  In recent years, plenty of metrics have been proposed to identify networks
that are free of gradient explosion and vanishing. However, due to the
diversity of network components and complex serial-parallel hybrid connections
in modern DNNs, the evaluation of existing metrics usually requires strong
assumptions, complex statistical analysis, or has limited application fields,
which constraints their spread in the community. In this paper, inspired by the
Gradient Norm Equality and dynamical isometry, we first propose a novel metric
called Block Dynamical Isometry, which measures the change of gradient norm in
individual block. Because our Block Dynamical Isometry is norm-based, its
evaluation needs weaker assumptions compared with the original dynamical
isometry. To mitigate the challenging derivation, we propose a highly
modularized statistical framework based on free probability. Our framework
includes several key theorems to handle complex serial-parallel hybrid
connections and a library to cover the diversity of network components.
Besides, several sufficient prerequisites are provided. Powered by our metric
and framework, we analyze extensive initialization, normalization, and network
structures. We find that Gradient Norm Equality is a universal philosophy
behind them. Then, we improve some existing methods based on our analysis,
including an activation function selection strategy for initialization
techniques, a new configuration for weight normalization, and a depth-aware way
to derive coefficients in SeLU. Moreover, we propose a novel normalization
technique named second moment normalization, which is theoretically 30% faster
than batch normalization without accuracy loss. Last but not least, our
conclusions and methods are evidenced by extensive experiments on multiple
models over CIFAR10 and ImageNet.
