Is Randomness Necessary for Adaptive Data Analysis?

Edith Cohen, Haim Kaplan, Yishay Mansour, Shay Sapir, Uri Stemmer

The Adaptive Data Analysis (ADA) problem formalizes the challenge of preventing false discovery and overfitting when a dataset is repeatedly reused. Formally, our input is a dataset containing $n$ i.i.d.\ samples from an unknown distribution $P$ over a domain $X$, and our goal is to answer a sequence of $k$ adaptively chosen statistical queries with respect to $P$. The main question is how many queries we can support (i.e., how large $k$ can be), primarily as a function of the number of samples $n$. This question has been intensively studied and is relatively well-understood for randomized mechanisms: there are computationally efficient mechanisms that support $k \approx n^2$ queries, and no computationally efficient mechanism can answer $k \gg n^2$ queries. In this paper, we address a fundamental question: is randomness necessary for ADA? Despite a decade of work on ADA, this question remains open. A folklore observation dating back to the initial works on ADA is that randomness is {\em not} necessary when the analyst is computationally bounded. Yet, the necessity of randomness against computationally unbounded analysts has remained elusive. Our main contribution resolves this gap in the information-theoretic setting. Perhaps surprisingly, we show that randomness is strictly necessary to answer a non-trivial number of adaptive queries: when the analyst is unbounded, any deterministic mechanism can be forced to fail after just $k = \tilde{O}(n)$ queries.