We develop data processing inequalities that describe how Fisher information
from statistical samples can scale with the privacy parameter $\varepsilon$
under local differential privacy constraints. These bounds are valid under
general conditions on the distribution of the score of the statistical model,
and they elucidate under which conditions the dependence on $\varepsilon$ is
linear, quadratic, or exponential. We show how these inequalities imply order
optimal lower bounds for private estimation for both the Gaussian location
model and discrete distribution estimation for all levels of privacy
$\varepsilon>0$. We further apply these inequalities to sparse Bernoulli models
and demonstrate privacy mechanisms and estimators with order-matching squared
$\ell^2$ error.

本研究提出了描述在局部差分隐私限制下，统计样本的费舍尔信息如何随隐私参数 ϵ 缩放的数据处理不等式，证明了这些不等式对于所有隐私水平 ϵ>0 的高斯位置模型和离散分布估计都具有高阶匹配的平方ℓ2 误差的隐私机制和估计器。

局部差分隐私下的费舍尔信息

Fisher information under local differential privacy

Data processing inequalities for $f$-divergences can be sharpened using
constants called "contraction coefficients" to produce strong data processing
inequalities. For any discrete source-channel pair, the contraction
coefficients for $f$-divergences are lower bounded by the contraction
coefficient for $\chi^2$-divergence. In this paper, we elucidate that this
lower bound can be achieved by driving the input $f$-divergences of the
contraction coefficients to zero. Then, we establish a linear upper bound on
the contraction coefficients for a certain class of $f$-divergences using the
contraction coefficient for $\chi^2$-divergence, and refine this upper bound
for the salient special case of Kullback-Leibler (KL) divergence. Furthermore,
we present an alternative proof of the fact that the contraction coefficients
for KL and $\chi^2$-divergences are equal for a Gaussian source with an
additive Gaussian noise channel (where the former coefficient can be power
constrained). Finally, we generalize the well-known result that contraction
coefficients of channels (after extremizing over all possible sources) for all
$f$-divergences with non-linear operator convex $f$ are equal. In particular,
we prove that the so called "less noisy" preorder over channels can be
equivalently characterized by any non-linear operator convex $f$-divergence.

本文旨在通过驱动输入 $f$- 散度达到零来实现对 $f$- 散度的收缩系数下界以及建立起特定类别下的高效改进上界，并证明了一些重要结论。