Riemannian Optimization for LoRA on the Stiefel Manifold

TL;DR

This paper introduces a Riemannian optimization approach for LoRA on the Stiefel manifold, significantly improving fine-tuning efficiency and performance of large language models by addressing basis redundancy issues.

Contribution

It proposes a novel geometric optimization method on the Stiefel manifold for LoRA, enhancing parameter efficiency and outperforming AdamW in LLM fine-tuning.

Findings

Stiefel manifold optimization achieves near-perfect orthogonality.

Our method outperforms AdamW across multiple benchmarks.

Enhanced representational capacity improves LLM fine-tuning results.

Abstract

While powerful, large language models (LLMs) present significant fine-tuning challenges due to their size. Parameter-efficient fine-tuning (PEFT) methods like LoRA provide solutions, yet suffer from critical optimizer inefficiencies; notably basis redundancy in LoRA's $B$ matrix when using AdamW, which fundamentally limits performance. We address this by optimizing the $B$ matrix on the Stiefel manifold, imposing explicit orthogonality constraints that achieve near-perfect orthogonality and full effective rank. This geometric approach dramatically enhances parameter efficiency and representational capacity. Our Stiefel optimizer consistently outperforms AdamW across benchmarks with both LoRA and DoRA, demonstrating that geometric constraints are the key to unlocking LoRA's full potential for effective LLM fine-tuning.