Unveiling The Matthew Effect Across Channels: Assessing Layer Width Sufficiency via Weight Norm Variance

The trade-off between cost and performance has been a longstanding and critical issue for deep neural networks. One key factor affecting the computational cost is the width of each layer. However, in practice, the width of layers in a neural network is mostly empirically determined. In this paper, we show that a pattern regarding the variance of weight norm corresponding to different channels can indicate whether the layer is sufficiently wide and may help us better allocate computational resources across the layers. Starting from a simple intuition that channels with larger weights would have larger gradients and the difference in weight norm enlarges between channels with similar weight, we empirically validate that wide and narrow layers show two different patterns with experiments across different data modalities and network architectures. Based on the two different patterns, we identify three stages during training and explain each stage with corresponding evidence. We further propose to adjust the width based on the identified pattern and show that conventional layer width settings for CNNs could be adjusted to reduce the number of parameters while boosting the performance.