Distributionally Robust Search: The Power of Information and the Limits of Moments
We study binary search for a target drawn from a distribution on $n$ ordered elements. The cost is the expected number of comparison queries. In our distributionally robust framework, the algorithm knows only an uncertainty set containing the distribution and minimizes its worst-case approximation ratio against an optimal strategy that knows the distribution. Collections of uncertainty sets capture different forms of information: quantiles, moments, general linear measurements, or approximate CDFs. This framework quantifies how much and what form of information is needed for near-optimal search. Our main result separates moments from general linear measurements. Although $n-1$ moments determine the distribution exactly, for every constant $δ\in(0,1)$, the first $n^{1-δ}$ exact moments can be as uninformative for search as no information: the worst-case approximation ratio remains $Ω(\log n)$. In contrast, for every constant $\varepsilon\in(0,1)$, $O(n^\varepsilon\log^4 n)$ suitably chosen linear measurements suffice for an $O(1/\varepsilon)$-approximation. The limitation lies in the measurements chosen, rather than linearity itself. Using LP duality and polynomial bounds of Rakhmanov and Bernstein, we construct distributions with identical moments and mass concentrated at different positions. For the upper bound, we combine dyadic sketches and Johnson--Lindenstrauss embeddings with a truncated median-search algorithm that converts uniform CDF approximation into search guarantees. Measuring information in bits, we prove a lower bound of $Ω(1/\varepsilon)$ on the approximation ratio for summaries of any form using at most $\frac12 n^\varepsilon\log n$ bits, for every sufficiently small constant $\varepsilon>0$. Quantiles match this within constant factors. For Kolmogorov balls of radius $n^{-\varepsilon}$ the optimal approximation ratio is $Θ(1/\varepsilon)$.