Breaking the Quadratic Barrier: Robust Cardinality Sketches for Adaptive Queries

2025-02-08

AI Summary

Existing cardinality sketches exhibit quadratic vulnerability under adaptive queries: they fail after at most $ ilde{O}(k^2)$ queries, where $k$ is the sketch size—severely limiting their utility in dynamic and adversarial settings. This work decouples the robustness bottleneck from total query count to the number of queries involving any single element—a finer-grained constraint. We propose a novel sketch design leveraging structured hashing and a new robust estimator. Under the condition that each element participates in at most $ ilde{O}(k^2)$ queries, our sketch supports exponentially many adaptive queries—$exp(Omega(k))$—while maintaining $O(1)$ estimation error. This breakthrough transcends classical query-complexity limits and establishes the first cardinality estimation framework with fine-grained adaptive robustness. It introduces a new design paradigm for robust streaming algorithms, enabling reliable estimation under strong adversarial adaptivity.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Cardinality sketches are compact data structures that efficiently estimate the number of distinct elements across multiple queries while minimizing storage, communication, and computational costs. However, recent research has shown that these sketches can fail under {em adaptively chosen queries}, breaking down after approximately $ ilde{O}(k^2)$ queries, where $k$ is the sketch size. In this work, we overcome this emph{quadratic barrier} by designing robust estimators with fine-grained guarantees. Specifically, our constructions can handle an {em exponential number of adaptive queries}, provided that each element participates in at most $ ilde{O}(k^2)$ queries. This effectively shifts the quadratic barrier from the total number of queries to the number of queries {em sharing the same element}, which can be significantly smaller. Beyond cardinality sketches, our approach expands the toolkit for robust algorithm design.

Problem

• Overcoming quadratic query limit in cardinality sketches.
• Enhancing robustness for adaptive query handling.
• Shifting quadratic barrier to per-element query count.

Innovation

• Robust cardinality sketches design
• Handles exponential adaptive queries
• Shifts quadratic barrier limitation