近年来，Transformer模型在自然语言处理领域掀起了一场革命，其强大的能力让世人惊叹。但随着模型规模不断扩大，训练和部署这些庞然大物也变得越来越困难，尤其对于个人用户和小型机构来说。
为了解决这一难题，研究者们提出了各种解决方案，其中低秩分解成为了一个重要的方向。LoRA[7] 就是一个典型的例子，它通过在预训练模型的每一层学习一个低秩增量来实现高效的微调。
本文则更进一步，提出了一个全新的模型设计范式——垂直LoRA (VLoRA)[7]。它基于一个全新的视角：将Transformer模型看作是密集型期望最大化（EM）算法[7]。
Transformer：隐藏的EM算法
在监督学习中，Transformer模型的目标是最大化后验概率 
，其中 
是输入，
是标签，
是模型参数。本文指出，Transformer模型的每一层实际上都是EM算法的一次迭代，前向传播对应于E步，而下一层与当前层权重差异则对应于M步。
这个发现揭示了Transformer模型中一个重要的规律：每一层都是基于前一层学习一个增量。而正是基于这一规律，VLoRA应运而生。
VLoRA：垂直分解，层层递进
VLoRA 首先定义一个全秩基层，然后每一层都基于上一层学习一个低秩增量，并使用LoRA分解来逼近这个增量。这种垂直分解的方式，使得模型参数数量大幅减少，同时保留了原始模型的性能。
与传统的水平LoRA相比，VLoRA 更加高效，因为它减少了模型的总体参数，而不是仅仅针对微调阶段。
实验验证：性能提升，更少参数
本文在图像分类任务上进行了实验，使用 CIFAR-10 数据集[31] 对 12 层的 Vision Transformer[32] 进行了训练，并比较了其 VLoRA 版本的性能。
实验结果表明：
- VLoRA 版本的训练损失和准确率虽然略低于原始模型，但在评估阶段却展现出更强的泛化能力，不容易过拟合。
- VLoRA 版本的最佳评估指标与原始模型几乎相同，但参数数量却大幅减少。
- 即使使用较小的低秩（例如 r=2），VLoRA 依然能有效地对每一层的权重增量进行建模。
未来展望：更轻盈，更强大
VLoRA 的出现，为构建更轻盈、更强大的 Transformer 模型提供了新的思路。它不仅可以用于降低模型的训练和部署成本，还可以提升模型的泛化能力，使其在更多场景下发挥作用。
参考文献
[1] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.
[2] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020.
[3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.
[4] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. Llama 2: Open foundation and fine-tuned chat models. arXiv preprint arXiv:2307.09288, 2023.
[5] Anthropic. The claude 3 model family: Opus, sonnet, haiku, 2024.
[6] Tim Dettmers, Artidoro Pagnoni, Ari Holtzman, and Luke Zettlemoyer. Qlora: Efficient finetuning of quantized llms. Advances in Neural Information Processing Systems, 36, 2024.
[7] Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. Lora: Low-rank adaptation of large language models. arXiv preprint arXiv:2106.09685, 2021.
[8] Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin De Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly. Parameter-efficient transfer learning for nlp. In International conference on machine learning, pages 2790–2799. PMLR, 2019.
[9] Jonas Pfeiffer, Aishwarya Kamath, Andreas Rückl, Kyunghyun Cho, and Iryna Gurevych. Adapterfusion: Non-destructive task composition for transfer learning. arXiv preprint arXiv:2005.00247, 2020.
[10] Qingru Zhang, Minshuo Chen, Alexander Bukharin, Pengcheng He, Yu Cheng, Weizhu Chen, and Tuo Zhao. Adaptive budget allocation for parameter-efficient fine-tuning. In The Eleventh International Conference on Learning Representations, 2023.
[11] Sinong Wang, Belinda Z Li, Madian Khabsa, Han Fang, and Hao Ma. Linformer: Self-attention with linear complexity. arXiv preprint arXiv:2006.04768, 2020.
[12] Xiang Lisa Li and Percy Liang. Prefix-tuning: Optimizing continuous prompts for generation. arXiv preprint arXiv:2101.00190, 2021.
