Localized Adaptive Risk Control

TL;DR

Localized Adaptive Risk Control (L-ARC) is introduced, an online calibration scheme that targets statistical localized risk guarantees ranging from conditional risk to marginal risk, while preserving the worst-case performance of ARC.

Abstract

Adaptive Risk Control (ARC) is an online calibration strategy based on set prediction that offers worst-case deterministic long-term risk control, as well as statistical marginal coverage guarantees. ARC adjusts the size of the prediction set by varying a single scalar threshold based on feedback from past decisions. In this work, we introduce Localized Adaptive Risk Control (L-ARC), an online calibration scheme that targets statistical localized risk guarantees ranging from conditional risk to marginal risk, while preserving the worst-case performance of ARC. L-ARC updates a threshold function within a reproducing kernel Hilbert space (RKHS), with the kernel determining the level of localization of the statistical risk guarantee. The theoretical results highlight a trade-off between localization of the statistical risk and convergence speed to the long-term risk target. Thanks to localization, L-ARC is demonstrated via experiments to produce prediction sets with risk guarantees across different data subpopulations, significantly improving the fairness of the calibrated model for tasks such as image segmentation and beam selection in wireless networks.

Figures (10)

• Figure 1: Calibration of a tumor segmentation model via ARC angelopoulos2024conformal and the proposed localized ARC, L-ARC. Calibration data comprises images from multiple sources, namely, the Kvasir data set jha2020kvasir and the ETIS-LaribPolypDB data set silva2014toward. Both ARC and L-ARC achieve worst-case deterministic long-term risk control in terms of false negative rate (FNR). However, ARC does so by prioritizing Kvasir samples at the detriment of the Larib data source, for which the model has poor FNR performance. In contrast, L-ARC can yield uniformly satisfactory performance for both data subpopulations.
• Figure 2: The degree of localization in L-ARC is dictated by the choice of the reweighting function class $\mathcal{W}$ via the marginal-to-conditional guarantee (\ref{['eq:relaxation']}). At the leftmost extreme, we illustrate constant reweighting functions, for which marginal guarantees are recovered. At the rightmost extreme, reweighting with maximal localization given by Dirac delta functions for which the criterion (\ref{['eq:relaxation']}) corresponds to a conditional guarantee. In between the two extremes lie function sets $\mathcal{W}$ with an intermediate level of localization yielding localized guarantees.
• Figure 3: Long-term coverage (left) and average miscoverage error (right), marginalized and conditioned on weekdays and weekends. for ARC and L-ARC with varying values of the localization parameter $l$ on the Elec2 dataset.
• Figure 4: Long-term FNR (left), average FNR across different data sources (center), and average mask size across different data sources (right) for ARC and L-ARC with varying values of the localization parameter $l$ for the task of tumor segmentation fan2020pranet.
• Figure 5: Long-term risk (left-top), average beam set size (left-bottom), and SNR level across the deployment area (right) for ARC, Mondrian ARC, and L-ARC. The transmitter is denoted as a green circle and obstacles to propagation are shown as grey rectangles.

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof