  The ''Propose-Test-Release'' (PTR) framework is a classic recipe for
designing differentially private (DP) algorithms that are data-adaptive, i.e.
those that add less noise when the input dataset is nice. We extend PTR to a
more general setting by privately testing data-dependent privacy losses rather
than local sensitivity, hence making it applicable beyond the standard
noise-adding mechanisms, e.g. to queries with unbounded or undefined
sensitivity. We demonstrate the versatility of generalized PTR using private
linear regression as a case study. Additionally, we apply our algorithm to
solve an open problem from ''Private Aggregation of Teacher Ensembles (PATE)''
-- privately releasing the entire model with a delicate data-dependent
analysis.
