The widespread collection and sharing of location data, even in aggregated form, raises major privacy concerns. Previous studies used meta-classifier-based membership inference attacks~(MIAs) with multi-layer perceptrons~(MLPs) to estimate privacy risks in location data, including when protected by differential privacy (DP). In this work, however, we show that a significant gap exists between the expected attack accuracy given by DP and the empirical attack accuracy even with informed attackers (also known as DP attackers), indicating a potential underestimation of the privacy risk. To explore the potential causes for the observed gap, we first propose two new metric-based MIAs: the one-threshold attack and the two-threshold attack. We evaluate their performances on real-world location data and find that different data distributions require different attack strategies for optimal performance: the one-threshold attack is more effective with Gaussian DP noise, while the two-threshold attack performs better with Laplace DP noise. Comparing their performance with one of the MLP-based attack models in previous works shows that the MLP only learns the one-threshold rule, leading to a suboptimal performance under the Laplace DP noise and an underestimation of the privacy risk. Second, we theoretically prove that MLPs can encode complex rules~(\eg, the two-threshold attack rule), which can be learned when given a substantial amount of training data. We conclude by discussing the implications of our findings in practice, including broader applications extending beyond location aggregates to any differentially private datasets containing multiple observations per individual and how techniques such as synthetic data generation and pre-training might enable MLP to learn more complex optimal rules.
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