基于密集-稀疏权衡的对抗性鲁棒流

Omri Ben-Eliezer, T. Eden, Krzysztof Onak
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引用次数: 18

摘要

如果流算法即使在存在自适应对手的情况下也能保证正确执行,那么它就是对抗鲁棒的。近年来,人们开发了一些复杂的框架来增强经典流算法的鲁棒性。该领域的一个主要开放问题是,对于允许插入和删除的旋转门流模型下的矩估计问题,是否存在有效的对抗鲁棒算法。到目前为止,使用基于差分隐私(DP)的技术实现的长度为$m$的流的最著名的空间复杂度为$\tilde{O}(m^{1/2})$阶,用于计算具有高常数概率的常数因子近似。在这项工作中,我们提出了一种新的简单方法,通过在两种不同的制度之间交替来跟踪矩:稀疏制度,我们可以明确地保持当前的频率矢量并使用标准的稀疏恢复技术;密集制度,我们利用现有的基于dp的鲁棒化框架。使用我们的技术获得的结果打破了先前任何固定$p$的$m^{1/2}$障碍。更具体地说,我们对于$F_2$ -估计的空间复杂度是$\tilde{O}(m^{2/5})$,对于$F_0$ -估计,即计算不同元素的数量,它是$\tilde O(m^{1/3})$。所有现有的鲁棒性框架都有其空间复杂度乘法依赖于一个参数$\lambda$称为流问题的\emph{翻转数},其中$\lambda = m$在旋转门矩估计中。这些框架中最著名的依赖关系(对于常数因子近似)是$\tilde{O}(\lambda^{1/2})$阶,并且对于某些问题它是已知的紧密关系。同样,我们的方法打破了这个障碍,实现了$F_p$ -估计的顺序$\tilde{O}(\lambda^{1/2 - c(p)})$的依赖,其中$c(p)>0$只依赖于$p$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Adversarially Robust Streaming via Dense-Sparse Trade-offs
A streaming algorithm is adversarially robust if it is guaranteed to perform correctly even in the presence of an adaptive adversary. Recently, several sophisticated frameworks for robustification of classical streaming algorithms have been developed. One of the main open questions in this area is whether efficient adversarially robust algorithms exist for moment estimation problems under the turnstile streaming model, where both insertions and deletions are allowed. So far, the best known space complexity for streams of length $m$, achieved using differential privacy (DP) based techniques, is of order $\tilde{O}(m^{1/2})$ for computing a constant-factor approximation with high constant probability. In this work, we propose a new simple approach to tracking moments by alternating between two different regimes: a sparse regime, in which we can explicitly maintain the current frequency vector and use standard sparse recovery techniques, and a dense regime, in which we make use of existing DP-based robustification frameworks. The results obtained using our technique break the previous $m^{1/2}$ barrier for any fixed $p$. More specifically, our space complexity for $F_2$-estimation is $\tilde{O}(m^{2/5})$ and for $F_0$-estimation, i.e., counting the number of distinct elements, it is $\tilde O(m^{1/3})$. All existing robustness frameworks have their space complexity depend multiplicatively on a parameter $\lambda$ called the \emph{flip number} of the streaming problem, where $\lambda = m$ in turnstile moment estimation. The best known dependence in these frameworks (for constant factor approximation) is of order $\tilde{O}(\lambda^{1/2})$, and it is known to be tight for certain problems. Again, our approach breaks this barrier, achieving a dependence of order $\tilde{O}(\lambda^{1/2 - c(p)})$ for $F_p$-estimation, where $c(p)>0$ depends only on $p$.
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