In this paper, we study the local well-posedness of the cubic Schrödinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an operator of degree $σ\geq 2$. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation (i.e. $\mathscr{L} = Δ$) to any dimension and obtain results under natural assumptions for general $\mathscr{L}$.