Expressiveness of Rectifier Networks

Xingyuan Pan, Vivek Srikumar
Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:2427-2435, 2016.

Abstract

Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a variety of applications. However, unlike threshold and sigmoid networks, ReLU networks are less explored from the perspective of their expressiveness. This paper studies the expressiveness of ReLU networks. We characterize the decision boundary of two-layer ReLU networks by constructing functionally equivalent threshold networks. We show that while the decision boundary of a two-layer ReLU network can be captured by a threshold network, the latter may require an exponentially larger number of hidden units. We also formulate sufficient conditions for a corresponding logarithmic reduction in the number of hidden units to represent a sign network as a ReLU network. Finally, we experimentally compare threshold networks and their much smaller ReLU counterparts with respect to their ability to learn from synthetically generated data.

Cite this Paper


BibTeX
@InProceedings{pmlr-v48-panb16, title = {Expressiveness of Rectifier Networks}, author = {Pan, Xingyuan and Srikumar, Vivek}, booktitle = {Proceedings of The 33rd International Conference on Machine Learning}, pages = {2427--2435}, year = {2016}, editor = {Balcan, Maria Florina and Weinberger, Kilian Q.}, volume = {48}, series = {Proceedings of Machine Learning Research}, address = {New York, New York, USA}, month = {20--22 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v48/panb16.pdf}, url = {http://proceedings.mlr.press/v48/panb16.html}, abstract = {Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a variety of applications. However, unlike threshold and sigmoid networks, ReLU networks are less explored from the perspective of their expressiveness. This paper studies the expressiveness of ReLU networks. We characterize the decision boundary of two-layer ReLU networks by constructing functionally equivalent threshold networks. We show that while the decision boundary of a two-layer ReLU network can be captured by a threshold network, the latter may require an exponentially larger number of hidden units. We also formulate sufficient conditions for a corresponding logarithmic reduction in the number of hidden units to represent a sign network as a ReLU network. Finally, we experimentally compare threshold networks and their much smaller ReLU counterparts with respect to their ability to learn from synthetically generated data.} }
Endnote
%0 Conference Paper %T Expressiveness of Rectifier Networks %A Xingyuan Pan %A Vivek Srikumar %B Proceedings of The 33rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2016 %E Maria Florina Balcan %E Kilian Q. Weinberger %F pmlr-v48-panb16 %I PMLR %P 2427--2435 %U http://proceedings.mlr.press/v48/panb16.html %V 48 %X Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a variety of applications. However, unlike threshold and sigmoid networks, ReLU networks are less explored from the perspective of their expressiveness. This paper studies the expressiveness of ReLU networks. We characterize the decision boundary of two-layer ReLU networks by constructing functionally equivalent threshold networks. We show that while the decision boundary of a two-layer ReLU network can be captured by a threshold network, the latter may require an exponentially larger number of hidden units. We also formulate sufficient conditions for a corresponding logarithmic reduction in the number of hidden units to represent a sign network as a ReLU network. Finally, we experimentally compare threshold networks and their much smaller ReLU counterparts with respect to their ability to learn from synthetically generated data.
RIS
TY - CPAPER TI - Expressiveness of Rectifier Networks AU - Xingyuan Pan AU - Vivek Srikumar BT - Proceedings of The 33rd International Conference on Machine Learning DA - 2016/06/11 ED - Maria Florina Balcan ED - Kilian Q. Weinberger ID - pmlr-v48-panb16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 48 SP - 2427 EP - 2435 L1 - http://proceedings.mlr.press/v48/panb16.pdf UR - http://proceedings.mlr.press/v48/panb16.html AB - Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a variety of applications. However, unlike threshold and sigmoid networks, ReLU networks are less explored from the perspective of their expressiveness. This paper studies the expressiveness of ReLU networks. We characterize the decision boundary of two-layer ReLU networks by constructing functionally equivalent threshold networks. We show that while the decision boundary of a two-layer ReLU network can be captured by a threshold network, the latter may require an exponentially larger number of hidden units. We also formulate sufficient conditions for a corresponding logarithmic reduction in the number of hidden units to represent a sign network as a ReLU network. Finally, we experimentally compare threshold networks and their much smaller ReLU counterparts with respect to their ability to learn from synthetically generated data. ER -
APA
Pan, X. & Srikumar, V.. (2016). Expressiveness of Rectifier Networks. Proceedings of The 33rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 48:2427-2435 Available from http://proceedings.mlr.press/v48/panb16.html.

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