Distribution-free binary classification: prediction sets, confidence intervals and calibration

Abstract

We study three notions of uncertainty quantification-calibration, confidence intervals and prediction sets-for binary classification in the distribution-free setting, that is without making any distributional assumptions on the data. With a focus towards calibration, we establish a 'tripod' of theorems that connect these three notions for score-based classifiers. A direct implication is that distribution-free calibration is only possible, even asymptotically, using a scoring function whose level sets partition the feature space into at most countably many sets. Parametric calibration schemes such as variants of Platt scaling do not satisfy this requirement, while nonparametric schemes based on binning do. To close the loop, we derive distribution-free confidence intervals for binned probabilities for both fixed-width and uniform-mass binning. As a consequence of our 'tripod' theorems, these confidence intervals for binned probabilities lead to distribution-free calibration. We also derive extensions to settings with streaming data and covariate shift.

Understanding the Under-Coverage Bias in Uncertainty Estimation

“…Uncertainty quantification for classification For classification problems, two main types of uncertainty quantification methods have been considered: outputting discrete prediction sets with guarantees of covering the true (discrete) label [70,71,38,7,12,18,17], or calibrating the predicted probabilities [51,72,73,37,26]. The connection between prediction sets and calibration was discussed in [27]. The sample complexity of calibration has been studied in a number of theoretical works [36,27,54,30,40,8].…”

“…The connection between prediction sets and calibration was discussed in [27]. The sample complexity of calibration has been studied in a number of theoretical works [36,27,54,30,40,8]. Our work is inspired by the recent work of Bai et al [8], which showed that logistic regression is over-confident even if the model is correctly specified and the sample size is larger than the dimension.…”

Distribution-free inference for regression: discrete, continuous, and in between

“…Recent work by Barber [2020], Gupta et al [2020], Medarametla and Candès [2021] proves that distribution-free coverage properties similar to Definition 1 lead to fundamental limits on the accuracy of inference-if we compute a distribution-free confidence interval C n on data drawn from a distribution P where the marginal distribution P X is nonatomic (meaning that there are no point masses, i.e., P P X {X = x} = 0 for all points x ∈ R d ), then C n (X n+1 ) cannot have vanishing length as sample size n tends to infinity, regardless of the smoothness of P , or any other "nice" properties of this distribution.…”

“…The problem of distribution-free coverage for the conditional mean or median of Y given X has been studied by Barber [2020], Gupta et al [2020] (for the mean) and Medarametla and Candès [2021] (for the median). As mentioned above, these works establish that, if the marginal distribution P X of the feature vector X is nonatomic, then it is impossible for a distribution-free confidence interval C n to have vanishing length even as n → ∞; in particular, if C n is guaranteed to cover the mean or median for all distributions P , then it also must be a predictive interval, that is, it must cover Y itself (which implies its length cannot be vanishing, since Y is inherently noisy).…”

Monitoring Object Detection Abnormalities via Data-Label and Post-Algorithm Abstractions

“…Currently, proper calibration with guaranteed error bounds is known to be hard for autonomous driving, as the data distribution for realworld driving is unknown and can be highly individualized. Even under the binary classification setup, the sharpness of calibrated confidence is hard to be guaranteed without prior knowledge of the distribution [12]. Abstraction-based methods build a data manifold that encloses all values from the training dataset.…”