离散信道上Rényi散度的收缩性
Contraction of Rényi Divergences for Discrete Channels
- School of Computer and Communication Sciences, EPFL（洛桑联邦理工学院计算机与通信科学学院）
- Okinawa Institute of Science and Technology (OIST)（冲绳科学技术大学院大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究有限空间上Rényi散度的SDPI常数，证明其单调性与凸性，给出支撑集限制、闭式表达式及张量化界，并应用于本地差分隐私和马尔可夫链，获得尖锐收缩保证与改进的收敛界。
AI中文摘要：
我们研究了有限空间上Rényi散度的强数据处理不等式（SDPI）常数。我们研究了它们对Rényi阶数$\alpha$的依赖性，证明了它们是非递减的，并且它们乘以$(\alpha-1)$后在$\alpha\geq1$时是凸的。我们还确定了在确定这些常数时所涉及的概率测度的若干支撑集限制。特别地，在分布无关的设置中，支撑在至多两个点的公共集合上的测度足以评估Rényi-SDPI常数。对于$\alpha\in[0,1]$，我们进一步证明了与$\chi^2$-SDPI常数的相等性，而在无穷阶时我们得到了一个闭式表达式。为了将乘积空间上的收缩与沿单个坐标的收缩联系起来，我们提供了任意乘积通道的张量化界，类似于已知的$\varphi$-散度的张量化界。在有限阶时，Rényi-SDPI常数通过比较$\chi^2$-散度和Hellinger散度而得到上下界，并在多种情况下建立了尖锐性。在无穷阶时，我们反而将这些常数与全变差距离的收缩联系起来。最后，我们的发现应用于本地差分隐私（LDP）和马尔可夫链的分析。这为纯LDP机制提供了尖锐的收缩保证，并将Rényi-LDP与Rényi-SDPI常数联系起来。对于马尔可夫链，我们推导了有限时间收敛界，并展示了一族链，对于这些链，Rényi-SDPI比经典的基于$\chi^2$的界改进了任意大的因子。
英文摘要：
We investigate Strong Data-Processing Inequality (SDPI) constants for Rényi Divergences on finite spaces. We study their dependence on the Rényi order $α$, proving that they are non-decreasing and that their scaling by $(α-1)$ is convex for $α\geq1$. We also identify several support restrictions on the probability measures involved in determining these constants. In particular, in the distribution-independent setting, measures supported on a common set of at most two points suffice to evaluate the Rényi-SDPI constant. For $α\in[0,1]$, we further prove equality with the $χ^2$-SDPI constant, while at order infinity we obtain a closed-form expression. In order to link contraction over product spaces to contraction along individual coordinates, we provide tensorisation bounds for arbitrary product channels analogous to those known for $φ$-Divergences. At finite orders, the Rényi-SDPI constants are bounded above and below through comparisons with the $χ^2$-Divergence and Hellinger Divergences, with sharpness established in multiple cases. At order infinity, we instead relate these constants to the contraction of Total Variation Distance. Finally, our findings are applied to local differential privacy (LDP) and the analysis of Markov chains. This yields sharp contraction guarantees for pure-LDP mechanisms, and connects Rényi-LDP to Rényi-SDPI constants. For Markov chains, we derive finite-time convergence bounds and exhibit a family of chains for which Rényi-SDPIs improve on classical $χ^2$-based bounds by arbitrarily large factors.
