  This work studies algorithms for learning from aggregate responses. We focus
on the construction of aggregation sets (called bags in the literature) for
event-level loss functions. We prove for linear regression and generalized
linear models (GLMs) that the optimal bagging problem reduces to
one-dimensional size-constrained $k$-means clustering. Further, we
theoretically quantify the advantage of using curated bags over random bags. We
then propose the PriorBoost algorithm, which adaptively forms bags of samples
that are increasingly homogeneous with respect to (unobserved) individual
responses to improve model quality. We study label differential privacy for
aggregate learning, and we also provide extensive experiments showing that
PriorBoost regularly achieves optimal model quality for event-level
predictions, in stark contrast to non-adaptive algorithms.
