Tight Approximate Differential Privacy for Discrete-Valued Mechanisms Using FFT

We propose a numerical accountant for evaluating the tight $(\varepsilon,\delta)$-privacy loss for algorithms with discrete one-dimensional output. The method is based on the privacy loss distribution formalism and it is able to exploit the recently introduced Fast Fourier Transform based accounting technique. We carry out a complete error analysis of the method in terms of the moment bounds for the numerical estimate of the privacy loss distribution. We demonstrate the performance on the binomial mechanism and show that our approach allows decreasing noise variance up to an order of magnitude at equal privacy compared to existing bounds in the literature. We also give a novel approach for evaluating $(\varepsilon,\delta)$-upper bound for the subsampled Gaussian mechanism. This completes the previously proposed analysis by giving a strict upper bound for $(\varepsilon,\delta)$. We also illustrate how to compute tight bounds for the exponential mechanism applied to counting queries.