  We consider the problem of constructing distribution-free prediction sets
with finite-sample conditional guarantees. Prior work has shown that it is
impossible to provide exact conditional coverage universally in finite samples.
Thus, most popular methods only provide marginal coverage over the covariates.
This paper bridges this gap by defining a spectrum of problems that interpolate
between marginal and conditional validity. We motivate these problems by
reformulating conditional coverage as coverage over a class of covariate
shifts. When the target class of shifts is finite dimensional, we show how to
simultaneously obtain exact finite sample coverage over all possible shifts.
For example, given a collection of protected subgroups, our algorithm outputs
intervals with exact coverage over each group. For more flexible, infinite
dimensional classes where exact coverage is impossible, we provide a simple
procedure for quantifying the gap between the coverage of our algorithm and the
target level. Moreover, by tuning a single hyperparameter, we allow the
practitioner to control the size of this gap across shifts of interest. Our
methods can be easily incorporated into existing split conformal inference
pipelines, and thus can be used to quantify the uncertainty of modern black-box
algorithms without distributional assumptions.
