# Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm

arXiv: https://arxiv.org/abs/2411.15660 · HTML: https://mlsanity.com/paper/2411.15660/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 54 references that are in the map.
- First posted on arXiv 23 Nov 2024.

## Builds on (most cited first, 7 of 7)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Collecting Telemetry Data Privately](https://mlsanity.com/paper/1712.01524/) (2017)
- [DP-PCA: Statistically Optimal and Differentially Private PCA](https://mlsanity.com/paper/2205.13709/) (2022)
- [Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices](https://mlsanity.com/paper/2401.03820/) (2024)
- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [On Mean Estimation for Heteroscedastic Random Variables](https://mlsanity.com/paper/2010.11537/) (2020)
- [Federated Principal Component Analysis](https://mlsanity.com/paper/1907.08059/) (2019)

## Built on by (most cited first, 3 of 3)

- [WinFLoRA: Incentivizing Client-Adaptive Aggregation in Federated LoRA under Privacy Heterogeneity](https://mlsanity.com/paper/2602.01126/) (2026)
- [A Van Trees Lower Bound for Fully Interactive Differentially Private Federated Learning](https://mlsanity.com/paper/2605.19813/) (2026)
- [Minimax Gaussian Mechanisms for Continual Machine Unlearning](https://mlsanity.com/paper/2610.11628/) (2026)

## Abstract (arXiv)

Federated Learning (FL) has gained significant recent attention in machine learning for its enhanced privacy and data security, making it indispensable in fields such as healthcare, finance, and personalized services. This paper investigates federated PCA and estimation for spiked covariance matrices under distributed differential privacy constraints. We establish minimax rates of convergence, with a key finding that the central server's optimal rate is the harmonic mean of the local clients' minimax rates. This guarantees consistent estimation at the central server as long as at least one local client provides consistent results. Notably, consistency is maintained even if some local estimators are inconsistent, provided there are enough clients. These findings highlight the robustness and scalability of FL for reliable statistical inference under privacy constraints. To establish minimax lower bounds, we derive a matrix version of van Trees' inequality, which is of independent interest. Furthermore, we propose an efficient algorithm that preserves differential privacy while achieving near-optimal rates at the central server, up to a logarithmic factor. We address significant technical challenges in analyzing this algorithm, which involves a three-layer spectral decomposition. Numerical performance of the proposed algorithm is investigated using both simulated and real data.