  We investigate different methods for regularizing quantile regression when
predicting either a subset of quantiles or the full inverse CDF. We show that
minimizing an expected pinball loss over a continuous distribution of quantiles
is a good regularizer even when only predicting a specific quantile. For
predicting multiple quantiles, we propose achieving the classic goal of
non-crossing quantiles by using deep lattice networks that treat the quantile
as a monotonic input feature, and we discuss why monotonicity on other features
is an apt regularizer for quantile regression. We show that lattice models
enable regularizing the predicted distribution to a location-scale family.
Lastly, we propose applying rate constraints to improve the calibration of the
quantile predictions on specific subsets of interest and improve fairness
metrics. We demonstrate our contributions on simulations, benchmark datasets,
and real quantile regression problems.