[13] Xiao Liu, Kaixuan Ji, Yicheng Fu, Weng Lam Tam, Zhengxiao Du, Zhilin Yang, and Jie Tang. P-tuning v2: Prompt tuning can be comparable to fine-tuning universally across scales and tasks. arXiv preprint arXiv:2110.07602, 2021.
[14] Brian Lester, Rami Al-Rfou, and Noah Constant. The power of scale for parameter-efficient prompt tuning. arXiv preprint arXiv:2104.08691, 2021.
[15] Nikita Kitaev, Łukasz Kaiser, and Anselm Levskaya. Reformer: The efficient transformer. arXiv preprint arXiv:2001.04451, 2020.
[16] Jialin Li, Qiang Nie, Weifu Fu, Yuhuan Lin, Guangpin Tao, Yong Liu, and Chengjie Wang. Lors: Low-rank residual structure for parameter-efficient network stacking. arXiv preprint arXiv:2403.04303, 2024.
[17] Misha Denil, Babak Shakibi, Laurent Dinh, Marc Aurelio Ranzato, and Nando De Freitas. Predicting parameters in deep learning. Advances in neural information processing systems, 26, 2013.
[18] Armen Aghajanyan, Luke Zettlemoyer, and Sonal Gupta. Intrinsic dimensionality explains the effectiveness of language model fine-tuning. arXiv preprint arXiv:2012.13255, 2020.
[19] Chunyuan Li, Heerad Farkhoor, Rosanne Liu, and Jason Yosinski. Measuring the intrinsic dimension of objective landscapes. arXiv preprint arXiv:1804.08838, 2018.
[20] Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding up convolutional neural networks with low rank expansions. arXiv preprint arXiv:1405.3866, 2014.
[21] Tara N Sainath, Brian Kingsbury, Vikas Sindhwani, Ebru Arisoy, and Bhuvana Ramabhadran. Low-rank matrix factorization for deep neural network training with high-dimensional output targets. In 2013 IEEE international conference on acoustics, speech and signal processing, pages 6655–6659. IEEE, 2013.
[22] Xiangyu Zhang, Jianhua Zou, Kaiming He, and Jian Sun. Accelerating very deep convolutional networks for classification and detection. IEEE transactions on pattern analysis and machine intelligence, 38(10):1943–1955, 2015.
[23] Jian Xue, Jinyu Li, and Yifan Gong. Restructuring of deep neural network acoustic models with singular value decomposition. In Interspeech, pages 2365–2369, 2013.
[24] Emily L Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting linear structure within convolutional networks for efficient evaluation. Advances in neural information processing systems, 27, 2014.
[25] Vadim Lebedev, Yaroslav Ganin, Maksim Rakhuba, Ivan Oseledets, and Victor Lempitsky. Speeding-up convolutional neural networks using fine-tuned cp-decomposition. arXiv preprint arXiv:1412.6553, 2014.
[26] Yong-Deok Kim, Eunhyeok Park, Sungjoo Yoo, Taelim Choi, Lu Yang, and Dongjun Shin. Compression of deep convolutional neural networks for fast and low power mobile applications. arXiv preprint arXiv:1511.06530, 2015.
[27] Xiangdi Meng, Damai Dai, Weiyao Luo, Zhe Yang, Shaoxiang Wu, Xiaochen Wang, Peiyi Wang, Qingxiu Dong, Liang Chen, and Zhifang Sui. Periodiclora: Breaking the low-rank bottleneck in lora optimization. arXiv preprint arXiv:2402.16141, 2024.
[28] Yang Lin, Xinyu Ma, Xu Chu, Yujie Jin, Zhibang Yang, Yasha Wang, and Hong Mei. Lora dropout as a sparsity regularizer for overfitting control. arXiv preprint arXiv:2404.09610, 2024.
[29] Soufiane Hayou, Nikhil Ghosh, and Bin Yu. Lora+: Efficient low rank adaptation of large models. arXiv preprint arXiv:2402.12354, 2024.
[30] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
[31] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[32] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16×16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.