# Scaling up deep neural networks: a capacity allocation perspective

## TL;DR

This paper investigates how capacity propagation in deep neural networks affects their ability to avoid the shattering problem, proposing scaling relations for weights and biases to ensure stable training as networks grow deeper.

## Contribution

It formulates a conjecture linking capacity propagation to the shattering problem and derives practical scaling rules for weights and biases in various architectures.

## Key findings

- Reveals conditions for capacity propagation to prevent shattering.
- Derives scaling rules consistent with Xavier initialization.
- Provides guidelines for scaling weights in residual and recurrent networks.

## Abstract

Following the recent work on capacity allocation, we formulate the conjecture that the shattering problem in deep neural networks can only be avoided if the capacity propagation through layers has a non-degenerate continuous limit when the number of layers tends to infinity. This allows us to study a number of commonly used architectures and determine which scaling relations should be enforced in practice as the number of layers grows large. In particular, we recover the conditions of Xavier initialization in the multi-channel case, and we find that weights and biases should be scaled down as the inverse square root of the number of layers for deep residual networks and as the inverse square root of the desired memory length for recurrent networks.