supervised learning with multidimensional output label

I am able to implement supervised learning algorithms when the output label ($Y$) is one dimensional. Just in case, the output label is multidimensional ($Y$= [$y_1$ $y_2$ ... $y_s$]), I believe that for such cases, we have to have $s$ different models with $i^{th}$ model predicting $y_i$ from the input data.

Am I right with this interpretation ? Any better alternative ?

Let's assume I am implementing linear regression algorithm on a labelled data with the output label being $s$ dimensional.