Before revision:  We then propose a mechanism such that the number of users scales as \mathcal{O}(k/(m\alpha^2) + k/m\epsilon\alpha) and further show that it is nearly-optimal under certain regimes. Thus the privacy penalty is \mathcal{O \Theta (m) times smaller compared to the standard mechanisms  .
After revision:  We then propose a mechanism such that the number of users scales as \mathcal{O}(k/(m\alpha^2) + k/m\epsilon\alpha) and hence the privacy penalty is \mathcal{O \Theta (m) times smaller compared to the standard mechanisms .