On the Convergence of Gradient Flow on Multi-layer Linear Models

Hancheng Min, Rene Vidal, Enrique Mallada
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:24850-24887, 2023.

Abstract

In this paper, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form $f(W_1W_2\cdots W_L)$. We show that when $f$ satisfies the gradient dominance property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific quantities that are controlled by the weight initialization: the imbalance matrices, which measure the difference between the weights of adjacent layers, and the least singular value of the weight product $W=W_1W_2\cdots W_L$. Our analysis exploits the fact that the gradient of the overparameterized loss can be written as the composition of the non-overparametrized gradient with a time-varying (weight-dependent) linear operator whose smallest eigenvalue controls the convergence rate. The key challenge we address is to derive a uniform lower bound for this time-varying eigenvalue that lead to improved rates for several multi-layer network models studied in the literature.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-min23d, title = {On the Convergence of Gradient Flow on Multi-layer Linear Models}, author = {Min, Hancheng and Vidal, Rene and Mallada, Enrique}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {24850--24887}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/min23d/min23d.pdf}, url = {https://proceedings.mlr.press/v202/min23d.html}, abstract = {In this paper, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form $f(W_1W_2\cdots W_L)$. We show that when $f$ satisfies the gradient dominance property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific quantities that are controlled by the weight initialization: the imbalance matrices, which measure the difference between the weights of adjacent layers, and the least singular value of the weight product $W=W_1W_2\cdots W_L$. Our analysis exploits the fact that the gradient of the overparameterized loss can be written as the composition of the non-overparametrized gradient with a time-varying (weight-dependent) linear operator whose smallest eigenvalue controls the convergence rate. The key challenge we address is to derive a uniform lower bound for this time-varying eigenvalue that lead to improved rates for several multi-layer network models studied in the literature.} }
Endnote
%0 Conference Paper %T On the Convergence of Gradient Flow on Multi-layer Linear Models %A Hancheng Min %A Rene Vidal %A Enrique Mallada %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-min23d %I PMLR %P 24850--24887 %U https://proceedings.mlr.press/v202/min23d.html %V 202 %X In this paper, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form $f(W_1W_2\cdots W_L)$. We show that when $f$ satisfies the gradient dominance property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific quantities that are controlled by the weight initialization: the imbalance matrices, which measure the difference between the weights of adjacent layers, and the least singular value of the weight product $W=W_1W_2\cdots W_L$. Our analysis exploits the fact that the gradient of the overparameterized loss can be written as the composition of the non-overparametrized gradient with a time-varying (weight-dependent) linear operator whose smallest eigenvalue controls the convergence rate. The key challenge we address is to derive a uniform lower bound for this time-varying eigenvalue that lead to improved rates for several multi-layer network models studied in the literature.
APA
Min, H., Vidal, R. & Mallada, E.. (2023). On the Convergence of Gradient Flow on Multi-layer Linear Models. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:24850-24887 Available from https://proceedings.mlr.press/v202/min23d.html.

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