I'm trying to prove a result but I'm stuck at the very end of it: I'm having troubles understanding how noise propagates when considering a probability distribution. In other words, if I inject some noise in a vector, how does it change its probability distribution? Let's give some notation:

Let $\sigma: \mathbb{R}^d \rightarrow (0,1)^d$ be the softmax operator which maps $x$ to the manifold of the probability distributions on $d$ variables defined in the following way: $$\sigma(x)_i = \frac{e^{x_i}}{\sum_j e^{x_j}}$$ I have a vector $x \in \mathbb{R}^d$ with norm one. I also have random noise vector $\eta \in \mathbb{R}^d$ with norm $1/k$ taken uniformely at random from the sphere $S^{d-1}$ of radius 1/k, where k is a large natural number.

I need to compute an upper bound of the following expected value, which is the norm of the difference between the vector without noise and the one with injected noise $\eta$:

$$ \mathbb{E}[||\sigma(x) - \sigma(x+\eta)||]$$

I gave a bounty to this question because I really need to understand this problem (I will have to solve several problems of this kind).

Thank you very much!