Robust Algorithms on Adaptive Inputs from Bounded Adversaries

We study dynamic algorithms robust to adaptive inputs 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 inputs 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. Together, we provide a unified framework 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. Our 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. Surprisingly, we show that these novel subroutines for each of these problems can be generally combined with the elegant use of differential privacy to hide the internal randomness of various subroutines, leading to robust algorithms across these different settings. In addition, 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.