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原文摘要

In this paper we study answering counting and other linear queries online under differential privacy constraints. Recent work by Nikolov, Tang, and Ullman [STOC 2026] gave the first online algorithm that answers \emph{obliviously chosen} online linear queries privately, and achieves poly-logarithmic competitiveness against an optimal offline factorization mechanism. This also implied poly-logarithmic competitiveness against arbitrary private offline mechanisms when the private dataset is large enough. The error guarantees of their algorithm, however, can fail when the queries are adaptive. We give a new algorithm that also achieves poly-logarithmic competitiveness, while being able to handle \emph{adaptively chosen} queries. This implies that there is no large separation between the optimal error bounds for privately answering, respectively, offline linear queries, obliviously chosen online linear queries, and adaptively chosen online linear queries. Our algorithm combines the online factorization approach from prior work with a novel layered Euclidean iterative database construction, presenting a new way to use factorization in private algorithms. We also give a lower bound that shows that some poly-logarithmic factor in the competitive ratio is unavoidable. Our lower bound first shows that any interactive private mechanism implies a deterministic online factorization algorithm, and then gives a new lower bound for the competitive ratio of any deterministic online factorization algorithm.