Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training

• URL: http://arxiv.org/abs/2311.18022v6
• Date: Tue, 04 Feb 2025 21:55:41 GMT
• Title: Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training
• Authors: Max Milkert, David Hyde, Forrest Laine,
• Abstract summary: In a neural network with ReLULU activations, the number of piecewise linear regions in the output can grow exponentially with depth. We introduce a novel parameterization of the network that restricts the network that restricts its weights to its regions throughout training. This approach allows us to learn approximations of convex convex functions that are several orders of magnitude more accurate than their randomly counterparts.
• Score: 1.7205106391379021
• License:
• Abstract: In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth. However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore often leads to the use of networks that are unnecessarily large. To address this problem, we introduce a novel parameterization of the network that restricts its weights so that a depth $d$ network produces exactly $2^d$ linear regions at initialization and maintains those regions throughout training under the parameterization. This approach allows us to learn approximations of convex, one dimensional functions that are several orders of magnitude more accurate than their randomly initialized counterparts. We further demonstrate how to extend our approach to multidimensional and non-convex functions, allowing it to replace the dense layers in other networks; preliminary improvements are shown for image classification on CIFAR-10 and ImageNet.

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