[Paper Review] Hypothesis Testing in the High Privacy Limit

Jiachun Liao, Lalitha Sankar+2 more|arXiv (Cornell University)|Jul 2, 2016

Wireless Communication Security Techniques6 references3 citations

TL;DR

This paper proposes a Euclidean information-theoretic (E-IT) approximation for optimizing privacy-preserving randomizing mechanisms in binary hypothesis testing under mutual information-based privacy constraints. It shows that in the high privacy regime, the optimal mechanism is independent of alphabet size and preserves privacy by perturbing statistically unlikely source symbols most, maximizing utility while minimizing leakage for both source distributions.

ABSTRACT

Binary hypothesis testing under the Neyman-Pearson formalism is a statistical inference framework for distinguishing data generated by two different source distributions. Privacy restrictions may require the curator of the data or the data respondents themselves to share data with the test only after applying a randomizing privacy mechanism. Using mutual information as the privacy metric and the relative entropy between the two distributions of the output (postrandomization) source classes as the utility metric (motivated by the Chernoff-Stein Lemma), this work focuses on finding an optimal mechanism that maximizes the chosen utility function while ensuring that the mutual information based leakage for both source distributions is bounded. Focusing on the high privacy regime, an Euclidean information-theoretic (E-IT) approximation to the tradeoff problem is presented. It is shown that the solution to the E-IT approximation is independent of the alphabet size and clarifies that a mutual information based privacy metric preserves the privacy of the source symbols in inverse proportion to their likelihood.

Motivation & Objective

• To design a privacy mechanism that maximizes statistical utility (relative entropy between output distributions) while constraining mutual information leakage for both source classes.
• To address the NP-hard nature of the exact utility-privacy tradeoff problem by approximating it in the high privacy regime.
• To show that the E-IT approximation yields a closed-form solution independent of alphabet size.
• To clarify how mutual information-based privacy metrics protect source symbols in inverse proportion to their likelihood.

Proposed method

• Uses Euclidean information theory (E-IT) to approximate the relative entropy and mutual information functions in the high privacy regime.
• Derives a convex optimization problem by linearizing the utility and privacy constraints around a near-perfect privacy mechanism.
• Applies a diagonal matrix transformation using the square root of the prior distribution and a uniform weighting vector to simplify the approximation.
• Solves the resulting convex program to obtain a perturbation-based privacy mechanism that balances utility and leakage.
• Validates the approximation numerically using multiple pairs of Bernoulli distributions across varying privacy regimes.
• Uses the Chernoff-Stein Lemma as motivation for using relative entropy as the utility metric.

Experimental results

Research questions

• RQ1 How can the utility-privacy tradeoff in binary hypothesis testing be approximated in the high privacy regime?
• RQ2 What is the structure of the optimal privacy mechanism under mutual information-based leakage constraints?
• RQ3 Does the E-IT approximation yield a solution independent of the input alphabet size?
• RQ4 How does the privacy mechanism distribute perturbations across source symbols based on their likelihood?
• RQ5 In what regime is the E-IT approximation accurate for different source distribution pairs?

Key findings

• The E-IT approximation is highly accurate in a high privacy regime where statistical leakage is less than 0.5% of the minimum entropy of the two source distributions.
• For pairs of distributions both close to uniform or far apart from uniform and each other, the approximation performs well over a broader leakage regime.
• The optimal mechanism derived from the E-IT approximation is independent of the alphabet size, simplifying design across different data types.
• The mechanism perturbs the least likely source symbols the most, preserving privacy for those most vulnerable to inference attacks.
• The relative entropy (utility) of the E-IT solution is nearly identical to the true optimal for all tested pairs in the high privacy regime.
• A positive error exponent can still be achieved in the high privacy regime, albeit with higher sample complexity, indicating feasibility under moderate deviations.