Efficient Algorithms for Adversarially Robust Approximate Nearest Neighbor Search

We study the Approximate Nearest Neighbor (ANN) problem under a powerful adaptive adversary that controls both the dataset and a sequence of $Q$ queries. Primarily, for the high-dimensional regime of $d = \omega(\sqrt{Q})$, we introduce a sequence of algorithms with progressively stronger guarantees. We first establish a novel connection between adaptive security and \textit{fairness}, leveraging fair ANN search to hide internal randomness from the adversary with information-theoretic guarantees. To achieve data-independent performance, we then reduce the search problem to a robust decision primitive, solved using a differentially private mechanism on a Locality-Sensitive Hashing (LSH) data structure. This approach, however, faces an inherent $\sqrt{n}$ query time barrier. To break the barrier, we propose a novel concentric-annuli LSH construction that synthesizes these fairness and differential privacy techniques. The analysis introduces a new method for robustly releasing timing information from the underlying algorithm instances and, as a corollary, also improves existing results for fair ANN. In addition, for the low-dimensional regime $d = O(\sqrt{Q})$, we propose specialized algorithms that provide a strong ``for-all'' guarantee: correctness on \textit{every} possible query with high probability. We introduce novel metric covering constructions that simplify and improve prior approaches for ANN in Hamming and $\ell_p$ spaces.