# Bidirectionally Self-Normalizing Neural Networks

arXiv: https://arxiv.org/abs/2006.12169 · HTML: https://mlsanity.com/paper/2006.12169/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2020.

## Built on by (most cited first, 3 of 3)

- [Deep Learning without Shortcuts: Shaping the Kernel with Tailored Rectifiers](https://mlsanity.com/paper/2203.08120/) (2022)
- [Contrastive Forward-Forward: A Training Algorithm of Vision Transformer](https://mlsanity.com/paper/2502.00571/) (2025)
- [GNN-VPA: A Variance-Preserving Aggregation Strategy for Graph Neural Networks](https://mlsanity.com/paper/2403.04747/) (2024)

## Abstract (arXiv)

The problem of vanishing and exploding gradients has been a long-standing obstacle that hinders the effective training of neural networks. Despite various tricks and techniques that have been employed to alleviate the problem in practice, there still lacks satisfactory theories or provable solutions. In this paper, we address the problem from the perspective of high-dimensional probability theory. We provide a rigorous result that shows, under mild conditions, how the vanishing/exploding gradients problem disappears with high probability if the neural networks have sufficient width. Our main idea is to constrain both forward and backward signal propagation in a nonlinear neural network through a new class of activation functions, namely Gaussian-Poincar'e normalized functions, and orthogonal weight matrices. Experiments on both synthetic and real-world data validate our theory and confirm its effectiveness on very deep neural networks when applied in practice.