Tight Differential Privacy for Discrete-Valued Mechanisms and for the Subsampled Gaussian Mechanism 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 uses the recently introduced Fast Fourier Transform based accounting technique. We carry out a complete error analysis of the method in terms of moment bounds of the privacy loss distribution which leads to rigorous lower and upper bounds for the true $(\varepsilon,\delta)$-values. As an application we give a novel approach to accurate privacy accounting of the subsampled Gaussian mechanism. This completes the previously proposed analysis by giving a strict lower and upper bounds for the $(\varepsilon,\delta)$-values. We also demonstrate the performance of the accountant on the binomial mechanism and show that our approach allows decreasing noise variance up to 75 percent at equal privacy compared to existing bounds in the literature. We also illustrate how to compute tight bounds for the exponential mechanism applied to counting queries.