Learning Nearest-Neighbor Maps from Adaptive Queries

2026-08-07

AI Summary

This work investigates the optimal query complexity for reconstructing a hidden point set via adaptive nearest neighbor queries in arbitrary normed spaces. It introduces a novel approach integrating geometric analysis, adaptive querying strategies, randomized dimensionality reduction, and kissing number theory, establishing—for the first time—the tight worst-case query complexity bound of Θ(nκ) under general norms. A key insight is the fundamental distinction between spherical and ball-shaped query domains: while reconstruction within a Euclidean ball admits complexity O(min(n,d)), exponential queries are necessary for balls and cones in general norms. The study also refines dimensionality reduction techniques on spheres, underscoring how intrinsic geometric structure critically governs the difficulty of learning.

Abstract

We study the problem of learning nearest-neighbor maps from adaptive queries, which is equivalent to the following problem of reconstructing a hidden set $H$ via a nearest-neighbor query oracle. Let $K \subset \mathbb{R}^d$ be a compact domain in a normed space $(\mathbb{R}^d,\| \cdot\|)$ and let $H \subset K$ be a hidden set of $n$ points. Upon querying $q \in K$, the oracle returns some $h \in H$ with minimum distance from $q$. How many queries are required to exactly recover $H$? Previous work has studied this question in specific domains, namely the Boolean hypercube and the $\ell_2$-unit sphere. We generalize previous work and prove the tight worst-case query complexity bound of $Θ(nκ)$, where $κ$ is the kissing number of the underlying norm. In the Euclidean norm, obtaining tight asymptotic bounds on $κ$ is a significant open question, although it is known that $κ= \exp(Θ(d))$. Our second set of results shows that an exponential dependence on $d$ is required even in natural Euclidean domains: $\exp(Ω(d))$ queries are needed in the ball, even when $n=2$, and $n\exp(Ω(d))$ queries are needed in the cone. Lastly, we prove a sharper upper bound in the Euclidean sphere. Here, $d$ can be replaced by $\min(n,d)$ via a dimension reduction preprocessing step. This is a randomized version of a procedure due to Prabhu-Woodruff (ICML 2024) where we improve the query complexity from $O(nd)$ to $O(\min(n,d))$. This reveals a striking contrast between the sphere and the ball: when $n = O(1)$, the sphere admits an $O(1)$ query algorithm, whereas the ball requires $\exp(Ω(d))$.

Problem

• nearest-neighbor queries
• hidden set reconstruction
• adaptive queries
• kissing number
• query complexity

Innovation

• nearest-neighbor queries
• query complexity
• kissing number
• dimension reduction
• adaptive learning