[Paper Review] Imputation under Differential Privacy

Soumojit Das, Jorg Dreschler+2 more|arXiv (Cornell University)|Jun 30, 2022

Privacy-Preserving Technologies in Data4 citations

TL;DR

This paper investigates the interplay between differential privacy (DP) and data imputation, showing that naive application of DP to imputed data can severely degrade privacy due to increased sensitivity from missing values. It proposes a two-stage DP approach—applying privacy during both imputation and analysis—demonstrating that this preserves privacy guarantees and reduces noise, leading to improved utility in mean estimation under a fixed privacy budget.

ABSTRACT

The literature on differential privacy almost invariably assumes that the data to be analyzed are fully observed. In most practical applications this is an unrealistic assumption. A popular strategy to address this problem is imputation, in which missing values are replaced by estimated values given the observed data. In this paper we evaluate various approaches to answering queries on an imputed dataset in a differentially private manner, as well as discuss trade-offs as to where along the pipeline privacy is considered. We show that if imputation is done without consideration to privacy, the sensitivity of certain queries can increase linearly with the number of incomplete records. On the other hand, for a general class of imputation strategies, these worst case scenarios can be greatly reduced by ensuring privacy already during the imputation stage. We use a simulated dataset to demonstrate these results across a number of imputation schemes (both private and non-private) and examine their impact on the utility of a private query on the data.

Motivation & Objective

• To investigate how differential privacy interacts with data imputation, especially when missing data are present.
• To identify privacy risks arising from applying DP only at the analysis stage after imputation.
• To propose a method that ensures differential privacy during both imputation and analysis to maintain strong privacy guarantees.
• To evaluate the utility trade-offs of different strategies for handling missing data under differential privacy.
• To demonstrate that privacy-preserving imputation leads to more accurate and less biased query results.

Proposed method

• The authors model imputation as a function ι that maps incomplete datasets to fully observed ones, ensuring observed values remain unchanged.
• They establish theoretical bounds using group privacy, showing that neighboring datasets can differ in up to n_mis + 1 entries after imputation, increasing sensitivity.
• They propose a two-stage DP mechanism: first, use an ε₁-differentially private method to estimate imputation model parameters; second, apply ε₂-differentially private query mechanisms to the imputed data.
• They apply the Sequential Composition Theorem to show that the overall mechanism is (ε₁ + ε₂)-DP, ensuring end-to-end privacy.
• They implement and compare three strategies: available-case analysis, DP only at analysis stage, and DP at both imputation and analysis stages.
• Experiments use simulated OLS regression imputation with Laplace mechanism for mean queries, comparing bias and variance across strategies under ε = 1.

Experimental results

Research questions

• RQ1 What are the privacy risks of applying differential privacy only at the analysis stage after imputation?
• RQ2 How does the number of missing values affect the sensitivity and privacy budget in imputed datasets?
• RQ3 Can enforcing differential privacy during the imputation stage improve overall privacy and utility?
• RQ4 How does the choice of privacy budget allocation between imputation and analysis impact the accuracy of query results?
• RQ5 What is the relative bias and variance of mean estimates under different strategies for handling missing data under differential privacy?

Key findings

• Applying differential privacy only at the analysis stage after imputation can lead to privacy degradation, as the sensitivity of the query increases with the number of missing values.
• The worst-case privacy loss from imputation is bounded by e^(n_mis + 1)ε, which can be substantial when many values are missing.
• Imputing missing values using a differentially private model parameter estimation followed by a differentially private query mechanism ensures (ε₁ + ε₂)-DP, preserving formal privacy guarantees.
• Simulation results show that available-case analysis introduces significant bias in mean estimates, while both imputation strategies produce unbiased results.
• The two-stage DP approach (privacy at both imputation and analysis) results in substantially lower uncertainty (tighter confidence intervals) than applying DP only at the analysis stage.
• The two-stage method achieves better utility under the same privacy budget, demonstrating that early privacy enforcement improves accuracy.