Auditing Privacy of Additive Noise Mechanisms Using Linear Predictive Models

We propose a privacy auditing framework using minimum mean-squared error (MMSE) estimation and linear auditing models. Our approach provides theoretical lower bounds on the true MMSE of inferring sensitive features from noisy observations of other correlated features. The bounds are in terms of the empirical MMSE under a restricted hypothesis class and a decomposable error term capturing finite sample and approximation effects. For linear auditing models, we derive order-optimal closed-form bounds for classes of relationships between the private and non-private features, including linear mappings, binary symmetric channels, and class-conditional Gaussian models. Through empirical evaluation, we demonstrate that our linear model-based auditing framework serves as a powerful yet tractable tool for MMSE-based privacy auditing that balances theoretical guarantees with practical efficiency.