  Estimating the Kullback-Leibler (KL) divergence between two distributions
given samples from them is well-studied in machine learning and information
theory. Motivated by considerations of multi-group fairness, we seek KL
divergence estimates that accurately reflect the contributions of
sub-populations to the overall divergence. We model the sub-populations coming
from a rich (possibly infinite) family $\mathcal{C}$ of overlapping subsets of
the domain. We propose the notion of multi-group attribution for $\mathcal{C}$,
which requires that the estimated divergence conditioned on every
sub-population in $\mathcal{C}$ satisfies some natural accuracy and fairness
desiderata, such as ensuring that sub-populations where the model predicts
significant divergence do diverge significantly in the two distributions. Our
main technical contribution is to show that multi-group attribution can be
derived from the recently introduced notion of multi-calibration for importance
weights [HKRR18, GRSW21]. We provide experimental evidence to support our
theoretical results, and show that multi-group attribution provides better KL
divergence estimates when conditioned on sub-populations than other popular
algorithms.
