If the underlying classifier is poor , CP results in too large and thus uninformative confidence sets . “ Uneven ” coverage is also a common issue , where lower coverage is achieved on more difficult classes . To address such problems , the threshold CP method of ( Sadinle et al. , 2019 ) explicitly minimizes inefficiency . Romano et al . ( 2020 ) and Cauchois et al . ( 2020 ) propose methods that perform favorably in terms of ( approximate ) conditional coverage . The adaptive prediction sets ( APS ) method of Romano et al . ( 2020 ) is further extended by Angelopoulos et al . ( 2021 ) to return smaller confidence sets . These various objectives are typically achieved by changing the definition of the conformity scores . In all cases , CP is used as a post-training calibration step . In contrast , our work does not focus on advancing CP itself , e.g. , through new conformity scores , but develops a novel training procedure for the classifier πθ . After training , any of the above CP methods can readily be applied . Indeed , while the flexibility of CP regarding the underlying model appears attractive , it is also a severe limitation : Learning the model parameters θ is not informed about the post-hoc “ conformalization ” , i.e. , they are are not tuned towards any specific objective such as reducing expected confidence set size ( inefficiency ) . During training , the model will typically be trained to minimize cross-entropy loss . At test time , in contrast , it is used to obtain a set predictor C ( X ) with specific properties such as low inefficiency . In concurrent work , Bellotti ( 2021 ) addresses this issue by learning a set predictor C ( X ) through thresholding logits : Classes with logits exceeding 1 are included in C ( X ) and training aims to minimize inefficiency while targeting coverage 1 − α . In experiments using linear models only , this approach is shown to decrease inefficiency . However , ( Bellotti , 2021 ) ignores the crucial calibration step of CP during training and does not allow to optimize losses beyond marginal coverage or inefficiency . In contrast , our work subsumes ( Bellotti , 2021 ) , but additionally considers the calibration step during training , which is crucial for further decreasing inefficiency . Furthermore , we aim to allow fine-grained control over class-conditional inefficiency or the composition of the confidence sets by allowing to optimize arbitrary losses defined on confidence sets . Our contributions can be summarized as follows : 1 . We propose conformal training ( ConfTr ) , a procedure allowing to train model and conformal wrapper end-to-end . This is achieved by developing smooth implementations of recent CP methods for use during training . On each mini-batch , ConfTr “ simulates ” conformalization , using half of the batch for calibration , and the other half for prediction and loss computation , c.f . Fig . 1 a© . After training , any existing CP method can provide a coverage guarantee . 2 . In experiments , using ConfTr for training consistently reduces the inefficiency of conformal predictors such as threshold CP ( THR ) ( Sadinle et al. , 2019 ) or APS ( Romano et al. , 2020 ) applied after training . We further improve over ( Bellotti , 2021 ) , illustrating the importance of the calibration step during training . 3 . Using carefully constructed losses , ConfTr allows to “ shape ” the confidence sets obtained at test time : We can reduce class-conditional inefficiency or “ coverage confusion ” , i.e. , the likelihood of two or more classes being included in the same confidence sets , c.f . Fig . 1 b© . Generally , in contrast to ( Bellotti , 2021 ) , ConfTr allows to optimize arbitrary losses on the confidence sets . Because ConfTr is agnostic to the CP method used at test time , our work is complementary to most related work , i.e. , any advancement in terms of CP is directly applicable to ConfTr . For example , this might include conditional or application-specific guarantees as in ( Sadinle et al. , 2016 ; Bates et al. , 2021 ) . Most importantly , ConfTr preserves the coverage guarantee obtained through CP . 2 DIFFERENTIABLE CONFORMAL PREDICTORS . We are interested in training the model πθ end-to-end with the conformal wrapper in order to allow fine-grained control over the confidence sets C ( X ) . Before developing differentiable CP methods for training in Sec . 2.2 , we review two recently proposed conformal predictors that we use at test time . These consist of two steps , see Sec . 2.1 : for prediction ( on the test set ) we need to define the confidence sets Cθ ( X ; τ ) which depend on the model parameters θ through the predictions πθ and where the threshold τ is determined during calibration on a held-out calibration set ( Xi , Yi ) , i ∈ Ical in order to obtain coverage . 