Locally Valid and Discriminative Prediction Intervals for Deep Learning Models

Crucial for building trust in deep learning models for critical real-world applications is efficient and theoretically sound uncertainty quantification , a task that continues to be challenging . Useful uncertainty information is expected to have two key properties : It should be valid ( guaranteeing coverage ) and discriminative ( more uncertain when the expected risk is high ) . Moreover , when combined with deep learning ( DL ) methods , it should be scalable and affect the DL model performance minimally . Most existing Bayesian methods lack frequentist coverage guarantees and usually affect model performance . The few available frequentist methods are rarely discriminative and/or violate coverage guarantees due to unrealistic assumptions . Moreover , many methods are expensive or require substantial modifications to the base neural network . Building upon recent advances in conformal prediction [ 13 , 33 ] and leveraging the classical idea of kernel regression , we propose Locally Valid and Discriminative prediction intervals ( LVD ) , a simple , efficient and lightweight method to construct discriminative prediction intervals ( PIs ) for almost any DL model . With no assumptions on the data distribution , such PIs also offer finite-sample local coverage guarantees ( contrasted to the simpler marginal coverage ) . We empirically verify , using diverse datasets , that besides being the only locally valid method for DL , LVD also exceeds or matches the performance ( including coverage rate and prediction accuracy ) of existing uncertainty quantification methods , while offering additional benefits in scalability and flexibility . 1 Introduction . Consider a training set Strain = { ( Xi , Yi ) } Ni=1 and a test example ( XN+1 , YN+1 ) , all drawn i.i.d from an arbitrary joint distribution P , with ( Xi , Yi ) ∈ X × Y for some X ⊆ Rd and Y ⊆ R. We are interested in the problem of predictive inference : On observing Strain and XN+1 , our task is to construct a prediction interval ( PI ) 1 estimate Ĉ ( XN+1 ) that contains the true value of YN+1 with a ( pre-specified ) high probability . 1Several recent deep learning papers use “ Confidence Interval ” and “ Prediction Interval ” interchangeably . We stick to the conventional statistical usage . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . The construction of actionable PIs involves two general challenges : First , Ĉ should be valid , meaning that if the specified probability is 1− α , we expect Ĉ ( XN+1 ) to cover YN+1 at least 1− α of the time . Moreover , Ĉ should be discriminative i.e. , we expect Ĉ ( XN+1 ) to be narrower for confident cases and vice-versa . The width of the PI Ĉ ( XN+1 ) is thus quantification of the uncertainty . Figure 1 illustrates these notions , with more details in Section 2.2 and 2.3 . While deep learning ( DL ) models have demonstrated impressive performance over a range of complicated tasks and data modalities , it has remained difficult to quantify the uncertainty for their predictions . For DL predictions to be actionable , uncertainty information is however indispensable , especially in domains like medicine and finance [ 2 ] . Apart from requiring validity and discrimination as discussed earlier , two additional challenges exist specifically for DL models . Obviously , any uncertainty estimation method needs to finish reasonably fast to be useful , so the third challenge is scalability . The fourth challenge is accuracy : The uncertainty estimation should not decrease the prediction accuracy of the DL model . Post-hoc methods are ideal because they usually do not interfere with the base NN prediction at all . These four requirements together constitute a set of essential desiderata for uncertainty quantification in DL . Existing uncertainty estimation methods for DL rarely address more than one or two of the above requirements . Credible intervals given by posteriors of approximate Bayesian methods such as [ 43 , 15 ] , deep ensemble [ 20 , 44 ] and Monte-Carlo dropout [ 12 ] are not valid in the frequentist sense [ 7 ] . Most existing methods also interfere with the original