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

Mahtab Alizadeh Vandchali, Fangshuo, Liao, Anastasios Kyrillidis

TL;DR

The paper investigates error propagation in sequential learning framed as low-rank linear regression with deflation, decomposing the process into rank-1 updates. It derives recursive, spectrum-aware bounds showing how numerical errors compound across depths, governed by singular-value gaps $\mathcal{T}_k^\star$ and the conditioning of $\mathbf{X}$, and provides generalization guarantees for both noiseless and noisy labels. Empirical results on synthetic data and LoRA-style adaptation demonstrate that front-loading budget to early components mitigates cascading errors and yields competitive performance, while enabling online rank determination. Collectively, the work offers principled guidelines for designing hierarchical learning systems and resource allocation strategies in sequential low-rank updates with practical relevance to PEFT methods like LoRA.

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.

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

TL;DR

The paper investigates error propagation in sequential learning framed as low-rank linear regression with deflation, decomposing the process into rank-1 updates. It derives recursive, spectrum-aware bounds showing how numerical errors compound across depths, governed by singular-value gaps and the conditioning of , and provides generalization guarantees for both noiseless and noisy labels. Empirical results on synthetic data and LoRA-style adaptation demonstrate that front-loading budget to early components mitigates cascading errors and yields competitive performance, while enabling online rank determination. Collectively, the work offers principled guidelines for designing hierarchical learning systems and resource allocation strategies in sequential low-rank updates with practical relevance to PEFT methods like LoRA.

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.

Paper Structure

This paper contains 22 sections, 13 theorems, 101 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Error propagation during training
  4. Generalization of sequential rank-$1$ update
  5. Experimental results
  6. Synthetic validation of theoretical setting.
  7. Experimental analysis using LoRA.
  8. Limitations
  9. Conclusion
  10. Proof of Theorem \ref{['thm:main_theorem_1']}
  11. Missing proofs from Appendix \ref{['app:1']} and a generalized theorem
  12. Proof of Lemma \ref{['lem:sequential_decomposition']}
  13. Proof of Lemma \ref{['lem:matrix_diff_propagate']}
  14. Proof of Lemma \ref{['lem:sum_of_rank_1_diff']}
  15. Proof of a generalization of Theorem \ref{['thm:main_theorem_1']}
  16. ...and 7 more sections

Key Result

Lemma 1

According to the Eckart-Young-Mirsky theorem, under our defined settings and deflation method, for each $k$, we have that $\mathbf{Y}^\star_{k}=\sum_{k' = k}^p \sigma_{k'}^\star \mathbf{u}_{k'}^\star \mathbf{v}_{k'}^{\star\top}$ and $\mathbf{b}_k^\star \mathbf{a}_k^{\star \top} \mathbf{X} = \sigma_k

Figures (13)

  • Figure 1: Impact of iteration allocation strategy under a fixed iteration budget. Left: $\mathbf{W}^\star$ reconstruction error. Right: Objective's training error.
  • Figure 2: Test accuracy of sequential rank-1 LoRA when adapting to new classes across the three datasets. Left: MNIST. Center: CIFAR10. Right: CIFAR100. Note that, on purpose, the pretrained models are trained with good (MNIST), mediocre (CIFAR10) and bad (CIFAR100) accuracy.
  • Figure 3: Marker sizes: relative efficiency of each config.
  • Figure 4: $\alpha \shortrightarrow \beta \shortrightarrow \gamma$ denotes sequential training with $\alpha, \beta$ and $\gamma$ epochs for each component. Not all combinations are shown.
  • Figure 5: Comparison of singular value decay under different profiles. Left: Singular values of $\mathbf{W}^\star$. Right: Singular values of $\mathbf{Y} = \mathbf{W}^\star \mathbf{X}$.
  • ...and 8 more figures

Theorems & Definitions (22)

  • Lemma 1
  • Definition 1: Numerical Error
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • proof
  • proof
  • ...and 12 more