  Deep learning models are often trained on datasets that contain sensitive
information such as individuals' shopping transactions, personal contacts, and
medical records. An increasingly important line of work therefore has sought to
train neural networks subject to privacy constraints that are specified by
differential privacy or its divergence-based relaxations. These privacy
definitions, however, have weaknesses in handling certain important primitives
(composition and subsampling), thereby giving loose or complicated privacy
analyses of training neural networks. In this paper, we consider a recently
proposed privacy definition termed \textit{$f$-differential privacy} [18] for a
refined privacy analysis of training neural networks. Leveraging the appealing
properties of $f$-differential privacy in handling composition and subsampling,
this paper derives analytically tractable expressions for the privacy
guarantees of both stochastic gradient descent and Adam used in training deep
neural networks, without the need of developing sophisticated techniques as [3]
did. Our results demonstrate that the $f$-differential privacy framework allows
for a new privacy analysis that improves on the prior analysis~[3], which in
turn suggests tuning certain parameters of neural networks for a better
prediction accuracy without violating the privacy budget. These theoretically
derived improvements are confirmed by our experiments in a range of tasks in
image classification, text classification, and recommender systems. Python code
to calculate the privacy cost for these experiments is publicly available in
the \texttt{TensorFlow Privacy} library.
