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.

Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC); Statistics Theory (math.ST)

arXiv:2609.22642 [stat.ML]

(or arXiv:2609.22642v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.22642 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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