Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests
This work addresses privacy-preserving hypothesis testing in federated settings, offering theoretical benchmarks and adaptive methods, though it is incremental in extending nonparametric testing to distributed DP frameworks.
The paper tackles federated nonparametric goodness-of-fit testing under differential privacy constraints, establishing matching lower and upper bounds on the minimax separation rate up to a logarithmic factor and revealing phase transitions and advantages of shared randomness. It also constructs an adaptive testing procedure that maintains privacy while adapting to unknown regularity parameters with minimal cost.
Federated learning has attracted significant recent attention due to its applicability across a wide range of settings where data is collected and analyzed across disparate locations. In this paper, we study federated nonparametric goodness-of-fit testing in the white-noise-with-drift model under distributed differential privacy (DP) constraints. We first establish matching lower and upper bounds, up to a logarithmic factor, on the minimax separation rate. This optimal rate serves as a benchmark for the difficulty of the testing problem, factoring in model characteristics such as the number of observations, noise level, and regularity of the signal class, along with the strictness of the $(ε,δ)$-DP requirement. The results demonstrate interesting and novel phase transition phenomena. Furthermore, the results reveal an interesting phenomenon that distributed one-shot protocols with access to shared randomness outperform those without access to shared randomness. We also construct a data-driven testing procedure that possesses the ability to adapt to an unknown regularity parameter over a large collection of function classes with minimal additional cost, all while maintaining adherence to the same set of DP constraints.