  In recent years, formal methods of privacy protection such as differential
privacy (DP), capable of deployment to data-driven tasks such as machine
learning (ML), have emerged. Reconciling large-scale ML with the closed-form
reasoning required for the principled analysis of individual privacy loss
requires the introduction of new tools for automatic sensitivity analysis and
for tracking an individual's data and their features through the flow of
computation. For this purpose, we introduce a novel \textit{hybrid} automatic
differentiation (AD) system which combines the efficiency of reverse-mode AD
with an ability to obtain a closed-form expression for any given quantity in
the computational graph. This enables modelling the sensitivity of arbitrary
differentiable function compositions, such as the training of neural networks
on private data. We demonstrate our approach by analysing the individual DP
guarantees of statistical database queries. Moreover, we investigate the
application of our technique to the training of DP neural networks. Our
approach can enable the principled reasoning about privacy loss in the setting
of data processing, and further the development of automatic sensitivity
analysis and privacy budgeting systems.
