We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While robust algorithms exist for insertion-only streams with only a polylogarithmic overhead in memory over non-robust algorithms, it was widely conjectured that turnstile streams of length polynomial in the universe size $n$ require space linear in $n$. We refute this conjecture, showing that robustness can be achieved using space which is significantly sublinear in $n$. Our framework combines multiple linear sketches in a novel estimator-corrector-learner framework, yielding the first insertion-deletion algorithms that approximate: (1) the second moment $F_2$ up to a $(1+\varepsilon)$-factor in polylogarithmic space, (2) any symmetric function $\cal{F}$ with an $\mathcal{O}(1)$-approximate triangle inequality up to a $2^{\mathcal{O}(C)}$ factor in $\tilde{\mathcal{O}}(n^{1/C}) \cdot S(n)$ bits of space, where $S$ is the space required to approximate $\cal{F}$ non-robustly; this includes a broad class of functions such as the $L_1$-norm, the support size $F_0$, and non-normed losses such as the $M$-estimators, and (3) $L_2$ heavy hitters. For the $F_2$ moment, our algorithm is optimal up to $\textrm{poly}((\log n)/\varepsilon)$ factors. Given the recent results of Gribelyuk et al. (STOC, 2025), this shows an exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams.

翻译：我们研究了插入-删除（旋转门）流上的对抗鲁棒算法，其中未来的更新可能依赖于过去的算法输出。尽管对于仅插入流存在鲁棒算法，其内存开销仅比非鲁棒算法多出多对数级别，但学界普遍猜想：当流长度与全集大小$n$呈多项式关系时，旋转门流需要与$n$线性相关的空间。我们否定了这一猜想，证明了鲁棒性可以通过显著小于$n$的空间实现。我们的框架通过新颖的估计器-校正器-学习器架构结合多个线性草图，首次实现了在插入-删除流中近似计算以下目标的算法：（1）在多项式对数空间内以$(1+\varepsilon)$因子近似二阶矩$F_2$；（2）对于满足$\mathcal{O}(1)$近似三角不等式的任意对称函数$\cal{F}$，在$\tilde{\mathcal{O}}(n^{1/C}) \cdot S(n)$比特空间内以$2^{\mathcal{O}(C)}$因子近似，其中$S$为非鲁棒近似$\cal{F}$所需的空间；此类函数涵盖$L_1$范数、支持度$F_0$以及$M$估计量等非范数损失函数；（3）$L_2$重击者检测。对于$F_2$矩，我们的算法在$\textrm{poly}((\log n)/\varepsilon)$因子内达到最优。结合Gribelyuk等人（STOC 2025）的最新研究成果，这揭示了在旋转门流中实现对抗鲁棒性时，线性草图与非线性草图之间存在指数级分离。