One Rank at a Time: Cascading Error Dynamics in Sequential Learning

TL;DR

This paper analyzes how errors propagate in sequential learning of low-rank subspaces, providing bounds on error accumulation and insights into algorithm stability in hierarchical AI tasks.

Contribution

It introduces a framework for understanding error dynamics in rank-1 sequential learning, with theoretical bounds on error propagation and stability implications.

Findings

Errors compound predictably in sequential low-rank learning.

Finite precision and computational limits impact overall accuracy.

The analysis informs better algorithm design for hierarchical learning.

Abstract

Sequential learning -- where complex tasks are broken down into simpler, hierarchical components -- has emerged as a paradigm in AI. This paper views sequential learning through the lens of low-rank linear regression, focusing specifically on how errors propagate when learning rank-1 subspaces sequentially. We present an analysis framework that decomposes the learning process into a series of rank-1 estimation problems, where each subsequent estimation depends on the accuracy of previous steps. Our contribution is a characterization of the error propagation in this sequential process, establishing bounds on how errors -- e.g., due to limited computational budgets and finite precision -- affect the overall model accuracy. We prove that these errors compound in predictable ways, with implications for both algorithmic design and stability guarantees.