  Manifold regularization is a commonly used technique in semi-supervised
learning. It enforces the classification rule to be smooth with respect to the
data-manifold. Here, we derive sample complexity bounds based on
pseudo-dimension for models that add a convex data dependent regularization
term to a supervised learning process, as is in particular done in Manifold
regularization. We then compare the bound for those semi-supervised methods to
purely supervised methods, and discuss a setting in which the semi-supervised
method can only have a constant improvement, ignoring logarithmic terms. By
viewing Manifold regularization as a kernel method we then derive Rademacher
bounds which allow for a distribution dependent analysis. Finally we illustrate
that these bounds may be useful for choosing an appropriate manifold
regularization parameter in situations with very sparsely labeled data.
