# Chain of LoRA: Efficient Fine-tuning of Language Models via Residual Learning

arXiv: https://arxiv.org/abs/2401.04151 · HTML: https://mlsanity.com/paper/2401.04151/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 8 Jan 2024.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension](https://mlsanity.com/paper/1910.13461/) (2019)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [CodeT5: Identifier-aware Unified Pre-trained Encoder-Decoder Models for Code Understanding and Generation](https://mlsanity.com/paper/2109.00859/) (2021)
- [Towards a Unified View of Parameter-Efficient Transfer Learning](https://mlsanity.com/paper/2110.04366/) (2021)
- [AdapterFusion: Non-Destructive Task Composition for Transfer Learning](https://mlsanity.com/paper/2005.00247/) (2020)
- [Fine-Tuning Language Models with Just Forward Passes](https://mlsanity.com/paper/2305.17333/) (2023)
- [Parameter-efficient Multi-task Fine-tuning for Transformers via Shared Hypernetworks](https://mlsanity.com/paper/2106.04489/) (2021)
- [Simple, Scalable Adaptation for Neural Machine Translation](https://mlsanity.com/paper/1909.08478/) (2019)
- [S-LoRA: Serving Thousands of Concurrent LoRA Adapters](https://mlsanity.com/paper/2311.03285/) (2023)
- [Tied-Lora: Enhancing parameter efficiency of LoRA with weight tying](https://mlsanity.com/paper/2311.09578/) (2023)
- [Convergence Rate of Frank-Wolfe for Non-Convex Objectives](https://mlsanity.com/paper/1607.00345/) (2016)
- [Stochastic Frank-Wolfe Methods for Nonconvex Optimization](https://mlsanity.com/paper/1607.08254/) (2016)
- [MultiLoRA: Democratizing LoRA for Better Multi-Task Learning](https://mlsanity.com/paper/2311.11501/) (2023)
- [Linear Convergence of a Frank-Wolfe Type Algorithm over Trace-Norm Balls](https://mlsanity.com/paper/1708.02105/) (2017)
- [Beyond Output Matching: Bidirectional Alignment for Enhanced In-Context Learning](https://mlsanity.com/paper/2312.17055/) (2023)

## Built on by (most cited first, 30 of 82)

- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [LoRA vs Full Fine-tuning: An Illusion of Equivalence](https://mlsanity.com/paper/2410.21228/) (2024)
- [Unveiling and Harnessing Hidden Attention Sinks: Enhancing Large Language Models without Training through Attention Calibration](https://mlsanity.com/paper/2406.15765/) (2024)
- [A Survey on LoRA of Large Language Models](https://mlsanity.com/paper/2407.11046/) (2024)
- [Beyond Chain-of-Thought: A Survey of Chain-of-X Paradigms for LLMs](https://mlsanity.com/paper/2404.15676/) (2024)
- [SLTrain: a sparse plus low-rank approach for parameter and memory efficient pretraining](https://mlsanity.com/paper/2406.02214/) (2024)
- [Fira: Can We Achieve Full-rank Training of LLMs Under Low-rank Constraint?](https://mlsanity.com/paper/2410.01623/) (2024)
- [Q-GaLore: Quantized GaLore with INT4 Projection and Layer-Adaptive Low-Rank Gradients](https://mlsanity.com/paper/2407.08296/) (2024)
- [LDAdam: Adaptive Optimization from Low-Dimensional Gradient Statistics](https://mlsanity.com/paper/2410.16103/) (2024)
- [Low-Rank Adaptation for Foundation Models: A Comprehensive Review](https://mlsanity.com/paper/2501.00365/) (2024)
- [AdaMoLE: Fine-Tuning Large Language Models with Adaptive Mixture of Low-Rank Adaptation Experts](https://mlsanity.com/paper/2405.00361/) (2024)
- [BAdam: A Memory Efficient Full Parameter Optimization Method for Large Language Models](https://mlsanity.com/paper/2404.02827/) (2024)
- [From Low Rank Gradient Subspace Stabilization to Low-Rank Weights: Observations, Theories, and Applications](https://mlsanity.com/paper/2407.11239/) (2024)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [MELoRA: Mini-Ensemble Low-Rank Adapters for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2402.17263/) (2024)
- [VeLoRA: Memory Efficient Training using Rank-1 Sub-Token Projections](https://mlsanity.com/paper/2405.17991/) (2024)
- [Grass: Compute Efficient Low-Memory LLM Training with Structured Sparse Gradients](https://mlsanity.com/paper/2406.17660/) (2024)
- [SubTrack++ : Gradient Subspace Tracking for Scalable LLM Training](https://mlsanity.com/paper/2502.01586/) (2025)
- [On the Crucial Role of Initialization for Matrix Factorization](https://mlsanity.com/paper/2410.18965/) (2024)
- [Alignment and Safety in Large Language Models: Safety Mechanisms, Training Paradigms, and Emerging Challenges](https://mlsanity.com/paper/2507.19672/) (2025)
- [Propulsion: Steering LLM with Tiny Fine-Tuning](https://mlsanity.com/paper/2409.10927/) (2024)
- [Randomized Asymmetric Chain of LoRA: The First Meaningful Theoretical Framework for Low-Rank Adaptation](https://mlsanity.com/paper/2410.08305/) (2024)
- [Mini-batch Coresets for Memory-efficient Language Model Training on Data Mixtures](https://mlsanity.com/paper/2407.19580/) (2024)
- [Gradient Weight-normalized Low-rank Projection for Efficient LLM Training](https://mlsanity.com/paper/2412.19616/) (2024)
- [Continual Gradient Low-Rank Projection Fine-Tuning for LLMs](https://mlsanity.com/paper/2507.02503/) (2025)
- [Parameter-Efficient Fine-Tuning of Large Language Models using Semantic Knowledge Tuning](https://mlsanity.com/paper/2410.08598/) (2024)
- [Implicit Regularization of Sharpness-Aware Minimization for Scale-Invariant Problems](https://mlsanity.com/paper/2410.14802/) (2024)
- [DeLoRA: Decoupling Angles and Strength in Low-rank Adaptation](https://mlsanity.com/paper/2503.18225/) (2025)

## Abstract (arXiv)

Fine-tuning is the primary methodology for tailoring pre-trained large language models to specific tasks. As the model's scale and the diversity of tasks expand, parameter-efficient fine-tuning methods are of paramount importance. One of the most widely used family of methods is low-rank adaptation (LoRA) and its variants. LoRA encodes weight update as the product of two low-rank matrices. Despite its advantages, LoRA falls short of full-parameter fine-tuning in terms of generalization error for certain tasks. We introduce Chain of LoRA (COLA), an iterative optimization framework inspired by the Frank-Wolfe algorithm, to bridge the gap between LoRA and full parameter fine-tuning, without incurring additional computational costs or memory overheads. COLA employs a residual learning procedure where it merges learned LoRA modules into the pre-trained language model parameters and re-initilize optimization for new born LoRA modules. We provide theoretical convergence guarantees as well as empirical results to validate the effectiveness of our algorithm. Across various models (OPT and llama-2) and seven benchmarking tasks, we demonstrate that COLA can consistently outperform LoRA without additional computational or memory costs.