Locally Private Inference for Riemannian Stochastic Optimization

2026-09-18

AI Summary

研究提出了一种在本地隐私保护下，通过随机化切线梯度和黎曼随机逼近方法解决流形值数据最小化问题的方法。

Abstract

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.

Problem

• Locally Private Inference
• Riemannian Stochastic Optimization
• Manifold-valued Population Minimizers

Innovation

• Locally Private Inference
• Riemannian Stochastic Optimization
• Symmetric-pair Regression (SPR)
• Conditional Centring
• Local Differential Privacy