  In adaptive data analysis, the user makes a sequence of queries on the data,
where at each step the choice of query may depend on the results in previous
steps. The releases are often randomized in order to reduce overfitting for
such adaptively chosen queries. In this paper, we propose a minimax framework
for adaptive data analysis. Assuming Gaussianity of queries, we establish the
first sharp minimax lower bound on the squared error in the order of
$O(\frac{\sqrt{k}\sigma^2}{n})$, where $k$ is the number of queries asked, and
$\sigma^2/n$ is the ordinary signal-to-noise ratio for a single query. Our
lower bound is based on the construction of an approximately least favorable
adversary who picks a sequence of queries that are most likely to be affected
by overfitting. This approximately least favorable adversary uses only one
level of adaptivity, suggesting that the minimax risk for 1-step adaptivity
with k-1 initial releases and that for $k$-step adaptivity are on the same
order. The key technical component of the lower bound proof is a reduction to
finding the convoluting distribution that optimally obfuscates the sign of a
Gaussian signal. Our lower bound construction also reveals a transparent and
elementary proof of the matching upper bound as an alternative approach to
Russo and Zou (2015), who used information-theoretic tools to provide the same
upper bound. We believe that the proposed framework opens up opportunities to
obtain theoretical insights for many other settings of adaptive data analysis,
which would extend the idea to more practical realms.
