Private data summaries improve accuracy in curved space learning
Locally Private Inference for Riemannian Stochastic Optimization
Machine Learning
Summary
When data points come from many people, protecting each person's privacy makes it tricky to understand patterns that lie on curved spaces, like shapes or directions. The authors created a way to send private, noisy data updates that still let a computer learn the overall trends accurately. They also developed a method to estimate how confident we can be in these learned trends using only the private messages without extra data. Their tests show this approach works well, even with privacy protections in place, including on body measurement data.
What this means in practice
- •For health data analysts: Privately estimate health-related shape trends to protect participant confidentiality while assessing population patterns.
- •For machine learning engineers: Build privacy-aware algorithms that optimize models with data naturally lying on curved spaces using private gradient updates.
Authors
Xiaotian Chang, Yangdi Jiang, Qirui Hu
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.