model design , loss function and/or training , which could be expensive and decrease the model performance ( as verified in our experiments ) [ 12 , 1 , 7 , 20 ] . To address these requirements , we leverage recent advances in conformal prediction and the classical idea of kernel regression . Conformal prediction , pioneered by Vovk [ 40 ] , is a powerful approach for constructing valid PIs . The most popular split conformal methods usually leverage prediction errors from a hold-out set to construct Ĉ ( XN+1 ) , which would be valid if a future data point ( XN+1 , YN+1 ) follows the same distribution as data in the hold-out set . This framework is particularly suitable for deep learning due to its distribution-free nature , and has motivated many recent uncertainty quantification efforts in deep learning for both classification and regression tasks [ 1 , 23 , 3 , 11 ] . However , most conformal methods are only marginally valid [ 25 , 40 , 21 , 4 ] . Moreover , less-thanmeticulous applications to DL can break distributional assumptions and theoretical validity , as in the case of [ 1 ] ( see discussion in Appendix ) . We however seek to construct a PI conditioning on the input ( similar to the third PI in Fig . 1 ) . Some recent advances ( [ 13 , 33 ] ) examine the possibility of “ approximately ” conditionally valid PIs . While these methods can not be directly applied to DL due to efficiency and performance considerations , their methodological and theoretical contributions serve as major inspirations for us to develop a highly flexible and practical method . Summary of Contributions : We propose Locally Valid Discriminative Prediction Intervals ( LVD ) , a simple uncertainty estimation method for deep learning which combines recent advances in conformal prediction and the classical idea of kernel regression . LVD applies to almost all DL models , and is the first method that satisfies all four aforementioned requirements : • Validity : LVD has frequentist coverage guarantee ( not just marginal , but approximately conditional ) . • Discrimination : The width of the PIs given by LVD adapts to the risk/uncertainty level of XN+1 . • Scalability : LVD is lightweight , adding limited overhead to the base DL model . • Accuracy : LVD is post-hoc without requiring model retraining , and does not affect the base performance of the DL model . We must note that while the theoretical foundation for guaranteeing `` validity '' is mostly based on [ 13 ] , LVD addresses several challenges to satisfy the other three requirements . The code to replicate all our results can be found at https : //github.com/zlin7/LVD . 2 Preliminaries . 2.1 Learning Setup and Assumptions . We assume data and response pairs ( X , Y ) ∈ X × Y have a joint distribution denoted P , with the marginal distributions of Y and X and the conditional distribution Y |X denoted as PY , PX , and PY |X , respectively . Further , we will define Zi : = ( Xi , Yi ) for concision . Assuming that we already have an algorithm ( with all the training protocols folded in ) , such as a Deep Neural Network ( DNN ) , that provides a mean estimator µ̂NN ( x ) : X 7→ Y . Given a target coverage level 1− α ∈ ( 0 , 1 ) , our task is to also construct a prediction interval estimator function Ĉα ( x ) : X 7→ { subset of Y } that has the validity and discrimination properties as defined below . 2.2 Validity ( Frequentist Coverage ) . There are several ( related ) notions for a PI to be valid – marginal , conditional , and local . Given target level 1−α , we say Ĉα has the marginal coverage guarantee ( or , equivalently , is marginally valid ) if P { YN+1 ∈ Ĉα ( XN+1 ) } ≥ 1− α ( 1 ) where the probability is taken over the training data and ( the unseen ) ( XN+1 , YN+1 ) . A limitation of marginal coverage is that it is not conditioned on XN+1 . A more desirable , albeit stronger , property would be conditional coverage at 1− α : P { YN+1 ∈ Ĉα ( XN+1 ) |XN+1 = x } ≥ 1− α for almost all x ∈ X . ( 2 ) Here the probability is taken over the training data and YN+1 ( with XN+1 fixed ) . It is thus clear that conditional coverage implies marginal coverage but not the other way around . Indeed , a Ĉα with marginal coverage property only implies a 1 − α chance of being accurate on average across all data points ( marginalizing over XN+1 ) i.e . there might be a sub-population in the data for which the coverage is completely missed . Unfortunately , it is impossible to achieve distribution-free finitesample conditional coverage ( Eq . 