  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.
