  Differentially Private algorithms often need to select the best amongst many
candidate options. Classical works on this selection problem require that the
candidates' goodness, measured as a real-valued score function, does not change
by much when one person's data changes. In many applications such as
hyperparameter optimization, this stability assumption is much too strong. In
this work, we consider the selection problem under a much weaker stability
assumption on the candidates, namely that the score functions are
differentially private. Under this assumption, we present algorithms that are
near-optimal along the three relevant dimensions: privacy, utility and
computational efficiency.
  Our result can be seen as a generalization of the exponential mechanism and
its existing generalizations. We also develop an online version of our
algorithm, that can be seen as a generalization of the sparse vector technique
to this weaker stability assumption. We show how our results imply better
algorithms for hyperparameter selection in differentially private machine
learning, as well as for adaptive data analysis.
