# Locally Private Inference for Riemannian Stochastic Optimization

arXiv: https://arxiv.org/abs/2609.22642 · HTML: https://mlsanity.com/paper/2609.22642/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 18 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Differential Privacy Over Riemannian Manifolds](https://mlsanity.com/paper/2111.02516/) (2021)
- [Differentially private Riemannian optimization](https://mlsanity.com/paper/2205.09494/) (2022)
- [Differentially Private Fr'echet Mean on the Manifold of Symmetric Positive Definite (SPD) Matrices with log-Euclidean Metric](https://mlsanity.com/paper/2208.04245/) (2022)
- [Shape And Structure Preserving Differential Privacy](https://mlsanity.com/paper/2209.12667/) (2022)
- [Gaussian Differential Privacy on Riemannian Manifolds](https://mlsanity.com/paper/2311.10101/) (2023)
- [Online Local Differential Private Quantile Inference via Self-normalization](https://mlsanity.com/paper/2306.10260/) (2023)
- [Online differentially private inference in stochastic gradient descent](https://mlsanity.com/paper/2505.08227/) (2025)
- [Federated Learning of Quantile Inference under Local Differential Privacy](https://mlsanity.com/paper/2509.21800/) (2025)
- [Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanisms](https://mlsanity.com/paper/2604.20761/) (2026)
- [Differentially private inference framework for Riemannian manifold data](https://mlsanity.com/paper/2605.14762/) (2026)

## Abstract (arXiv)

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.