Global Optimality Beyond Two Layers: Training Deep ReLU Networks via Convex Programs

Tolga Ergen, Mert Pilanci
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:2993-3003, 2021.

Abstract

Understanding the fundamental mechanism behind the success of deep neural networks is one of the key challenges in the modern machine learning literature. Despite numerous attempts, a solid theoretical analysis is yet to be developed. In this paper, we develop a novel unified framework to reveal a hidden regularization mechanism through the lens of convex optimization. We first show that the training of multiple three-layer ReLU sub-networks with weight decay regularization can be equivalently cast as a convex optimization problem in a higher dimensional space, where sparsity is enforced via a group $\ell_1$-norm regularization. Consequently, ReLU networks can be interpreted as high dimensional feature selection methods. More importantly, we then prove that the equivalent convex problem can be globally optimized by a standard convex optimization solver with a polynomial-time complexity with respect to the number of samples and data dimension when the width of the network is fixed. Finally, we numerically validate our theoretical results via experiments involving both synthetic and real datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-ergen21a, title = {Global Optimality Beyond Two Layers: Training Deep ReLU Networks via Convex Programs}, author = {Ergen, Tolga and Pilanci, Mert}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {2993--3003}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/ergen21a/ergen21a.pdf}, url = {https://proceedings.mlr.press/v139/ergen21a.html}, abstract = {Understanding the fundamental mechanism behind the success of deep neural networks is one of the key challenges in the modern machine learning literature. Despite numerous attempts, a solid theoretical analysis is yet to be developed. In this paper, we develop a novel unified framework to reveal a hidden regularization mechanism through the lens of convex optimization. We first show that the training of multiple three-layer ReLU sub-networks with weight decay regularization can be equivalently cast as a convex optimization problem in a higher dimensional space, where sparsity is enforced via a group $\ell_1$-norm regularization. Consequently, ReLU networks can be interpreted as high dimensional feature selection methods. More importantly, we then prove that the equivalent convex problem can be globally optimized by a standard convex optimization solver with a polynomial-time complexity with respect to the number of samples and data dimension when the width of the network is fixed. Finally, we numerically validate our theoretical results via experiments involving both synthetic and real datasets.} }
Endnote
%0 Conference Paper %T Global Optimality Beyond Two Layers: Training Deep ReLU Networks via Convex Programs %A Tolga Ergen %A Mert Pilanci %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-ergen21a %I PMLR %P 2993--3003 %U https://proceedings.mlr.press/v139/ergen21a.html %V 139 %X Understanding the fundamental mechanism behind the success of deep neural networks is one of the key challenges in the modern machine learning literature. Despite numerous attempts, a solid theoretical analysis is yet to be developed. In this paper, we develop a novel unified framework to reveal a hidden regularization mechanism through the lens of convex optimization. We first show that the training of multiple three-layer ReLU sub-networks with weight decay regularization can be equivalently cast as a convex optimization problem in a higher dimensional space, where sparsity is enforced via a group $\ell_1$-norm regularization. Consequently, ReLU networks can be interpreted as high dimensional feature selection methods. More importantly, we then prove that the equivalent convex problem can be globally optimized by a standard convex optimization solver with a polynomial-time complexity with respect to the number of samples and data dimension when the width of the network is fixed. Finally, we numerically validate our theoretical results via experiments involving both synthetic and real datasets.
APA
Ergen, T. & Pilanci, M.. (2021). Global Optimality Beyond Two Layers: Training Deep ReLU Networks via Convex Programs. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:2993-3003 Available from https://proceedings.mlr.press/v139/ergen21a.html.

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