Generalization bounds for multiclass learning when the output is vector space?
Sounds like you're trying to learn a map from vector space $X$ to vector space $Y$. The first thing that comes to mind is regression, which is a map from $X$ to $\mathbb{R}$. You can of course perform multiple regressions in parallel -- one for each dimension of $Y$ -- to learn a map from $X$ to $Y$. This has the shortcoming of ignoring any "interaction" or dependencies between the labels. You could first perform PCA on both the inputs in $X$ and the outputs in $Y$, and learn a map from the principal vectors of the input to the principle vectors of the output.

Another idea -- one that I'm particularly fond of -- is to perform Lipschitz extension. This assumes that $X$ and $Y$ are both metric spaces with known metrics. (A much more general setting than vectors spaces!) See the pioneering paper of von Luxburg and Bousquet, for example:
http://www.jmlr.org/papers/volume5/luxburg04b/luxburg04b.pdf