2 ) in a non-trivial way . Indeed , it is known that a finite-sample estimated Ĉα ( x ) can not achieve conditional coverage , unless it produces infinitely wide prediction intervals in expectation under any non-discrete distribution P [ 39 , 22 , 5 ] . It is thus reasonable to instead seek approximate conditional coverage . As might be apparent , there is considerable freedom in defining an appropriate notion of “ approximate ” , depending on specific tasks and domains . However , a sufficiently general-purpose and natural notion involves using a kernel function K : X × X 7→ R and a center x′ ∈ X , like the relaxation given in [ 33 ] : ∫ P { YN+1 ∈ Ĉα ( x′ ) |XN+1 = x } K ( x , x′ ) dPX ( x ) ∫ K ( x , x′ ) dPX ( x ) ≥ 1− α ( 3 ) with the probability ( P in the integral ) taken over all training samples and YN+1 , with ( XN+1 , YN+1 ) ∼ P̃ = P̃X × PY |X . Here P̃X is just the distribution re-weighted by the kernel with a center x′ , defined by dP̃X ( x ) dx ∝ dPX ( x ) dx K ( x ′ , x ) . Instead of choosing x′ beforehand , if we let the center be XN+1 and fold the integral into P like in [ 13 ] , we arrive at the definition of local coverage : P { ỸN+1 ∈ Ĉα ( XN+1 ) |XN+1 = x′ } ≥ 1− α . ( 4 ) Here the probability integrates over all training data and an additional ( X̃N+1 , ỸN+1 ) ∼ P̃ defined above . Intuitively , this definition means Ĉα ( XN+1 ) is valid “ on average ” within a small neighborhood of XN+1 . Note that Eqs . 1 and 2 reduce to Eq . 4 with K being constant and delta functions , respectively . In the rest of the paper , we will call Ĉα marginally/conditionally/locally valid if it satisfies Eq . 1/2/4 respectively , and we will pursue finite-sample local validity . 2.3 Discrimination . The idea of discrimination is simple : If the error of our prediction µ̂ ( x ) is high for an input x , the PI should be wide , and vice versa . Formally , following [ 1 ] , we require E [ W ( Ĉ ( x ) ) ] ≥ E [ W ( Ĉ ( x′ ) ) ] ⇔ E [ ` ( y , µ̂ ( x ) ) ] ≥ E [ ` ( y′ , µ̂ ( x′ ) ) ] . ( 5 ) Here the expectation is taken over the training data , W is a measure of the width of the PI , and ` is a loss function such as MSE . This property can be verified ( as shown in Section 4 ) by checking how well W ( Ĉ ( x ) ) could predict the magnitude of the error . Discrimination could be considered a measure of efficiency , as a good Ĉ could “ save ” some width when the expected risk is low . However , it only makes sense to compare efficiency if all else is equal ( i.e . two marginally valid PIs estimators with the same error ) . Note that although discrimination could be related to conditional/local validity , they are not the same - e.g. , a PI that is always infinitely wide is conditionally valid , but not discriminative . Our goal is to achieve both local validity and discrimination without making any assumptions about the underlying distribution P ( i.e. , in a distribution-free setting ) . As noted in Section 1 , our method should also run fast and not affect the performance of underlying neural network model µ̂NN .

Summary: The authors propose a new method for locally valid (e.g. marginal coverage guarantees) and discriminative prediction* intervals by connecting two ideas: kernel regression and conformal inference. They evaluate their method on a suite of small to medium scale regression tasks and find that the proposed approach is locally valid   *note that the authors refer to these intervals as _confidence_ intervals which I believe to be an error. CIs seek to cover a population parameter, which for regression would be the true, unknown, and underlying mean function (which can only be assessed via a simulation). What the authors propose are _prediction_ intervals which seek to cover draws from this distribution and are inclusive of distributional noise, c.f. this reference for more:  https://www.stat.cmu.edu/~ryantibs/papers/conformal.pdf