Gaussian differential privacy: the trade-off function

Privacy is expressed as the trade-off between the two error rates of the hypothesis test that distinguishes neighbouring databases; a mechanism is mu-GDP when that trade-off is at least that of testing N(0, 1) against N(mu, 1). mu = 0 gives the diagonal 1 - alpha, perfect privacy, and delta = 0 for every epsilon; both limits are exact and are what the tests check.

Usage

Gdpf(mech = NULL, mu = NULL, alpha = NULL, epsilon = 1)

Arguments

mech: Optional c(sensitivity, sigma) of a Gaussian mechanism (l2 sensitivity, noise sd), which is (sensitivity / sigma)-GDP; used when mu is NULL.
mu: Non-negative GDP parameter.
alpha: Type I error rates at which to evaluate the trade-off.
epsilon: Epsilon at which the (eps, delta) conversion is given.

Value

List with estimate (delta), trade_off, alpha, delta, mu, epsilon, n, method.

Details

Formula: G_mu(alpha) = Phi(Phi^{-1}(1 - alpha) - mu); delta(eps) = Phi(-eps/mu + mu/2) - exp(eps) Phi(-eps/mu - mu/2).

References

Dong, Roth and Su (2022), Gaussian differential privacy, JRSS B 84(1):3-37, Corollary 2.13. doi:10.1111/rssb.12454

Examples

Gdpf(mu = 1.3, alpha = c(0.05, 0.5))$trade_off
#> [1] 0.63489779 0.09680048
Gdpf(mech = c(1, 2))$mu
#> [1] 0.5