I think I understand each of these concepts (cross-validation, regularization) independently, but I'm not quite clear on how they can be put together in practice.
Loosely speaking, in cross-validation I will train my models on subsets of my data, and then choose the model that performs best on the reserved portion of data. In regularization I will heuristically choose some sort of regularizer function and then try to find the parameter $\lambda$ that gives the best results. Can we use cross-validation to pick $\lambda$? I think each different value of $\lambda$ can be seen as yielding a new model, but then don't we have infinitely many models to choose from?