Information Geometry of Orthogonal Initializations and Training

By Piotr A. Sokól, Il-Su Park. 2018

17 citations recorded by the source corpus.

Recently mean field theory has been successfully used to analyze properties of wide, random neural networks. It gave rise to a prescriptive theory for initializing feed-forward neural networks with orthogonal weights, which ensures that both the forward propagated activations and the backpropagated gradients are near $\ell_2$ isometries and as a consequence training is orders of magnitude faster. Despite strong empirical performance, the mechanisms by which critical initializations confer an advantage in the optimization of deep neural networks are poorly understood. Here we show a novel connection between the maximum curvature of the optimization landscape (gradient smoothness) as measured by the Fisher information matrix (FIM) and the spectral radius of the input-output Jacobian, which partially explains why more isometric networks can train much faster. Furthermore, given that orthogonal weights are necessary to ensure that gradient norms are approximately preserved at initialization, we experimentally investigate the benefits of maintaining orthogonality throughout training, from which we conclude that manifold optimization of weights performs well regardless of the smoothness of the gradients. Moreover, motivated by experimental results we show that a low condition number of the FIM is not predictive of faster learning.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Ling Shao
2. Fan Zhu
3. Li Liu
4. James Lucas
5. Yi Zhou
6. Lei Huang
7. Jie Qin
8. Roger Baker Grosse

In Computer Vision and Pattern Recognition

1. Ling Shao
2. Li Liu
3. Fan Zhu
4. Yi Zhou
5. Jie Qin
6. Lei Huang
7. James Lucas
8. Luc Van Gool

Citations in the corpus

• Sorting out Lipschitz function approximation (2018, methodology): “This property has been shown to speed up training by orders of magnitude when enforced during weight initialization [37, 43], and explored in the contexts of training RNNs [9] and very deep convolutional neural networks [54].”
• Normalization Techniques in Training DNNs: Methodology, Analysis and Application (2020, background): “entially. For example, the weight initialization techniques aim to satisfy Criteria 1, obtaining nearly equal variances for layer input/output-gradients across different layers [3], [44], [45], [46], [47] by designing initial weight matrices. However, the equal-variance property across layers can be”