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Adversarial Robustness on Insertion-Deletion Streams

Elena Gribelyuk, Honghao Lin, David P. Woodruff, Huacheng Yu, Samson Zhou

TL;DR

An exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams is shown, showing that robustness can be achieved using space which is significantly sublinear in $n$.

Abstract

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.

Adversarial Robustness on Insertion-Deletion Streams

TL;DR

An exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams is shown, showing that robustness can be achieved using space which is significantly sublinear in .

Abstract

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 require space linear in . We refute this conjecture, showing that robustness can be achieved using space which is significantly sublinear in . 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 up to a -factor in polylogarithmic space, (2) any symmetric function with an -approximate triangle inequality up to a factor in bits of space, where is the space required to approximate non-robustly; this includes a broad class of functions such as the -norm, the support size , and non-normed losses such as the -estimators, and (3) heavy hitters. For the moment, our algorithm is optimal up to 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.
Paper Structure (21 sections, 19 theorems, 56 equations, 5 figures, 3 algorithms)

This paper contains 21 sections, 19 theorems, 56 equations, 5 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Adversarial robustness.
  3. Insertion-only streams.
  4. Insertion-deletion streams.
  5. Our Contributions
  6. Technical Overview
  7. Technical Overview for F2 Moment Estimation
  8. Estimator, corrector, learner framework.
  9. Robustness and progress.
  10. Recursive estimator.
  11. Two levels and beyond.
  12. Approximate triangle inequality
  13. F2 Algorithm
  14. Algorithm description.
  15. The ${\mathsf{EstLevel}}\xspace$ subroutine.
  16. ...and 6 more sections

Key Result

Theorem 1.3

Given any $\varepsilon\in(0,1)$, there exists an adversarially robust insertion-deletion streaming algorithm on a stream of length $m$ that with high probability, outputs a $(1+\varepsilon)$-approximation to the $F_2$ moment at all times, for the underlying frequency vector of universe size $n$. For

Figures (5)

  • Figure 1: Flowchart for learner, corrector, and estimator framework. The estimator outputs a current estimate, the corrector verifies its accuracy, and the learner updates its internal state ${\mathbf{z}}\xspace'$ based on incorrect estimates.
  • Figure 2: If $\|{\mathbf{z}}\xspace-{\mathbf{z}}\xspace'\|_2^2+\|{\mathbf{z}}\xspace'+{\mathbf{q}}\xspace\|_2^2<(1-\varepsilon)\|{\mathbf{z}}\xspace+{\mathbf{q}}\xspace\|_2^2$, moving ${\mathbf{z}}\xspace'$ towards $-{\mathbf{q}}\xspace$ reduces the distance to ${\mathbf{z}}\xspace$.
  • Figure 3: Example of recursive tree structure on stream with $B=2$ blocks and $H$ levels. Level $H-1$ outputs $\|{\mathbf{z}}\xspace-{\mathbf{z}}\xspace'\|_2^2+\|{\mathbf{z}}\xspace'+{\mathbf{q}}\xspace\|_2^2$ as estimate for $\|{\mathbf{x}}\xspace\|_2^2=\|{\mathbf{z}}\xspace+{\mathbf{q}}\xspace\|_2^2$. Level $H-2$ outputs $\|{\mathbf{z}}\xspace'+{\mathbf{q}}\xspace_0-{\mathbf{z}}\xspace"\|_2^2+\|{\mathbf{z}}\xspace"+{\mathbf{q}}\xspace_1\|_2^2$ as estimate for $\|{\mathbf{z}}\xspace'+{\mathbf{q}}\xspace\|_2^2=\|{\mathbf{z}}\xspace'+{\mathbf{q}}\xspace_0+{\mathbf{q}}\xspace_1\|_2^2$.
  • Figure 4: Adversarially robust $F_2$ norm estimation algorithm on insertion-deletion streams
  • Figure 5: Algorithm for adversarially robust heavy-hitters on insertion-deletion streams.

Theorems & Definitions (36)

  • Definition 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: AMS Algorithm for $F_2$ estimation
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 26 more