Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Diving into Kronecker Adapters: Component Design Matters

Jiayu Bai, Danchen Yu, Zhenyu Liao, TianQi Hou, Feng Zhou, Robert C. Qiu, Zenan Ling

TL;DR

This work analyzes how the component configuration of Kronecker adapters (defined by $(r_1,r_2,r)$) governs their ability to approximate full fine-tuning, revealing that high attainable rank does not guarantee better performance. Through a Kronecker-SVD-based theoretical alignment analysis, the authors derive principled guidelines that favor large $r_2$, small $r_1$, and moderate $r$ within a fixed parameter budget, and introduce a training-stabilization factor $\\lambda$ to ensure stable learning across configurations. They propose Component Designed Kronecker Adapters (CDKA) and validate them across NLP tasks, achieving state-of-the-art performance in mathematical reasoning and strong results in code generation and NLU while using substantially fewer trainable parameters than competing PEFT methods. The results demonstrate that principled component design substantially boosts Kronecker adapters, making them competitive with the best PEFT approaches in practice and scalable to large language models. These insights provide actionable design principles for practitioners deploying Kronecker-based PEFT in real-world systems.

Abstract

Kronecker adapters have emerged as a promising approach for fine-tuning large-scale models, enabling high-rank updates through tunable component structures. However, existing work largely treats the component structure as a fixed or heuristic design choice, leaving the dimensions and number of Kronecker components underexplored. In this paper, we identify component structure as a key factor governing the capacity of Kronecker adapters. We perform a fine-grained analysis of both the dimensions and number of Kronecker components. In particular, we show that the alignment between Kronecker adapters and full fine-tuning depends on component configurations. Guided by these insights, we propose Component Designed Kronecker Adapters (CDKA). We further provide parameter-budget-aware configuration guidelines and a tailored training stabilization strategy for practical deployment. Experiments across various natural language processing tasks demonstrate the effectiveness of CDKA. Code is available at https://github.com/rainstonee/CDKA.

Diving into Kronecker Adapters: Component Design Matters

TL;DR

This work analyzes how the component configuration of Kronecker adapters (defined by ) governs their ability to approximate full fine-tuning, revealing that high attainable rank does not guarantee better performance. Through a Kronecker-SVD-based theoretical alignment analysis, the authors derive principled guidelines that favor large , small , and moderate within a fixed parameter budget, and introduce a training-stabilization factor to ensure stable learning across configurations. They propose Component Designed Kronecker Adapters (CDKA) and validate them across NLP tasks, achieving state-of-the-art performance in mathematical reasoning and strong results in code generation and NLU while using substantially fewer trainable parameters than competing PEFT methods. The results demonstrate that principled component design substantially boosts Kronecker adapters, making them competitive with the best PEFT approaches in practice and scalable to large language models. These insights provide actionable design principles for practitioners deploying Kronecker-based PEFT in real-world systems.

Abstract

Kronecker adapters have emerged as a promising approach for fine-tuning large-scale models, enabling high-rank updates through tunable component structures. However, existing work largely treats the component structure as a fixed or heuristic design choice, leaving the dimensions and number of Kronecker components underexplored. In this paper, we identify component structure as a key factor governing the capacity of Kronecker adapters. We perform a fine-grained analysis of both the dimensions and number of Kronecker components. In particular, we show that the alignment between Kronecker adapters and full fine-tuning depends on component configurations. Guided by these insights, we propose Component Designed Kronecker Adapters (CDKA). We further provide parameter-budget-aware configuration guidelines and a tailored training stabilization strategy for practical deployment. Experiments across various natural language processing tasks demonstrate the effectiveness of CDKA. Code is available at https://github.com/rainstonee/CDKA.
Paper Structure (38 sections, 12 theorems, 82 equations, 1 figure, 13 tables)

This paper contains 38 sections, 12 theorems, 82 equations, 1 figure, 13 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Adapter-based methods.
  4. Kronecker Adapters
  5. Notations
  6. Preliminaries
  7. Low-Rank Adapters
  8. Kronecker Adapters
  9. Main Results
  10. Component Design Matters for Kronecker Adapters
  11. Kronecker adapters enable higher rank.
  12. Unleash the high rank potential through component design.
  13. Component Designed Kronecker Adapters
  14. Problem settings.
  15. Validations for Our Proposed Principles.
  16. ...and 23 more sections

Key Result

Theorem 3.2

Under the settings described in Section sec:alignment and taking $r^*\leq r< 2r^*$, we consider random Gaussian initialization for $\widetilde{{\bm{A}}}_{0}$ with $[\widetilde{{\bm{A}}}_{0}]_{ij} \sim \mathcal{N}(0, \alpha^2)$ and zero initialization for $\widetilde{{\bm{B}}}_{0}$ with $[\widetilde{ where $\kappa$ is the condition number of $\widetilde{{\bm{G}}}_0$. Then if we run gradient descent

Figures (1)

  • Figure 1: Gradient norm during training with (a) $\lambda \equiv1$ and (b) $\lambda = \frac{16}{\sqrt{r\cdot r_2}}$ under different $r_1$, $r_2$ and $r$. Identical colors indicate identical component configurations. Our method maintains gradient norms at the same scale across different component configurations.

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 3.1
  • Theorem 3.2: A simplified version of Theorem \ref{['theorem:alignmentA']} with $r^*\leq r< 2r^*$.
  • Remark 3.3: Principles for component design.
  • Theorem 3.4
  • Theorem 2.1: The top-$r^*$ left singular subspace alignment between $\widetilde{{\bm{A}}_t}$ and $\widetilde{{\bm{G}}}_0$.
  • proof
  • Lemma 2.2: Linear dynamic of the Kronecker adapter. Adapted from Lemma C.5 in zhanglora.
  • Theorem 2.3: The error bound of the linear approximation
  • proof
  • ...and 10 more