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

Jingyang Li, T. Tony Cai, Dong Xia, Anru R. Zhang

TL;DR

An efficient algorithm is proposed that preserves differential privacy while achieving near-optimal rates at the central server, up to a logarithmic factor, and a matrix version of van Trees' inequality is derived, which is of independent interest.

Abstract

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.

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

TL;DR

An efficient algorithm is proposed that preserves differential privacy while achieving near-optimal rates at the central server, up to a logarithmic factor, and a matrix version of van Trees' inequality is derived, which is of independent interest.

Abstract

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.

Paper Structure

This paper contains 25 sections, 10 theorems, 270 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main contribution
  4. Related work
  5. Organization of the paper
  6. Optimal Federated PCA by Gaussian Mechanism
  7. Minimax Lower Bound
  8. Numerical Experiments
  9. Simulations
  10. The Lung Cancer Data
  11. Discussions
  12. Acknowledgment
  13. Proofs
  14. Proof of Theorem \ref{['thm:highprob:upperbound']}
  15. Proof of Theorem \ref{['thm:covariance:upperbound']}
  16. ...and 10 more sections

Key Result

Lemma 1

Suppose that $X_i^{(j)}\stackrel{{\rm i.i.d.}}{\sim} N(0, \Sigma)$ with $\Sigma\in\Theta(\lambda, \sigma^2)$ for $j\in[m]$ and $i\in[n_j]$. For any weight vectors ${\boldsymbol v}=(v_1,\cdots,v_m)^{\top}$ and ${\boldsymbol w}=(w_1,\cdots,w_m)^{\top}$, the output $\widehat{U}\widehat{U}^{\top}$ and $

Figures (3)

  • Figure 1: Numerical simulations comparing our method with existing methods and their variations. The performance is assessed using the projection distance $\| \widehat{U}\widehat{U}^\top - UU^\top \|_{\rm{F}}$.
  • Figure 2: We compare our proposed method with the equal-weight aggregation approach and the Fed-DP-Oja algorithm using the Lung Cancer dataset. For simplicity, the entire dataset is unevenly divided between two clients, each allocated a privacy budget of $\varepsilon = 0.4,\delta= 0.1$.
  • Figure 3: Layer by layer decomposition of $\| \widehat{U}\widehat{U}^\top - UU^\top \|_{\rm{F}}^2$, with the building blocks $\{G^{(j)},H^{(j)}, Z_{j,1}\enspace Z_{j,2}\enspace Z_{j,3}\}_{j=1}^M$

Theorems & Definitions (15)

  • Lemma 1
  • Theorem 1
  • proof : Proof sketch of Theorem \ref{['thm:highprob:upperbound']}
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 5 more