Direct Prediction Set Minimization via Bilevel Conformal Classifier Training

Conformal prediction (CP) is a promising uncertainty quantification framework which works as a wrapper around a black-box classifier to construct prediction sets (i.e., subset of candidate classes) with provable guarantees. However, standard calibration methods for CP tend to produce large prediction sets which makes them less useful in practice. This paper considers the problem of integrating conformal principles into the training process of deep classifiers to directly minimize the size of prediction sets. We formulate conformal training as a bilevel optimization problem and propose the {\em Direct Prediction Set Minimization (DPSM)} algorithm to solve it. The key insight behind DPSM is to minimize a measure of the prediction set size (upper level) that is conditioned on the learned quantile of conformity scores (lower level). We analyze that DPSM has a learning bound of $O(1/\sqrt{n})$ (with $n$ training samples), while prior conformal training methods based on stochastic approximation for the quantile has a bound of $\Omega(1/s)$ (with batch size $s$ and typically $s \ll \sqrt{n}$). Experiments on various benchmark datasets and deep models show that DPSM significantly outperforms the best prior conformal training baseline with $20.46\%\downarrow$ in the prediction set size and validates our theory.

Conformal Prediction for Class-wise Coverage via Augmented Label Rank Calibration

Conformal prediction (CP) is an emerging uncertainty quantification framework that allows us to construct a prediction set to cover the true label with a pre-specified marginal or conditional probability. Although the valid coverage guarantee has been extensively studied for classification problems, CP often produces large prediction sets which may not be practically useful. This issue is exacerbated for the setting of class-conditional coverage on imbalanced classification tasks. This paper proposes the Rank Calibrated Class-conditional CP (RC3P) algorithm to reduce the prediction set sizes to achieve class-conditional coverage, where the valid coverage holds for each class. In contrast to the standard class-conditional CP (CCP) method that uniformly thresholds the class-wise conformity score for each class, the augmented label rank calibration step allows RC3P to selectively iterate this class-wise thresholding subroutine only for a subset of classes whose class-wise top-k error is small. We prove that agnostic to the classifier and data distribution, RC3P achieves class-wise coverage. We also show that RC3P reduces the size of prediction sets compared to the CCP method. Comprehensive experiments on multiple real-world datasets demonstrate that RC3P achieves class-wise coverage and 26.25% reduction in prediction set sizes on average.

Probabilistically robust conformal prediction

Conformal prediction (CP) is a framework to quantify uncertainty of machine learning classifiers including deep neural networks. Given a testing example and a trained classifier, CP produces a prediction set of candidate labels with a user-specified coverage (i.e., true class label is contained with high probability). Almost all the existing work on CP assumes clean testing data and there is not much known about the robustness of CP algorithms w.r.t natural/adversarial perturbations to testing examples. This paper studies the problem of probabilistically robust conformal prediction (PRCP) which ensures robustness to most perturbations around clean input examples. PRCP generalizes the standard CP (cannot handle perturbations) and adversarially robust CP (ensures robustness w.r.t worst-case perturbations) to achieve better trade-offs between nominal performance and robustness. We propose a novel adaptive PRCP (aPRCP) algorithm to achieve probabilistically robust coverage. The key idea behind aPRCP is to determine two parallel thresholds, one for data samples and another one for the perturbations on data (aka "quantile-of-quantile" design). We provide theoretical analysis to show that aPRCP algorithm achieves robust coverage. Our experiments on CIFAR-10, CIFAR-100, and ImageNet datasets using deep neural networks demonstrate that aPRCP achieves better trade-offs than state-of-the-art CP and adversarially robust CP algorithms.