Suppose that $x \sim \cal{N}(0,I_d)$ be a $d$-dimensional standard Gaussian vector and let $x_1,\ldots,x_n$ denote $n$ i.i.d. samples drawn from the same distribution.
For some fixed vector $\theta \in \mathbb{R}^d$, I am trying to obtain a concentration inequality as follows: $$ \mathbb{P}\left(\|\frac{1}{n} \sum_{i=1}^n f(\theta^\top x_i) x_i - \mathbb{E}[f(\theta^\top x) x] \| \geq \epsilon \right) \leq f(n,\epsilon), $$ where $f(z)=\frac{1}{1+e^{-z}}$ is the logistic-sigmoid function. Can anyone provide me a suggestion or reference to how one would approach this problem?
