Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

2025-03-25

AI Summary

This work investigates dimension lower bounds and adversarial robustness of linear sketches for integer data streams. Addressing classical tasks—including frequency moment estimation, operator norm approximation, and compressed sensing—the paper establishes, for the first time, optimal sketch-length lower bounds under discrete (integer) inputs. To achieve this, the authors lift existing real-valued sketch lower bounds to the integer setting, introduce orthogonal lattice smoothing as a preprocessing technique, and construct adaptive adversarial attacks. Crucially, they prove that no integer linear sketch can robustly approximate arbitrary $L_p$ norms—even approximately—thereby refuting the existence of such robust sketches. Their results demonstrate that, under the insertion/deletion stream model, every integer linear sketch fails to provide adversarial-robust $L_p$-norm approximation. This resolves a long-standing open problem in streaming algorithms and sketching theory.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Lifting linear sketch bounds from real to integer inputs
• Proving no adversarially robust linear sketch exists for Lp norms
• Developing techniques for optimal sketching lower bounds

Innovation

• Lifting real-valued sketches to integer-valued bounds
• Adversarial attack on integer-valued linear sketches
• Pre-processing for smooth orthogonal lattice