黎曼随机优化的本地私有推断
Locally Private Inference for Riemannian Stochastic Optimization
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- Nanyang Technological University（南洋理工大学）
- Shanghai University of Finance and Economics（上海财经大学）
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中文总结 AI 辅助
本文提出一种基于本地差分隐私的黎曼随机优化推断方法，通过条件居中的随机切线梯度与对称对回归估计方差，实现流形上总体最小值的有效推断，并在模拟和NHANES数据上验证了性能。
中文摘要 AI 辅助
我们针对流形上的总体最小值开发了推断方法，其中每个观测值属于不同的参与者，分析师只能收到本地私有的消息。该方法发布随机化的切线梯度，并通过黎曼随机逼近和Polyak-Ruppert平均进行组合。直接将私有数据替代项插入非线性损失函数可能会改变其总体目标，而对所发布梯度进行条件居中则保留了其一阶方程。我们引入了对称对回归（SPR），以从用于点估计的同一私有消息中估计渐近方差，而无需保留参与者或请求第二次发布。我们证明了在本地差分隐私下，基于完整转录的三明治协方差和内在Wald区域的中心极限定理和一致性。跨各种统计问题和流形的模拟支持在适度隐私下预测的估计误差减少和接近名义覆盖率。对NHANES人体测量数据的应用展示了主要体型方向及其不确定性的私有估计。
英文摘要
We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.
