Monarch: Expressive Structured Matrices for Efficient and Accurate Training (2204.00595v1)

Abstract: Large neural networks excel in many domains, but they are expensive to train and fine-tune. A popular approach to reduce their compute or memory requirements is to replace dense weight matrices with structured ones (e.g., sparse, low-rank, Fourier transform). These methods have not seen widespread adoption (1) in end-to-end training due to unfavorable efficiency--quality tradeoffs, and (2) in dense-to-sparse fine-tuning due to lack of tractable algorithms to approximate a given dense weight matrix. To address these issues, we propose a class of matrices (Monarch) that is hardware-efficient (they are parameterized as products of two block-diagonal matrices for better hardware utilization) and expressive (they can represent many commonly used transforms). Surprisingly, the problem of approximating a dense weight matrix with a Monarch matrix, though nonconvex, has an analytical optimal solution. These properties of Monarch matrices unlock new ways to train and fine-tune sparse and dense models. We empirically validate that Monarch can achieve favorable accuracy-efficiency tradeoffs in several end-to-end sparse training applications: speeding up ViT and GPT-2 training on ImageNet classification and Wikitext-103 LLMing by 2x with comparable model quality, and reducing the error on PDE solving and MRI reconstruction tasks by 40%. In sparse-to-dense training, with a simple technique called "reverse sparsification," Monarch matrices serve as a useful intermediate representation to speed up GPT-2 pretraining on OpenWebText by 2x without quality drop. The same technique brings 23% faster BERT pretraining than even the very optimized implementation from Nvidia that set the MLPerf 1.1 record. In dense-to-sparse fine-tuning, as a proof-of-concept, our Monarch approximation algorithm speeds up BERT fine-tuning on GLUE by 1.7x with comparable accuracy.

An Analysis of "Monarch: Expressive Structured Matrices for Efficient and Accurate Training"

The paper "Monarch: Expressive Structured Matrices for Efficient and Accurate Training" introduces a novel matrix class termed Monarch matrices, designed to optimize the training and fine-tuning processes in large-scale neural networks. By leveraging the computational efficiencies afforded by structured matrices, Monarch matrices provide a solution to reduce the high memory and resource demands typically encountered with dense weight matrices. This paper explores the application of Monarch matrices within different neural network architectures and quantifies their impact on training speed and model accuracy across several benchmarks.

Monarch matrices offer a unique combination of hardware efficiency and expressiveness. They are characterized as products of two block-diagonal matrices with an associated permutation, enabling rapid computations on modern GPU architectures. Notably, they retain the ability to represent a wide range of linear transforms, including convolutions and the Fourier transform, ensuring applicability across diverse domains. A key advantage of Monarch matrices is the analytically optimal solution for approximating dense matrices, a hurdle for many structured matrices, facilitating their use in dense-to-sparse fine-tuning processes.

The empirical results demonstrate Monarch matrices' utility in accelerating end-to-end training. For instance, replacing dense matrices in Vision Transformers (ViT) and GPT-2 models with Monarch matrices yielded a 2× speedup in training on ImageNet and Wikitext-103 datasets while maintaining comparable model accuracy. Furthermore, Monarch matrices improved the efficiency of solving Partial Differential Equations (PDEs) and MRI reconstruction tasks, reducing reconstruction errors by 40% and increasing pSNR and SSIM metrics by significant margins over traditional methods.

The Monarch framework is also positioned to address dense-to-sparse (D2S) transitions in models, which are crucial for fine-tuning pretrained models like BERT. The results suggest a 1.7× speedup in fine-tuning on the GLUE benchmark with minimal impact on the model's accuracy. This capability underscores Monarch matrices as effective intermediates, enhancing the sparsification strategies without extensive retraining requirements.

The paper addresses the computational constraints of sparse-to-dense transitions through a reverse sparsification process. Experiments with GPT-2 on the OpenWebText dataset reveal that training with Monarch matrices for the majority of the epochs, followed by transitioning to dense matrices, achieves a 2× speedup without compromising performance.

Implications of this research are significant. As machine learning models grow in complexity and size, exemplified by transformers and LLMs, the need for efficient training regimes becomes urgent. Monarch matrices provide a pathway to achieve scalable model training without sacrificing computational efficiency or accuracy. This advance enables broader applicability of structured matrices in real-world machine learning applications, potentially influencing future developments in hardware-aware model design.

Continued exploration and refinement of the Monarch framework could further optimize various neural architectures, impacting both the theoretical underpinnings and practical implementations of AI systems. As such, Monarch matrices could catalyze new research directions in efficient neural network training, including hybrid methods combining structured and unstructured components, which leverage the respective strengths of each approach.