[Paper Review] Differentially Private Analysis of Outliers

Rina Okada, Kazuto Fukuchi+2 more|arXiv (Cornell University)|Jul 24, 2015

Privacy-Preserving Technologies in Data21 references4 citations

TL;DR

This paper proposes differentially private methods for analyzing distance-based outliers, using smooth sensitivity and the exponential mechanism to reduce noise in outlier counting and subspace discovery. It achieves significantly lower noise (up to 30× less) than global sensitivity-based approaches, especially at higher privacy budgets, enabling practical utility in high-dimensional data.

ABSTRACT

This paper investigates differentially private analysis of distance-based outliers. The problem of outlier detection is to find a small number of instances that are apparently distant from the remaining instances. On the other hand, the objective of differential privacy is to conceal presence (or absence) of any particular instance. Outlier detection and privacy protection are thus intrinsically conflicting tasks. In this paper, instead of reporting outliers detected, we present two types of differentially private queries that help to understand behavior of outliers. One is the query to count outliers, which reports the number of outliers that appear in a given subspace. Our formal analysis on the exact global sensitivity of outlier counts reveals that regular global sensitivity based method can make the outputs too noisy, particularly when the dimensionality of the given subspace is high. Noting that the counts of outliers are typically expected to be relatively small compared to the number of data, we introduce a mechanism based on the smooth upper bound of the local sensitivity. The other is the query to discovery top-$h$ subspaces containing a large number of outliers. This task can be naively achieved by issuing count queries to each subspace in turn. However, the variation of subspaces can grow exponentially in the data dimensionality. This can cause serious consumption of the privacy budget. For this task, we propose an exponential mechanism with a customized score function for subspace discovery. To the best of our knowledge, this study is the first trial to ensure differential privacy for distance-based outlier analysis. We demonstrated our methods with synthesized datasets and real datasets. The experimental results show that out method achieve better utility compared to the global sensitivity based methods.

Motivation & Objective

• To address the intrinsic conflict between outlier detection and differential privacy, where identifying outliers risks exposing individual data.
• To develop differentially private queries that reveal statistical behavior of outliers without disclosing individual instances.
• To improve utility in high-dimensional data by reducing noise in outlier counting and subspace discovery.
• To design efficient algorithms for smooth sensitivity evaluation and subspace scoring under privacy constraints.
• To demonstrate practical applicability through experiments on synthetic and real-world datasets.

Proposed method

• Proposes a smooth sensitivity-based mechanism for differentially private outlier counting, leveraging data-dependent noise that adapts to the actual number of outliers.
• Introduces an efficient algorithm to compute the smooth upper bound of local sensitivity for outlier counts, reducing computational cost.
• Employs the exponential mechanism with a customized score function to discover top-h subspaces with high outlier density, minimizing privacy budget consumption.
• Uses sequential composition to iteratively select subspaces, balancing accuracy and privacy under differential privacy guarantees.
• Applies the smallest enclosing ball algorithm to compute outlier status in subspaces, enabling robust distance-based outlier detection.
• Employs Laplace and Gaussian noise mechanisms with privacy parameters ε to ensure (ε, δ)-differential privacy.

Experimental results

Research questions

• RQ1 Can differentially private outlier counting be made more accurate than global sensitivity-based methods in high-dimensional data?
• RQ2 How can smooth sensitivity be efficiently computed and applied to outlier counting under differential privacy?
• RQ3 Can the exponential mechanism with a tailored score function improve subspace discovery while preserving privacy in high-dimensional settings?
• RQ4 What is the trade-off between privacy budget (ε) and utility (precision/recall) in differentially private outlier analysis?
• RQ5 How does the proposed method compare in noise magnitude and utility to standard global sensitivity-based approaches?

Key findings

• The smooth sensitivity-based mechanism reduced noise standard deviation by up to 30 times compared to global sensitivity-based methods, even with lower bounds.
• For ε ≥ 0.7, the smooth sensitivity mechanism achieved noise standard deviation below 7, demonstrating strong utility in practical privacy regimes.
• The method’s noise depends only on the number of outliers, not on data dimensionality, making it scalable to high-dimensional subspaces.
• Precision decreased with increasing h in top-h subspace discovery, while recall improved due to higher probability of selecting true subspaces.
• Sequential composition effects dominated at higher h, reducing recall despite increased detection probability, indicating a trade-off in utility.
• The proposed method achieved better detection accuracy than baseline methods, especially in high-dimensional settings, due to efficient privacy budget usage.