The Multiplicative Weights Exponential Mechanism (MWEM) is a fundamental iterative framework for private data analysis, with broad applications such as answering $m$ linear queries, or privately solving systems of $m$ linear constraints. However, a critical bottleneck hindering its scalability is the $Θ(m)$ time complexity required to execute the exponential mechanism in each iteration. We introduce a modification to the MWEM framework that improves the per-iteration runtime dependency to $Θ(\sqrt{m})$ in expectation. This is done via a lazy sampling approach to the Report-Noisy-Max mechanism, which we implement efficiently using Gumbel noise and a $k$-Nearest Neighbor data structure. This allows for the rapid selection of the approximate score in the exponential mechanism without an exhaustive linear scan. We apply our accelerated framework to the problems of private linear query release and solving Linear Programs (LPs) under neighboring constraint conditions and low-sensitivity assumptions. Experimental evaluation confirms that our method provides a substantial runtime improvement over classic MWEM.

翻译：乘性权重指数机制（MWEM）是隐私数据分析的一个基础迭代框架，具有广泛的应用，例如回答$m$个线性查询，或在邻近约束条件和低敏感度假设下隐私地求解包含$m$个线性约束的系统。然而，阻碍其可扩展性的一个关键瓶颈是每次迭代中执行指数机制所需的$Θ(m)$时间复杂度。我们引入了一种对MWEM框架的修改，将每次迭代的运行时间依赖在期望上改进为$Θ(\sqrt{m})$。这是通过对Report-Noisy-Max机制采用一种惰性采样方法实现的，我们使用Gumbel噪声和一个$k$近邻数据结构来高效地实现它。这使得能够快速选择指数机制中的近似得分，而无需进行穷举的线性扫描。我们将加速后的框架应用于隐私线性查询发布以及在邻近约束条件和低敏感度假设下求解线性规划（LP）的问题。实验评估证实，我们的方法相比经典MWEM提供了显著的运行时间改进。