Tight Bounds for Answering Adaptively Chosen Concentrated Queries

2025-07-18

AI Summary

This work investigates the utility limits of concentrated queries in adaptive data analysis under sample-correlated data. While existing frameworks suffer significant performance degradation on correlated data, we establish—for the first time—that, under natural assumptions, the number of adaptively answerable concentrated queries is inherently bounded by $O(n)$, drastically lower than the $O(n^2)$ bound achievable under independence; this gap is fundamental and unavoidable. We further design a simple, implementable algorithm that provably achieves this $O(n)$ bound, thereby attaining theoretical optimality. Our analysis integrates probabilistic reasoning, concentration inequalities, and formal modeling of adaptive query access, accompanied by rigorous complexity characterization. The results uncover a fundamental limitation imposed by data correlation on adaptive differentially private analysis, providing a critical theoretical benchmark for future framework design.

Technology Category

• Data Mining & Knowledge Management: Intelligent Query Processing
• Machine Learning: Privacy
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Most work on adaptive data analysis assumes that samples in the dataset are independent. When correlations are allowed, even the non-adaptive setting can become intractable, unless some structural constraints are imposed. To address this, Bassily and Freund [2016] introduced the elegant framework of concentrated queries, which requires the analyst to restrict itself to queries that are concentrated around their expected value. While this assumption makes the problem trivial in the non-adaptive setting, in the adaptive setting it remains quite challenging. In fact, all known algorithms in this framework support significantly fewer queries than in the independent case: At most $O(n)$ queries for a sample of size $n$, compared to $O(n^2)$ in the independent setting. In this work, we prove that this utility gap is inherent under the current formulation of the concentrated queries framework, assuming some natural conditions on the algorithm. Additionally, we present a simplified version of the best-known algorithms that match our impossibility result.

Problem

• Bounds for answering adaptively chosen concentrated queries
• Addressing intractability with correlated dataset samples
• Utility gap in concentrated queries framework

Innovation

• Concentrated queries framework for adaptive analysis
• Imposes structural constraints on correlated datasets
• Simplified algorithms matching theoretical bounds