How sensitive are Neural Networks to weight change?

Question

Let's consider the space of feedforward neural networks with a given structure: $L$ layers, $m$ neurons per layer, ReLu activation, input dimension $d$, output dimension $k$.

Which means I'm considering the map $F: \mathcal{W}_1 \times \mathcal{W}_2 \times \dots \times \mathcal{W}_L \times \mathbb{R}^d \to \mathbb{R}^k$, where $\mathcal{W}_i$ is the space of possible weights for layer $i$. We also assume, per simplicity, that every weight matrix has norm bounded by a constant $M$. Let's now fix the parameters so that we obtain $v = F(W_1, \dots, W_L, x^*) \in \mathbb{R}^k$ (note that $x^*$ is fixed as well).

Image now that I inject some random noise $\eta \in \mathbb{R}^{m \times m} $ in a weight matrix $W_i$, where the norm of the noise is 10% the norm of the matrix, e.g. $\|\eta\| = \|W_i\|/10$ . How does it affect my final output?

which means, what's the expected value of $\|v - v_*\|$, where $v_*$ is the output of the network obtained after the small change in the weights describer before?

Answer 1

Output of the ReLU network is $$v = \sum_{ij} X_i A_{ij} w^{(1)}_{ij} \cdots w^{(L)}_{ij} $$ where $i$ is the input $j$ is the path and $A_{ij}$ is $1$ if the path is open and $0$ otherwise. Now if you take the expectation and use independence of the noise and also assume that the noise has mean $0$ we get that asymptotically when $m$ is large we have $$\mathbb{E}(\lVert v - v^* \rVert_2) \leq \sqrt{\mathbb{E}(\lVert v - v^* \rVert_2^2)} \leq 1/10^{L}$$ since $A_{ij}$ and the product of parameters are asymptotically independent. Note that the expectation of $v^*$ squared is the sum of the expectation of the original network squared and the expectation of the network that contains only the noise squared.