Locally Private Inference for Riemannian Stochastic Optimization
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.
Related
- Self-Normalized Inference from Riemannian Stochastic Gradient Paths
- Bootstrap Inference for Riemannian Optimization Problems
- A Stochastic Riemannian Alternating Descent Ascent Method for Nonsmooth Composite Expectation Optimization on Riemannian Manifolds
- Riemannian Optimization on Unknown Manifolds via Stochastic Proximal Refinement
- Optimal quantum locally differentially private mechanisms in the high-privacy regime
BibTeX
@misc{chang2026locally,
title = {{Locally Private Inference for Riemannian Stochastic Optimization}},
author = {Xiaotian Chang and Yangdi Jiang and Qirui Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22642},
eprint = {2609.22642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22642},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Locally Private Inference for Riemannian Stochastic Optimization
AU - Xiaotian Chang
AU - Yangdi Jiang
AU - Qirui Hu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.22642
UR - https://arxiv.org/abs/2609.22642
AB - 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.
ER -