The Cost of Privacy in Generalized Linear Models: Algorithms and Minimax Lower Bounds
This work addresses privacy-preserving statistical inference for researchers and practitioners, offering a novel lower bound technique that is broadly applicable, though the algorithmic improvements are incremental.
The authors tackled the problem of parameter estimation in generalized linear models under differential privacy constraints by developing private projected gradient descent algorithms, achieving nearly rate-optimal performance as shown through minimax lower bounds and validated with experiments.
We propose differentially private algorithms for parameter estimation in both low-dimensional and high-dimensional sparse generalized linear models (GLMs) by constructing private versions of projected gradient descent. We show that the proposed algorithms are nearly rate-optimal by characterizing their statistical performance and establishing privacy-constrained minimax lower bounds for GLMs. The lower bounds are obtained via a novel technique, which is based on Stein's Lemma and generalizes the tracing attack technique for privacy-constrained lower bounds. This lower bound argument can be of independent interest as it is applicable to general parametric models. Simulated and real data experiments are conducted to demonstrate the numerical performance of our algorithms.