Riemannian Optimization for LoRA on the Stiefel Manifold

• URL: http://arxiv.org/abs/2508.17901v1
• Date: Mon, 25 Aug 2025 11:15:52 GMT
• Title: Riemannian Optimization for LoRA on the Stiefel Manifold
• Authors: Juneyoung Park, Minjae Kang, Seongbae Lee, Haegang Lee, Seongwan Kim, Jaeho Lee,
• Abstract summary: Large language models (LLMs) present significant fine-tuning challenges due to their size.<n>We show that geometric constraints are key to unlocking LoRA's full potential for effective fine-tuning.
• Score: 11.1808022633589
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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