通过投影私密学习超立方体上的光滑分布
Privately Learning Smooth Distributions on the Hypercube by Projections
AI总结：
本文针对d维超立方体上Sobolev光滑概率密度的中心私密估计问题，将一维结果推广到非整数光滑度与高维场景，还提出带新增惩罚项的自适应Lepskii方法，可在未知真实光滑度时私密选取最优投影估计器。
摘要：
在对能保障训练集隐私的统计方法需求日益增长的背景下，本文研究了d维超立方体上Sobolev光滑概率密度的中心私密估计问题。本文的贡献分为两方面：首先，它将(Lalanne等人，2023)的一维结果推广到非整数光滑度阶与高维场景，这一推广具备两点重要意义：更适配现代学习任务，且有助于厘清隐私、维度与光滑度之间的关联——这是差分隐私领域的核心问题。其次，本文提出了一种数据驱动（统计学中通常称为自适应）的私密估计策略，可在无需知晓真实光滑度$β$的前提下，从有限个私密投影估计器构成的族中私密选取能实现良好偏差-方差权衡的估计器。该策略通过改造用于私密选择的Lepskii方法实现，具体是新增了一个惩罚项，使估计过程具备隐私感知能力。
英文摘要：
Fueled by the ever-increasing need for statistics that guarantee the privacy of their training sets, this article studies the centrally-private estimation of Sobolev-smooth densities of probability over the hypercube in dimension d. The contributions of this article are two-fold : Firstly, it generalizes the one dimensional results of (Lalanne et al., 2023) to non-integer levels of smoothness and to a high-dimensional setting, which is important for two reasons : it is more suited for modern learning tasks, and it allows understanding the relations between privacy, dimensionality and smoothness, which is a central question with differential privacy. Secondly, this article presents a private strategy of estimation that is data-driven (usually referred to as adaptive in Statistics) in order to privately choose an estimator that achieves a good bias-variance trade-off among a finite family of private projection estimators without prior knowledge of the ground-truth smoothness $β$. This is achieved by adapting the Lepskii method for private selection, by adding a new penalization term that makes the estimation privacy-aware.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
