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Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

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

TL;DR

This work introduces a lifting framework that converts optimal linear-sketch lower bounds from continuous, real-valued inputs to discrete, integer-input settings in data streams. The authors leverage lattice theory and discrete Gaussian distributions to relate discrete and continuous sketches, enabling optimal lower bounds for problems such as $L_p$ estimation, operator and Ky Fan norms, eigenvalue estimation, PSD testing, and compressed sensing under discrete inputs. They also develop a substantial adversarial attack against adaptive, turnstile streaming with finite-precision integer sketches, proving that no sublinear-space, adversarially robust linear sketch can exist for $L_p$-estimation in this model. The combination of lifting, discrete Gaussian techniques, and adaptive attacks resolves key open questions from Banff and STOC/FOCS workshops, and provides a unified, quantitative picture of the limitations of sublinear-space sketching in discrete, possibly adversarial environments.

Abstract

We introduce a novel technique for ``lifting'' dimension lower bounds for linear sketches in the real-valued setting to dimension lower bounds for linear sketches with polynomially-bounded integer entries when the input is a polynomially-bounded integer vector. Using this technique, we obtain the first optimal sketching lower bounds for discrete inputs in a data stream, for classical problems such as approximating the frequency moments, estimating the operator norm, and compressed sensing. Additionally, we lift the adaptive attack of Hardt and Woodruff (STOC, 2013) for breaking any real-valued linear sketch via a sequence of real-valued queries, and show how to obtain an attack on any integer-valued linear sketch using integer-valued queries. This shows that there is no linear sketch in a data stream with insertions and deletions that is adversarially robust for approximating any $L_p$ norm of the input, resolving a central open question for adversarially robust streaming algorithms. To do so, we introduce a new pre-processing technique of independent interest which, given an integer-valued linear sketch, increases the dimension of the sketch by only a constant factor in order to make the orthogonal lattice to its row span smooth. This pre-processing then enables us to leverage results in lattice theory on discrete Gaussian distributions and reason that efficient discrete sketches imply efficient continuous sketches. Our work resolves open questions from the Banff '14 and '17 workshops on Communication Complexity and Applications, as well as the STOC '21 and FOCS '23 workshops on adaptivity and robustness.

Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

TL;DR

This work introduces a lifting framework that converts optimal linear-sketch lower bounds from continuous, real-valued inputs to discrete, integer-input settings in data streams. The authors leverage lattice theory and discrete Gaussian distributions to relate discrete and continuous sketches, enabling optimal lower bounds for problems such as estimation, operator and Ky Fan norms, eigenvalue estimation, PSD testing, and compressed sensing under discrete inputs. They also develop a substantial adversarial attack against adaptive, turnstile streaming with finite-precision integer sketches, proving that no sublinear-space, adversarially robust linear sketch can exist for -estimation in this model. The combination of lifting, discrete Gaussian techniques, and adaptive attacks resolves key open questions from Banff and STOC/FOCS workshops, and provides a unified, quantitative picture of the limitations of sublinear-space sketching in discrete, possibly adversarial environments.

Abstract

We introduce a novel technique for ``lifting'' dimension lower bounds for linear sketches in the real-valued setting to dimension lower bounds for linear sketches with polynomially-bounded integer entries when the input is a polynomially-bounded integer vector. Using this technique, we obtain the first optimal sketching lower bounds for discrete inputs in a data stream, for classical problems such as approximating the frequency moments, estimating the operator norm, and compressed sensing. Additionally, we lift the adaptive attack of Hardt and Woodruff (STOC, 2013) for breaking any real-valued linear sketch via a sequence of real-valued queries, and show how to obtain an attack on any integer-valued linear sketch using integer-valued queries. This shows that there is no linear sketch in a data stream with insertions and deletions that is adversarially robust for approximating any norm of the input, resolving a central open question for adversarially robust streaming algorithms. To do so, we introduce a new pre-processing technique of independent interest which, given an integer-valued linear sketch, increases the dimension of the sketch by only a constant factor in order to make the orthogonal lattice to its row span smooth. This pre-processing then enables us to leverage results in lattice theory on discrete Gaussian distributions and reason that efficient discrete sketches imply efficient continuous sketches. Our work resolves open questions from the Banff '14 and '17 workshops on Communication Complexity and Applications, as well as the STOC '21 and FOCS '23 workshops on adaptivity and robustness.

Paper Structure

This paper contains 28 sections, 48 theorems, 135 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions: Lifting Framework and Applications
  3. Our Contributions: Streaming against Adaptive Adversaries
  4. Technical Overview
  5. Smoothing parameter and successive minima.
  6. Pre-processing to upper bound the successive minima.
  7. A better pre-processing.
  8. Cell lemma.
  9. Adversarial attack for $L_2$ estimation.
  10. Preliminaries
  11. Continuous and Discrete Gaussians
  12. Lattice Theory
  13. Lifting Framework
  14. Adversarially Robust Streaming: Lp Estimation using Linear Sketches
  15. Preliminaries for Adversarial Attack
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let ${\mathcal{I}}\xspace$ denote an arbitrary distribution on $\mathbb{Z}^n$, ${\mathbf{S}}\xspace^\top{\mathbf{S}}\xspace$ be a full-rank covariance matrix, and $\delta\ge\frac{1}{\mathop{\mathrm{poly}}\limits(n)}$ be a failure parameter. Let $f$ be a $\frac{\delta}{3}$-smooth function on the dist There exists a post-processing function $h$ with another sketching matrix ${\mathbf{A}}\xspace' \in

Figures (1)

  • Figure 1: Algorithm that creates an adaptive attack using a turnstile data stream.

Theorems & Definitions (95)

  • Theorem 1.1: Informal, see \ref{['thm:thm:lifting']}
  • Definition 1.2: Integer sketch
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6: Adaptive attack for $L_p$ estimation for $p > 1$
  • Proposition 2.1
  • Definition 2.2: Continuous Gaussian
  • Definition 2.3: Discrete Gaussian
  • Lemma 2.5
  • ...and 85 more