  We investigate the contraction coefficients derived from strong data
processing inequalities for the $E_\gamma$-divergence. By generalizing the
celebrated Dobrushin's coefficient from total variation distance to
$E_\gamma$-divergence, we derive a closed-form expression for the contraction
of $E_\gamma$-divergence. This result has fundamental consequences in two
privacy settings. First, it implies that local differential privacy can be
equivalently expressed in terms of the contraction of $E_\gamma$-divergence.
This equivalent formula can be used to precisely quantify the impact of local
privacy in (Bayesian and minimax) estimation and hypothesis testing problems in
terms of the reduction of effective sample size. Second, it leads to a new
information-theoretic technique for analyzing privacy guarantees of online
algorithms. In this technique, we view such algorithms as a composition of
amplitude-constrained Gaussian channels and then relate their contraction
coefficients under $E_\gamma$-divergence to the overall differential privacy
guarantees. As an example, we apply our technique to derive the differential
privacy parameters of gradient descent. Moreover, we also show that this
framework can be tailored to batch learning algorithms that can be implemented
with one pass over the training dataset.
