AI 中文总结
本文研究从自适应查询学习最近邻映射的问题，推广相关研究得到查询复杂度紧界，证明欧氏域中查询次数对$d$呈指数依赖，并改进欧氏球域的查询复杂度。
AI 中文摘要
我们研究从自适应查询中学习最近邻映射的问题，这等价于通过最近邻查询神谕重构隐藏集合$H$的问题。设$K \subset \mathbb{R}^d$是赋范空间$(\mathbb{R}^d,\\| \cdot\\|)$中的紧域，$H \subset K$是含$n$个点的隐藏集合。查询$q \in K$时，神谕返回$H$中与$q$距离最小的点$h$，我们需要确定准确恢复$H$所需的查询次数。已有研究在布尔超立方体和$\ell_2$单位球等特定域中探讨过该问题，本文将其推广并证明了最坏情况下查询复杂度的紧界为$\Theta(n\kappa)$，其中$\kappa$是基础范数的吻数。对于欧氏范数，尽管已知$\kappa = \exp(\Theta(d))$，但获取其紧渐近界仍是重要开放问题。本文第二组结果表明，即使在自然欧氏域中也需要指数级依赖$d$：即使$n=2$，球域中也需要$\exp(\Omega(d))$次查询，锥域中则需要$n\exp(\Omega(d))$次查询。最后，我们在欧氏球域中证明了更精确的上界：通过降维预处理步骤，可将$d$替换为$\min(n,d)$，这是Prabhu-Woodruff（ICML 2024）提出的方法的随机版本，我们将查询复杂度从$O(nd)$改进为$O(\min(n,d))$。这揭示了球域与球之间的显著差异：当$n = O(1)$时，球域存在$O(1)$次查询的算法，而球域需要$\exp(\Omega(d))$次查询。
英文摘要
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 $\exp(Ω(d))$.
