Minimax signal detection in regression under differential privacy

Abstract

We establish minimax separation rates across sparsity levels for regression signal detection under $(ε, δ)$-central differential privacy, considering linear regression with an unknown error distribution and generalised linear models (GLMs) with an unknown intercept. Without sparsity restrictions, the separation rates for linear regression and GLMs with binary responses match the private Gaussian mean testing rate up to polylogarithmic factors. For GLMs with more than two possible response outcomes, additional information in the marginal response distribution permits faster detection of sufficiently small signals when the dimension exceeds the sample size $n$. Our upper bounds are based on a reduction to private mean testing, tests of dependence between the response and sparse covariate projections, and a response-only test that is adjusted for the unknown intercept. The resulting procedure adapts to unknown sparsity and matches our lower bounds up to polylogarithmic factors. We also construct polynomial-time tests that attain these rates when $ε\gtrsim n^{-1/2}$, and compare their numerical performance with existing private procedures.