A note on time-uniform concentration inequality for matrix products

TL;DR

This paper presents a straightforward method to extend fixed-time concentration bounds to any-time bounds for matrix products, motivated by applications in analyzing Oja's algorithms.

Contribution

It introduces a simple argument that enables time-uniform concentration inequalities for matrix products, enhancing analysis tools for algorithms like Oja's.

Findings

Provides a simple approach for time-uniform concentration bounds.

Applicable to the analysis of Oja's algorithms.

Enables bounds valid at any time, not just fixed points.

Abstract

This short note contains a simple argument that allows us to go from fixed-time to any-time bounds for the concentration of matrix products. The result presented here is motivated by the analysis of Oja's algorithms.