We study dynamic algorithms robust to adaptive input generated from sources with bounded capabilities, such as sparsity or limited interaction. For example, we consider robust linear algebraic algorithms when the updates to the input are sparse but given by an adversary with access to a query oracle. We also study robust algorithms in the standard centralized setting, where an adversary queries an algorithm in an adaptive manner, but the number of interactions between the adversary and the algorithm is bounded. We first recall a unified framework of [HKM+20, BKM+22, ACSS23] for answering $Q$ adaptive queries that incurs $\widetilde{\mathcal{O}}(\sqrt{Q})$ overhead in space, which is roughly a quadratic improvement over the na\"{i}ve implementation, and only incurs a logarithmic overhead in query time. Although the general framework has diverse applications in machine learning and data science, such as adaptive distance estimation, kernel density estimation, linear regression, range queries, and point queries and serves as a preliminary benchmark, we demonstrate even better algorithmic improvements for (1) reducing the pre-processing time for adaptive distance estimation and (2) permitting an unlimited number of adaptive queries for kernel density estimation. Finally, we complement our theoretical results with additional empirical evaluations.

翻译：我们研究对来自有限能力源（如稀疏性或有限交互）的自适应输入具有鲁棒性的动态算法。例如，我们考虑当输入更新是稀疏的但由具有查询预言机访问权限的对手提供时的鲁棒线性代数算法。我们还研究标准集中式设置下的鲁棒算法，其中对手以自适应方式查询算法，但对手与算法之间的交互次数是有限的。我们首先回顾[HKM+20, BKM+22, ACSS23]的统一框架，该框架用于回答$Q$个自适应查询，在空间上产生$\widetilde{\mathcal{O}}(\sqrt{Q})$的开销，这比朴素实现大致有二次改进，且仅在查询时间上产生对数级开销。尽管该通用框架在机器学习和数据科学中有广泛应用，例如自适应距离估计、核密度估计、线性回归、范围查询和点查询，并作为初步基准，但我们展示了在以下方面更优的算法改进：(1) 减少自适应距离估计的预处理时间，以及(2) 允许对核密度估计进行无限次自适应查询。最后，我们通过额外的实验评估补充了理论结果。