深层神经网络输出趋于确定值，与权重分布无关
A concentration result for multilayer feedforward neural networks
- 固定层间权重分布曲线，输入独立同分布
- 随着神经元数增加，输出几乎必然收敛到某常数
- 为深层网络行为提供理论保证，适合理论研究者
考虑任意固定的ρ层结构，第一层有n个神经元（输入层），最后一层仅1个神经元（输出层）。若层间连接权重的分布对所有大n均近似于一个不依赖n的固定连续曲线，且输入神经元取值独立同分布并具有连续概率密度函数，则存在某个常数ψ，使得对任意ε > 0，当n趋于无穷时，输出神经元值落在[ψ−ε, ψ+ε]的概率趋近于1。
原文摘要 · Abstract (English)
We consider for an arbitrary fixed $ρ$ and for each positive integer $n$ a multilayer feedforward artificial neural network with $ρ$ layers, $n$ neurons in the first layer (the input layer) and only one neuron, the output neuron, in the last layer. Very roughly formulated, the main result is that if the distribution of weights of connections from a layer to the next are, for all large $n$, approximated well by a fixed continuous (but otherwise arbitrary) curve which does not depend on $n$, and if the values of the $n$ input neurons are independently and identically distributed with a continuous probability density function, then there is a number $ψ$ such that for all $\varepsilon > 0$ the probability that the value of the output neuron is in $[ψ- \varepsilon, ψ+ \varepsilon]$ tends to 1 as $n$ tends to infinity.
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