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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.