Let $\epsilon_1,...,\epsilon_n$ be i.i.d. random signs, $\mathbf{u}_1,...,\mathbf{u}_n$ i.i.d. uniform random vectors on the unit sphere $\mathbb{S}^{d-1}$, assuming $d$ even, and $\mathbf{v}_1,...,\mathbf{v}_n$ be their half-truncations, that is $\mathbf{v}_i[j] = \mathbf{u}_i[j]$ for all $j \in \left \{1,...,d/2\right\}$, else $\mathbf{v}_i[j] = 0$, where $\mathbf{v}_i[j]$ denotes the $j$-th entry of $\mathbf{v}_i$. We assume that the $\epsilon_i$'s are independent from the $\mathbf{u}_i$'s.
I am looking for nontrivial lower bounds on the following quantity. $$\mathbb{E}\left |\sum_{i = 1}^n \epsilon_i \| \mathbf{v}_i\|^2_2 \right |$$ Any help would be greatly appreciated!