分析分步学习中误差如何逐级累积，揭示其可预测的传播规律。
One Rank at a Time: Cascading Error Dynamics in Sequential Learning
- 将分步学习拆解为一系列秩1子空间估计问题
- 发现误差按可预测方式逐级放大，影响整体精度
- 适合关注模型稳定性与算法设计的研究者
顺序学习——将复杂任务分解为更简单的层次化组件——已成为人工智能中的范式。本文从低秩线性回归的角度审视顺序学习，重点关注在依次学习秩1子空间时误差的传播机制。我们提出一个分析框架，将学习过程分解为一系列秩1估计问题，其中后续估计依赖于前序步骤的准确性。我们的贡献在于刻画了该顺序过程中的误差传播特性，建立了误差（如计算预算有限和有限精度所导致）对整体模型精度影响的边界。我们证明这些误差以可预测的方式累积，对算法设计和稳定性保证具有重要意义。
原文摘要 · Abstract (English)
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.
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