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Adaptively Robust Resettable Streaming

Edith Cohen, Elena Gribelyuk, Jelani Nelson, Uri Stemmer

TL;DR

The paper tackles adaptively robust streaming in the resettable model, where updates can increment or reset, and adversaries may adapt based on past outputs. It introduces dedicated robust sketches that achieve prefix-max accuracy with polylogarithmic space in the stream length, for cardinality, sum, and Bernstein statistics, by integrating differential privacy (DP) with the Binary Tree Mechanism for continual observation. The authors show that prior wrappers and linear/union-composable sketches are insufficient under adaptivity, and they provide both fixed-rate and adjustable-rate robust designs that maintain accuracy despite adaptive inputs. They further extend robustness to Bernstein statistics through a reduction that combines robust sum and cardinality sketches with independent randomness in the output mappings, yielding a unified framework for a broad class of sublinear statistics. This work enables reliable active-resource monitoring and machine unlearning in adversarial streaming settings, offering polylog-space solutions with formal prefix-max guarantees and outlining directions for future improvement on super-linear statistics and broader ReLU models.

Abstract

We study algorithms in the resettable streaming model, where the value of each key can either be increased or reset to zero. The model is suitable for applications such as active resource monitoring with support for deletions and machine unlearning. We show that all existing sketches for this model are vulnerable to adaptive adversarial attacks that apply even when the sketch size is polynomial in the length of the stream. To overcome these vulnerabilities, we present the first adaptively robust sketches for resettable streams that maintain polylogarithmic space complexity in the stream length. Our framework supports (sub) linear statistics including $L_p$ moments for $p\in[0,1]$ (in particular, Cardinality and Sum) and Bernstein statistics. We bypass strong impossibility results known for linear and composable sketches by designing dedicated streaming sketches robustified via Differential Privacy. Unlike standard robustification techniques, which provide limited benefits in this setting and still require polynomial space in the stream length, we leverage the Binary Tree Mechanism for continual observation to protect the sketch's internal randomness. This enables accurate prefix-max error guarantees with polylogarithmic space.

Adaptively Robust Resettable Streaming

TL;DR

The paper tackles adaptively robust streaming in the resettable model, where updates can increment or reset, and adversaries may adapt based on past outputs. It introduces dedicated robust sketches that achieve prefix-max accuracy with polylogarithmic space in the stream length, for cardinality, sum, and Bernstein statistics, by integrating differential privacy (DP) with the Binary Tree Mechanism for continual observation. The authors show that prior wrappers and linear/union-composable sketches are insufficient under adaptivity, and they provide both fixed-rate and adjustable-rate robust designs that maintain accuracy despite adaptive inputs. They further extend robustness to Bernstein statistics through a reduction that combines robust sum and cardinality sketches with independent randomness in the output mappings, yielding a unified framework for a broad class of sublinear statistics. This work enables reliable active-resource monitoring and machine unlearning in adversarial streaming settings, offering polylog-space solutions with formal prefix-max guarantees and outlining directions for future improvement on super-linear statistics and broader ReLU models.

Abstract

We study algorithms in the resettable streaming model, where the value of each key can either be increased or reset to zero. The model is suitable for applications such as active resource monitoring with support for deletions and machine unlearning. We show that all existing sketches for this model are vulnerable to adaptive adversarial attacks that apply even when the sketch size is polynomial in the length of the stream. To overcome these vulnerabilities, we present the first adaptively robust sketches for resettable streams that maintain polylogarithmic space complexity in the stream length. Our framework supports (sub) linear statistics including moments for (in particular, Cardinality and Sum) and Bernstein statistics. We bypass strong impossibility results known for linear and composable sketches by designing dedicated streaming sketches robustified via Differential Privacy. Unlike standard robustification techniques, which provide limited benefits in this setting and still require polynomial space in the stream length, we leverage the Binary Tree Mechanism for continual observation to protect the sketch's internal randomness. This enables accurate prefix-max error guarantees with polylogarithmic space.
Paper Structure (86 sections, 26 theorems, 174 equations, 2 figures)

This paper contains 86 sections, 26 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Frequency statistics.
  3. Prefix-max Accuracy Benchmark.
  4. The Adaptive Streaming Model
  5. Main Contribution
  6. Review of Prior Work
  7. Overview and Road map
  8. Robust Resettable $\ell_0$ estimation
  9. Standard Cardinality with Deletions Sketch
  10. Vulnerability to Adaptive Attacks
  11. The Re-Insertion Attack (Insertion-Only)
  12. Analysis.
  13. Sample-and-Delete Attack (resettable)
  14. Analysis.
  15. Intuition.
  16. ...and 71 more sections

Key Result

Theorem 1.2

For any $\varepsilon,\delta \in (0,1)$ and a resettable adaptive stream with $T_{\mathrm{Inc}}$ increments,The bounds allow for an arbitrary number of resets. there exist sketches for cardinality, sum, and Bernstein statistics of size $k=O(\mathrm{poly}(\varepsilon^{-1}, \log(T_{\mathrm{Inc}}/\delta

Figures (2)

  • Figure 1: Element Mapping $\mathcal{M}(x, \Delta)$
  • Figure 2: Stream Element Mapping $\mathcal{M^*}(x, \Delta)$

Theorems & Definitions (61)

  • Theorem 1.2: Robust resettable cardinality, sum, and Bernstein; informal
  • Lemma 2.1: Guarantees for non-adaptive streams
  • Remark 2.2: Prefix-max error guarantee via rate adjustments for the standard sketch
  • Claim 2.3: Failure of standard estimator for \ref{['alg:basic-card-fixedp']}
  • Theorem 2.4: Robust Cardinality with Deletions
  • Theorem 3.1: Robust Resettable Sum Sketch
  • Definition 3.2: Entry-Threshold Formulation
  • Definition 4.0: Bernstein Functions
  • Theorem 4.0: Robust Resettable Sketches for Bernstein Statistics
  • Lemma 2.1: Robust sampling via DP generalization
  • ...and 51 more