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Provable Model-Parallel Distributed Principal Component Analysis with Parallel Deflation

Fangshuo Liao, Wenyi Su, Anastasios Kyrillidis

TL;DR

This work tackles scalable distributed PCA by introducing Parallel Deflation, a model-parallel framework where each worker computes a distinct principal component in parallel using intermediate updates from peers. By reformulating deflation into an iterative, multi-round procedure, the method eliminates strict sequential dependencies and supports asynchronous communication with modest overhead. The authors prove a convergence guarantee under a contraction assumption for the Top1 subroutine, showing nearly-linear convergence after a data-dependent warmup via a Davis–Kahan based analysis. Empirical results on synthetic data, MNIST, and ImageNet demonstrate competitive performance with the state-of-the-art EigenGame variants and confirm scalability to large-scale datasets.

Abstract

We study a distributed Principal Component Analysis (PCA) framework where each worker targets a distinct eigenvector and refines its solution by updating from intermediate solutions provided by peers deemed as "superior". Drawing intuition from the deflation method in centralized eigenvalue problems, our approach breaks the sequential dependency in the deflation steps and allows asynchronous updates of workers, while incurring only a small communication cost. To our knowledge, a gap in the literature -- the theoretical underpinning of such distributed, dynamic interactions among workers -- has remained unaddressed. This paper offers a theoretical analysis explaining why, how, and when these intermediate, hierarchical updates lead to practical and provable convergence in distributed environments. Despite being a theoretical work, our prototype implementation demonstrates that such a distributed PCA algorithm converges effectively and in scalable way: through experiments, our proposed framework offers comparable performance to EigenGame-$μ$, the state-of-the-art model-parallel PCA solver.

Provable Model-Parallel Distributed Principal Component Analysis with Parallel Deflation

TL;DR

This work tackles scalable distributed PCA by introducing Parallel Deflation, a model-parallel framework where each worker computes a distinct principal component in parallel using intermediate updates from peers. By reformulating deflation into an iterative, multi-round procedure, the method eliminates strict sequential dependencies and supports asynchronous communication with modest overhead. The authors prove a convergence guarantee under a contraction assumption for the Top1 subroutine, showing nearly-linear convergence after a data-dependent warmup via a Davis–Kahan based analysis. Empirical results on synthetic data, MNIST, and ImageNet demonstrate competitive performance with the state-of-the-art EigenGame variants and confirm scalability to large-scale datasets.

Abstract

We study a distributed Principal Component Analysis (PCA) framework where each worker targets a distinct eigenvector and refines its solution by updating from intermediate solutions provided by peers deemed as "superior". Drawing intuition from the deflation method in centralized eigenvalue problems, our approach breaks the sequential dependency in the deflation steps and allows asynchronous updates of workers, while incurring only a small communication cost. To our knowledge, a gap in the literature -- the theoretical underpinning of such distributed, dynamic interactions among workers -- has remained unaddressed. This paper offers a theoretical analysis explaining why, how, and when these intermediate, hierarchical updates lead to practical and provable convergence in distributed environments. Despite being a theoretical work, our prototype implementation demonstrates that such a distributed PCA algorithm converges effectively and in scalable way: through experiments, our proposed framework offers comparable performance to EigenGame-, the state-of-the-art model-parallel PCA solver.

Paper Structure

This paper contains 17 sections, 7 theorems, 84 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem statement and background
  4. The Parallel Deflation Algorithm
  5. Convergence Guarantee for the Parallel Deflation Algorithm
  6. Experiments
  7. Conclusion
  8. Appendix
  9. Missing Proof from Section \ref{['sec:algorithm']}
  10. Missing Proof from Section \ref{['sec:conv_guarantee']}
  11. Proof of Theorem \ref{['theo:conv_theo']}
  12. Proof of Lemma \ref{['lem:davis-kahan-app']}
  13. Proof of Lemma \ref{['lem:ub_seq_conv']}
  14. Proof of Lemma \ref{['lem:lambert_bound']}
  15. Ablation studies.
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that the covariance matrix $\bm{\Sigma}$ has positive and strictly decreasing eigenvalues $\lambda_1^\star > \dots > \lambda_K^\star > 0$. Then, $\{{\mathbf{u}}_k^\star\}_{k=1}^K$ is the unique strict Nash Equilibrium defined by the utilities in (eq:deflation_util) up to sign perturbation, i.

Figures (5)

  • Figure 1: Illustration of the parallel deflation algorithm.
  • Figure 2: Comparison of the convergence behavior of parallel deflation, EigenGame-$\alpha$, and EigenGame-$\mu$ in deterministic setting on (a). synthetic dataset with power-law decaying eigenvalues, (b). synthetic dataset with exponentially decaying eigenvalues, and (c). MNIST dataset.
  • Figure 3: Comparison of the convergence behavior of parallel deflation, EigenGame-$\alpha$, and EigenGame-$\mu$ in stochastic setting on (a). synthetic dataset with power-law decaying eigenvalues, (b). MNIST dataset, and (c) ImageNet dataset.
  • Figure 4: Ablation study of the parallel deflation algorithm. (a) shows the benefit of the run-time by increasing the parallelism. (b) shows the benefit of decreasing the communication cost by increasing the number of local iterations.
  • Figure : Parallel Deflation

Theorems & Definitions (11)

  • Definition 1: Power Iteration
  • Definition 2: Hebb's Rule
  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['theo:nash_eq']}
  • Lemma 1: $\sin\Theta$ Theorem davis1970rotation
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 1 more