Let's say we have a real vector $\theta \in \mathbb{R}^n$ and a function $f: \mathbb{R}^n\rightarrow \mathbb{R}$.
Is there a way to simplify the following expectation
$$ \mathbb{E}_{m\sim Bern(p)^n}\left[|\nabla f(\theta)^T(-\theta \otimes m)|\right] $$
where $m$ is a vector of i.i.d. bernoulli variables of size $n$ and $\otimes$ is the element wise product.
Without the absolute value we can simplify to
$$ \mathbb{E}_{m\sim Bern(p)^n}\left[\nabla f(\theta)^T(-\theta \otimes m)\right] = -p \nabla f(\theta)^T\theta $$
But I'm wondering if we can simplify the absolute value object without bounds (e.g. without using cauchy-schwarz, Lipschitz)?