发表机构
Center for Data Science, Zhejiang University; Department of Statistics and Data Science, The Wharton School, University of Pennsylvania（浙江大学数据科学中心; 宾夕法尼亚大学沃顿商学院统计与数据科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出饱和LASSO算法，在差分隐私下实现稀疏高维线性回归的精确支持恢复，无需稀疏性先验，并证明其极小极大最优性，兼顾隐私保护与统计效率。
AI 中文摘要
我们研究在($\epsilon,\delta$)-差分隐私下稀疏高维线性回归中的精确支持恢复问题。我们引入了饱和提出-测试-发布（Saturated Propose-test-release）机制，这是一种通用机制，一旦其稳定性证书达到有限阈值，便以概率一私下发布离散选择器的输出。利用LASSO的坐标方向几何结构，我们构造了一个可计算的支持稳定性分数。由此产生的计算高效的饱和LASSO满足最坏情况下的($\epsilon,\delta$)-差分隐私，并在明确的规律性和beta-min条件下以高概率实现精确支持恢复。最大化稀疏性索引证书产生一个自适应过程，该过程无需稀疏性知识，并且在常见的公共调优参数下，其有限样本精确恢复风险与预言式固定稀疏性过程完全相同。我们还建立了一个显式跟踪$\delta$的极小极大下界：在其恢复条件下，饱和LASSO在$0<\delta\leq\epsilon/16$范围内，在$n$和$1/\delta$的对数因子内均匀地达到极小极大最优；在广泛的中等$\delta$区域，它进一步匹配下界的$\delta$依赖性。一个补充的信息论构造在已知稀疏性下，在独立高斯设计下达到下界速率（常数因子内），但计算成本是指数级的。模拟实验和一项使用美国社区调查公共协变量的半合成研究展示了所提出程序的数值性能。
英文摘要
We study exact support recovery under $(ε,δ)$-differential privacy in sparse high-dimensional linear regression. We introduce Saturated Propose-test-release, a general mechanism that privately releases the output of a discrete selector with probability one once its stability certificate reaches a finite threshold. Exploiting the coordinatewise geometry of the LASSO, we construct a computable support-stability score. The resulting computationally efficient Saturated LASSO satisfies worst-case $(ε,δ)$-differential privacy and achieves exact support recovery with high probability under explicit regularity and beta-min conditions. Maximizing sparsity-indexed certificates yields an adaptive procedure requiring no sparsity knowledge and having exactly the same finite-sample exact-recovery risk as the oracle fixed-sparsity procedure under common public tuning parameters. We also establish a minimax lower bound explicitly tracking $δ$: under its recovery conditions, Saturated LASSO is minimax optimal up to logarithmic factors in $n$ and $1/δ$ uniformly over $0 < δ\leq ε/16$; in the broad moderate-$δ$ regime, it further matches the lower-bound $δ$-dependence. A complementary information-theoretic construction with known sparsity attains the lower-bound rates up to constant factors under independent Gaussian design, at exponential computational cost. Simulations and a semi-synthetic study using public American Community Survey covariates illustrate the numerical performance of the proposed procedures.
