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# The Saddle-Point Method in Differential Privacy

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:508-528, 2023.

#### Abstract

We characterize the differential privacy guarantees of privacy mechanisms in the large-composition regime, i.e., when a privacy mechanism is sequentially applied a large number of times to sensitive data. Via exponentially tilting the privacy loss random variable, we derive a new formula for the privacy curve expressing it as a contour integral over an integration path that runs parallel to the imaginary axis with a free real-axis intercept. Then, using the method of steepest descent from mathematical physics, we demonstrate that the choice of saddle-point as the real-axis intercept yields closed-form accurate approximations of the desired contour integral. This procedure—dubbed the saddle-point accountant (SPA)—yields a constant-time accurate approximation of the privacy curve. Theoretically, our results can be viewed as a refinement of both Gaussian Differential Privacy and the moments accountant method found in Rényi Differential Privacy. In practice, we demonstrate through numerical experiments that the SPA provides a precise approximation of privacy guarantees competitive with purely numerical-based methods (such as FFT-based accountants), while enjoying closed-form mathematical expressions.