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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通过密集——稀疏权衡的对抗性强流

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，那么它就是具有对抗性的鲁棒性。最近，已经开发了几个用于增强经典流算法的复杂框架。该领域的主要开放问题之一是，在允许插入和删除的旋转门流模型下，是否存在有效的对抗性鲁棒算法来解决矩估计问题。到目前为止，使用基于差分隐私 (DP) 的技术实现的长度为 $m$ 的流的最广为人知的空间复杂度是 $\tilde{O}(m^{1/2})$ 用于计算常数 -具有高常数概率的因子近似。在这项工作中，我们提出了一种新的简单方法，通过在两种不同的机制之间交替来跟踪矩：稀疏机制，在其中我们可以明确地维护当前频率向量并使用标准的稀疏恢复技术，以及在其中我们利用现有的基于 DP 的稳健化框架的密集机制。使用我们的技术获得的结果打破了任何固定 $p$ 之前的 $m^{1/2}$ 障碍。更具体地说，我们对 $F_2$-estimation 的空间复杂度是 $\tilde{O}(m^{2/5})$ 而对于 $F_0$-estimation，即计算不同元素的数量，它是 $\波浪号 O(m^{1/3})$。所有现有的鲁棒性框架的空间复杂度都乘性地取决于称为流问题的 \emph{flip number} 的参数 $\lambda$，其中 $\lambda = m$ 在转门力矩估计中。这些框架中最著名的依赖关系（对于常数因子近似）是 $\tilde{O}(\lambda^{1/2})$，众所周知，它对某些问题很紧。同样，我们的方法打破了这个障碍，实现了对 $F_p$-estimation 的顺序 $\tilde{O}(\lambda^{1/2 - c(p)})$ 的依赖，其中 $c(p) > 0 $ 仅取决于 $p$。