  Conformal Prediction (CP) algorithms estimate the uncertainty of a prediction
model by calibrating its outputs on labeled data. The same calibration scheme
usually applies to any model and data without modifications. The obtained
prediction intervals are valid by construction but could be inefficient, i.e.
unnecessarily big, if the prediction errors are not uniformly distributed over
the input space.
  We present a general scheme to localize the intervals by training the
calibration process. The standard prediction error is replaced by an optimized
distance metric that depends explicitly on the object attributes. Learning the
optimal metric is equivalent to training a Normalizing Flow that acts on the
joint distribution of the errors and the inputs. Unlike the Error Reweighting
CP algorithm of Papadopoulos et al. (2008), the framework allows estimating the
gap between nominal and empirical conditional validity. The approach is
compatible with existing locally-adaptive CP strategies based on re-weighting
the calibration samples and applies to any point-prediction model without
retraining.