2.1 CONFORMAL PREDICTORS . The threshold conformal predictor ( THR ) ( Sadinle et al. , 2019 ) constructs the confidence sets by thresholding probabilities : Cθ ( x ; τ ) : = { k : πθ , k ( x ) = : Eθ ( x , k ) ≥ τ } . Here , the subscript Cθ makes the dependence on the model πθ and its parameters θ explicit . During calibration , τ is computed as the α ( 1 + 1/|Ical| ) -quantile of the so-called conformity scores Eθ ( xi , yi ) = πθ , yi ( xi ) . The conformity scores indicate , for each example , the threshold that ensures coverage . Marginal coverage of ( 1 − α ) is guaranteed on a test example ( X , Y ) . In practice , THR can also be applied on logits ( THRL ) or log-probabilities ( THRLP ) instead of probabilities . Adaptive Prediction Sets ( APS ) ( Romano et al. , 2020 ) constructs confidence sets based on the ordered probabilities . Specifically , Cθ ( x ; τ ) : = { k : Eθ ( x , k ) ≤ τ } with : Eθ ( x , k ) : = πθ , y ( 1 ) ( x ) + . . .+ πθ , y ( k−1 ) ( x ) + Uπθ , y ( k ) ( x ) , ( 1 ) where πθ , y ( 1 ) ( x ) ≥ . . . ≥ πθ , y ( K ) ( x ) and U is a uniform random variable in [ 0 , 1 ] to break ties . Similar to THR , the conformity scores Eθ ( xi , yi ) w.r.t . the true classes yi are used for calibration , but the ( 1− α ) ( 1 + 1/|Ical| ) -quantile is required to ensure marginal coverage on test examples . Performance of CP is then measured using two metrics : ( empirical and marginal ) coverage ( Cover ) as well as inefficiency ( Ineff ) . Letting Itest be a test set of size |Itest| , these metrics are computed as Cover : = 1 |Itest| ∑ i∈Itest δ [ yi ∈ C ( xi ) ] and Ineff : = 1 |Itest| ∑ i∈Itest |C ( xi ) | , ( 2 ) where δ denotes an indicator function that is 1 when its argument is true and 0 otherwise . Due to the marginal coverage guarantee provided by CP ( c.f . ( Romano et al. , 2020 ) or App . C ) , the empirical coverage , when averaged across several calibration/test splits , is Cover ≈ 1− α . Thus , we concentrate on inefficiency as the main metric to compare across CP methods and models . With accuracy , we refer to the ( top-1 ) accuracy with respect to the arg max-predictions , i.e. , arg maxk πθ , k ( x ) , obtained by the underlying model π . As shown in Fig . 2 ( left ) , THR clearly outperforms THRL and APS w.r.t . inefficiency ( lower is better ) averaged across random Ical/Itest splits ( details in Sec . F ) . CP is intended to be used as a “ wrapper ” around πθ . “ Better ” CP methods generally result in lower inefficiency for a fixed model πθ . For example , following Fig . 2 ( left ) , regularized APS ( RAPS ) ( Angelopoulos et al. , 2021 ) recently showed how to improve inefficiency compared to APS by modifying the conformity score – without outperforming THR , however . Fine-grained control over inefficiency , e.g. , conditioned on the class or the composition of the C ( X ) is generally not possible . Integrating CP into the training procedure promises a higher degree of control , however , requires differentiable CP implementations , e.g. , for THR or APS . 1 : function PREDICT ( πθ ( x ) , τ ) 2 : compute Eθ ( x , k ) , k∈ [ K ] 3 : return Cθ ( x ; τ ) = { k : Eθ ( x , k ) ≥ τ } 1 : function CALIBRATE ( { ( πθ ( xi ) , yi } ni=1 , α ) 2 : compute Eθ ( xi , yi ) , i=1 , . . . , n 3 : return QUANTILE ( { Eθ ( xi , yi ) } , α ( 1 + 1/n ) ) 1 : function SMOOTHPRED ( πθ ( x ) , τ , T=1 ) 2 : return Cθ , k ( x ; τ ) =σ ( ( Eθ ( x , k ) −τ ) T ) , k∈ [ K ] 3 : function SMOOTHCAL ( { ( πθ ( xi ) , yi } ni=1 , α ) 4 : return SMOOTHQUANT ( { Eθ ( xi , yi ) } , α ( 1+ 1n ) ) 1 : function CONFORMALTRAINING ( α , λ=1 ) 2 : for mini-batch B do 3 : randomly split batch Bcal ] Bpred = B 4 : { “ On-the-fly ” calibration on Bcal : } 5 : τ = SMOOTHCAL ( { ( πθ ( xi ) , yi ) } i∈Bcal , α ) 6 : { Prediction only on i ∈ Bpred : } 7 : Cθ ( xi ; τ ) = SMOOTHPRED ( πθ ( xi ) , τ ) 8 : { Optional classification loss : } 9 : LB = 0 or ∑ i∈Bpred L ( Cθ ( xi ; τ ) , yi ) 10 : ΩB = ∑ i∈Bpred Ω ( Cθ ( xi ; τ ) ) 11 : ∆ =∇θ1/|Bpred| ( LB + λΩB ) 12 : update parameters θ using ∆ Algorithm 1 : Smooth CP and Conformal Training ( ConfTr ) : Top left : At test time , for THR , PREDICT computes the conformity scores Eθ ( x , k ) for each k∈ [ K ] and constructs the confidence sets Cθ ( x ; τ ) by thresholding with τ . CALIBRATE determines the threshold τ as the α ( 1 + 1/n ) quantile of the conformity scores w.r.t . the true classes yi on a calibration set { ( xi , yi ) } of size n : =|Ical| . THR and APS use different conformity scores . Right and bottom left : ConfTr calibrates on a part of each mini-batch , Bcal . Thereby , we obtain guaranteed coverage on the other part , Bpred ( in expectation across batches ) . Then , the inefficiency on Bpred is minimized to update the model parameters θ . Smooth implementations of calibration and prediction are used .

Summary: **Summary:** * The paper considers putting conformal inference into the process of fitting deep neural networks, i.e., the paper considers "differentiable conformal prediction" / back-prop'ing through conformal inference. * The goal is to reduce the size of the prediction sets generated by conformal inference -- and more specifically to reduce the size variably in different parts of the input space. * Some experiments are given to show that the confidence set sizes are indeed smaller w/ the proposal than w/o it.