One Rank at a Time: Cascading Error Dynamics in Sequential Learning
This work addresses error dynamics in sequential learning, which is incremental and relevant for algorithmic design and stability in AI.
The paper tackles the problem of error propagation in sequential learning by analyzing low-rank linear regression, specifically focusing on rank-1 subspaces, and establishes bounds on how errors compound to affect overall model accuracy.
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.