Monarch: Expressive Structured Matrices for Efficient and Accurate Training

Tri Dao, Beidi Chen, Nimit S Sohoni, Arjun Desai, Michael Poli, Jessica Grogan, Alexander Liu, Aniruddh Rao, Atri Rudra, Christopher Re

Proceedings of the 39th International Conference on Machine Learning, PMLR 162:4690-4721, 2022.

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 language modeling 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.

Cite this Paper

BibTeX

@InProceedings{pmlr-v162-dao22a,
title = {Monarch: Expressive Structured Matrices for Efficient and Accurate Training},
author = {Dao, Tri and Chen, Beidi and Sohoni, Nimit S and Desai, Arjun and Poli, Michael and Grogan, Jessica and Liu, Alexander and Rao, Aniruddh and Rudra, Atri and Re, Christopher},
booktitle = {Proceedings of the 39th International Conference on Machine Learning},
pages = {4690--4721},
year = {2022},
editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
volume = {162},
series = {Proceedings of Machine Learning Research},
month = {17--23 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v162/dao22a/dao22a.pdf},
url = {https://proceedings.mlr.press/v162/dao22a.html},
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 language modeling 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.}
}

Endnote

%0 Conference Paper
%T Monarch: Expressive Structured Matrices for Efficient and Accurate Training
%A Tri Dao
%A Beidi Chen
%A Nimit S Sohoni
%A Arjun Desai
%A Michael Poli
%A Jessica Grogan
%A Alexander Liu
%A Aniruddh Rao
%A Atri Rudra
%A Christopher Re
%B Proceedings of the 39th International Conference on Machine Learning
%C Proceedings of Machine Learning Research
%D 2022
%E Kamalika Chaudhuri
%E Stefanie Jegelka
%E Le Song
%E Csaba Szepesvari
%E Gang Niu
%E Sivan Sabato
%F pmlr-v162-dao22a
%I PMLR
%P 4690--4721
%U https://proceedings.mlr.press/v162/dao22a.html
%V 162
%X 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 language modeling 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.

APA

Dao, T., Chen, B., Sohoni, N.S., Desai, A., Poli, M., Grogan, J., Liu, A., Rao, A., Rudra, A. & Re, C.. (2022). Monarch: Expressive Structured Matrices for Efficient and Accurate Training. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:4690-4721 Available from https://proceedings.mlr.press/v162/dao22a.html.