Polynomial Expansion Rank Adaptation: Enhancing Low-Rank Fine-Tuning with High-Order Interactions

2026-04-12

AI Summary

Existing low-rank fine-tuning methods, such as LoRA, are constrained by their linear structure and thus struggle to capture higher-order nonlinear interactions among parameters, limiting their representational capacity. This work proposes a novel efficient fine-tuning framework that enriches the low-rank adaptation space by introducing structured polynomial expansions—explicitly modeling quadratic terms—to transform the adapter into a polynomial manifold. Crucially, this enhancement boosts model expressiveness without increasing rank or inference overhead. To the best of our knowledge, this is the first approach to explicitly incorporate higher-order nonlinear interactions into low-rank fine-tuning. Extensive experiments across multiple benchmark tasks consistently demonstrate superior performance over current state-of-the-art methods, highlighting the critical role of higher-order components in improving both accuracy and robustness.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Low-rank adaptation (LoRA) is a widely used strategy for efficient fine-tuning of large language models (LLMs), but its strictly linear structure fundamentally limits expressive capacity. The bilinear formulation of weight updates captures only first-order dependencies between low-rank factors, restricting the modeling of nonlinear and higher-order parameter interactions. In this paper, we propose Polynomial Expansion Rank Adaptation (PERA), a novel method that introduces structured polynomial expansion directly into the low-rank factor space. By expanding each low-rank factor to synthesize high-order interaction terms before composition, PERA transforms the adaptation space into a polynomial manifold capable of modeling richer nonlinear coupling without increasing rank or inference cost. We provide theoretical analysis demonstrating that PERA offers enhanced expressive capacity and more effective feature utilization compare to existing linear adaptation approaches. Empirically, PERA consistently outperforms state-of-the-art methods across diverse benchmarks. Notably, our experiments show that incorporating high-order nonlinear components particularly square terms is crucial for enhancing expressive capacity and maintaining strong and robust performance under various rank settings. Our code is available at https://github.com/zhangwenhao6/PERA

Problem

• Low-rank adaptation
• High-order interactions
• Nonlinear modeling
• Expressive capacity
• Large language models

Innovation

• Polynomial Expansion
• Low-Rank Adaptation
• High-Order Interactions
• Nonlinear Modeling
• Parameter-Efficient Fine-Tuning