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.

翻译：我们研究了基于强数据处理不等式的$E_γ$-散度收缩系数。通过将著名的多勃鲁申系数从全变差距离推广到$E_γ$-散度，我们推导了$E_γ$-散度收缩的闭式表达式。该结果在两个隐私设置中具有基础性意义。首先，它表明局部差分隐私可以等价地用$E_γ$-散度的收缩来表达。这一等价公式可用于精确量化局部隐私在（贝叶斯和极小化极大）估计及假设检验问题中通过有效样本量缩减产生的影响。其次，它提出了一种分析在线算法隐私保证的新信息论技术。在该技术中，我们将此类算法视为振幅受限高斯通道的复合结构，并通过其在$E_γ$-散度下的收缩系数关联整体差分隐私保证。作为示例，我们应用该技术推导了梯度下降算法的差分隐私参数。此外，我们还证明该框架可适用于单次遍历训练数据集即可实现的批量学习算法。