Incremental Aggregation on the Grassmannian for Asynchronous Eigenspace Computation

2026-08-04

AI Summary

This work addresses the challenge of asynchronous subspace updates on Grassmann manifolds in heterogeneous distributed systems, where communication delays hinder synchronization. The authors propose an incremental aggregation method that avoids global synchronization by updating only arrived components, reusing cached gradients, and employing an extrinsic polar-coordinate update to preserve the geometric structure of the subspace. Theoretically, they establish—for the first time—an angle-dependent gradient dominance condition and basin invariance under delayed aggregation, yielding a two-stage linear convergence guarantee that combines wide-basin attraction with locally sharp convergence. Experiments demonstrate that the proposed approach significantly outperforms baseline methods in both serial and distributed PCA tasks, achieving marked improvements in sample efficiency and convergence speed under practical runtime constraints.

Abstract

We study asynchronous optimization for finite-sum eigenspace computation in heterogeneous distributed systems. The theoretical foundations for asynchronous eigenspace computation remain scarce, with existing approaches offering limited coverage of dynamics directly on the Grassmannian under stale information. In this paper, we propose a Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients, retaining low per-update cost without global synchronization. The method employs an extrinsic polar update that preserves the intrinsic subspace geometry without requiring parallel transport of stale tangent vectors. Our analysis establishes a tight angle-dependent gradient-dominance characterization of the objective and a basin-invariance property for stale aggregated updates. These yield two-phase linear convergence, comprising an explicit broad-basin regime and a sharper local regime, with constants controlled by component spectral spreads. Experiments on serial and distributed PCA demonstrate improved sample efficiency and wall-clock convergence over representative baselines.

Problem

• asynchronous eigenspace computation
• Grassmannian
• heterogeneous distributed systems
• stale information
• finite-sum optimization

Innovation

• Grassmannian optimization
• asynchronous eigenspace computation
• incremental aggregation
• polar update
• linear convergence