对抗性鲁棒的近似最远邻居
Adversarially Robust Approximate Furthest Neighbor
AI总结：
本文提出一种在对抗性查询模型下能高效处理近似最远邻居查询的鲁棒数据结构，其查询时间与经典结果匹配并优于现有方法，同时揭示了传统算法在对抗性查询下的脆弱性。
摘要：
我们在适应性查询模型中研究，给定点集P⊆R^d，寻求构造能正确高效回答一系列适应性查询的数据结构。在此模型中，对手观察到数据结构对先前查询q1,…,qi−1的响应，并基于此信息选择下一个查询点qi。此设置捕捉了现代机器学习管道中自然出现的强适应性形式，并排除了许多经典随机技术所假设的无偏查询。我们的关注点是此适应性设置下的最远邻居搜索问题，这是多个学习任务中的基本问题，包括多样性最大化、异常检测、对抗性示例生成等。我们提出了首个对抗性鲁棒的数据结构，用于c-近似最远邻居查询，查询时间约为~O(min(dn^{1/c²}, n^{2/c²}+d))。这与Indyk[SODA'03]在无偏设置下c-近似最远邻居查询时间的n依赖性相匹配，并优于Cherapanamjeri和Nelson[NeurIPS'20]适应性距离估计框架所实现的~O(n+d)查询时间，适用于广泛自然参数范围。为补充此结果，我们提出了针对无偏近似最远邻居算法的对抗性攻击。具体而言，我们证明了Indyk算法的数据结构在对抗性查询下无法维持其保证。
英文摘要：
We work in the adaptive query model, where one is given a point set $P \subset \mathbb{R}^d$ and seeks to construct a data structure that can answer correctly and efficiently a sequence of adaptive queries. In this model, an adversary observes the answers returned by the data structure to previous queries $q_1, \ldots, q_{i-1}$ and, based on this information, chooses the next query point $q_i$. This setting captures strong forms of adaptivity that naturally arise in modern machine learning pipelines, and rules out many classical randomized techniques that assume oblivious queries. Our focus is the problem of furthest neighbor search in this adaptive setting, a fundamental problem in several learning tasks, including diversity maximization, outlier and anomaly detection, adversarial example generation, and more. We present the first adversarially robust data structure for $c$-approximate furthest neighbor queries that achieves query time $\tilde{O}( \min( d n^{1/c^2}, n^{2/c^2} + d))$. This matches the $n$ dependency in the query time of the seminal result by Indyk~[SODA'03] for $c$-approximate furthest neighbor in the oblivious setting, and improves upon the $\tilde{O}(n + d)$ query time achieved via the adaptive distance estimation framework of Cherapanamjeri and Nelson~[NeurIPS'20] for a wide range of natural parameters. To complement this result, we present an adversarial attack against oblivious approximate furthest neighbor algorithms. Specifically, we show that the data structure from the algorithm by Indyk fails to maintain its guarantees against adaptive queries.
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