Chengwu Liu, Yichun Yin, Ye Yuan, Jiaxuan Xie, Botao Li, Siqi Li, Jianhao Shen, Yan Xu, Lifeng Shang, Ming Zhang. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2026.

This work finds the analytical expression of the global minima of a deep linear network with weight decay and stochastic neurons, a fundamental model for understanding the landscape of neural networks. Our result implies that zero is a special point in deep neural network architecture. We show that weight decay strongly interacts with the model architecture and can create bad minima at zero in a network with more than $1$ hidden layer, qualitatively different from a network with only $1$ hidden layer. Practically, our result implies that common deep learning initialization methods are insufficient to ease the optimization of neural networks in general.