  Datasets are often reused to perform multiple statistical analyses in an
adaptive way, in which each analysis may depend on the outcomes of previous
analyses on the same dataset. Standard statistical guarantees do not account
for these dependencies and little is known about how to provably avoid
overfitting and false discovery in the adaptive setting. We consider a natural
formalization of this problem in which the goal is to design an algorithm that,
given a limited number of i.i.d.~samples from an unknown distribution, can
answer adaptively-chosen queries about that distribution.
  We present an algorithm that estimates the expectations of $k$ arbitrary
adaptively-chosen real-valued estimators using a number of samples that scales
as $\sqrt{k}$. The answers given by our algorithm are essentially as accurate
as if fresh samples were used to evaluate each estimator. In contrast, prior
work yields error guarantees that scale with the worst-case sensitivity of each
estimator. We also give a version of our algorithm that can be used to verify
answers to such queries where the sample complexity depends logarithmically on
the number of queries $k$ (as in the reusable holdout technique).
  Our algorithm is based on a simple approximate median algorithm that
satisfies the strong stability guarantees of differential privacy. Our
techniques provide a new approach for analyzing the generalization guarantees
of differentially private algorithms.
