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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, for simplicity, that every weight matrix has a norm upper bounded by a constant $M$. Let's now assume that I have fixed parameters so that we obtain $v = F(W_1, \dots, W_L, x^*) \in \mathbb{R}^k$ (note that $x^*$ is fixed as well).

Now imagine 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% of 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 changes in the weights described before?

Note that this has nothing to do with the learning process, it's just about the sensitivity/resistance of a neural network with respect to random noise injected in a weight matrix.

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Its very hard to determine how exactly it would affect the learning of the network but from my experience, these are the possible scenarios. on how it can affect the output:

  1. It might take your network a longer time to converge
  2. Your network might never really reach the accuracy values as seen before adding the noise.
  3. Certain weights can be dropped during learning or made to 0 when using a regularizer.

I am not quite aware if the affect on the output can be really modeled by a mathematical equation when a certain type of noise is added, if that is what you are looking for.

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  • $\begingroup$ I'm not talking about the learning process, I'm assuming that the parameters of the neural network are fixed. I'm looking at the NN as a mathematical function which takes as an input points in a $d$-dimensional vector space and outputs points in a $k$ dimensional one. Now, once I have this function, which has fixed parameters, I add the noise mentioned above. My question is how this noise affects the output of the function. I'll edit because I think I wasn't clear enough $\endgroup$ – Alfred Jan 13 '19 at 19:49
  • $\begingroup$ IMO, the function will not change, just the parameters will be tuned to either learn the noise or learn to ignore the noise based on regularization methods used. Neural network as a function is not modeled around the dataset but on the problem the network needs to solve. $\endgroup$ – Nischal Hp Jan 14 '19 at 15:10
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A lot depends on the nature of the noise you inject (Sparse and spiky or white noise or uniformly spread DC shift) and the type of regularizer used (TV regularizer, Lasso, ridge, Elastic net). But I guess, the Neural network will start to learn the noise as well with the data to some extent if the norm of the noise is 10% of the norm of the weight matrix.

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