On Adaptive Distance Estimation

We provide a static data structure for distance estimation which supports {\it adaptive} queries. Concretely, given a dataset $X = \{x_i\}_{i = 1}^n$ of $n$ points in $\mathbb{R}^d$ and $0 < p \leq 2$, we construct a randomized data structure with low memory consumption and query time which, when later given any query point $q \in \mathbb{R}^d$, outputs a $(1+\epsilon)$-approximation of $\lVert q - x_i \rVert_p$ with high probability for all $i\in[n]$. The main novelty is our data structure's correctness guarantee holds even when the sequence of queries can be chosen adaptively: an adversary is allowed to choose the $j$th query point $q_j$ in a way that depends on the answers reported by the data structure for $q_1,\ldots,q_{j-1}$. Previous randomized Monte Carlo methods do not provide error guarantees in the setting of adaptively chosen queries. Our memory consumption is $\tilde O((n+d)d/\epsilon^2)$, slightly more than the $O(nd)$ required to store $X$ in memory explicitly, but with the benefit that our time to answer queries is only $\tilde O(\epsilon^{-2}(n + d))$, much faster than the naive $\Theta(nd)$ time obtained from a linear scan in the case of $n$ and $d$ very large. Here $\tilde O$ hides $\log(nd/\epsilon)$ factors. We discuss applications to nearest neighbor search and nonparametric estimation. Our method is simple and likely to be applicable to other domains: we describe a generic approach for transforming randomized Monte Carlo data structures which do not support adaptive queries to ones that do, and show that for the problem at hand, it can be applied to standard nonadaptive solutions to $\ell_p$ norm estimation with negligible overhead in query time and a factor $d$ overhead in memory.