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Let's consider a classic feedforward neural network $F$ with input dimension $d$, output dimension $k$, $L$ layers $l_i$ with $m$ neurons each. ReLu activation.

This means that, given a point $x \in R^d$ its image $F(x) \in R^k$. Let's now assume i add some gaussian noise $\eta_i$ in EVERY hidden layer $l_i(x)$ at the same time, where the norm of this noise is 5% the norm of its layer computed on the point $x$. Let's call this new neural network $F_*$

I know that, empirically, neural networks are resistant to this kind of noise, especially on the first layers. How can i show this theoretically?

The question i'm trying to answer is the following:

After having injected this noise $\eta_i$ in every layer $l_i(x)$, how far the output $F_{*}(x)$ will be from the output of the original neural network $F(x)$?

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