Generalisations of the Fundamental Theorem of Statistical Learning to different tasks and losses

The fundamental theorem of statistical learning gives an equivalence between uniform convergence of the empirical risk to learning in the PAC framework.

I have only seen this stated in the case of binary classification with the 0-1 loss.
Does a result of this form hold in more general settings? For example: margin-based classification rules, regression, multi-class classification, …?

Another statement of this question could be: under what circumstances does uniform convergence of the empirical risk imply PAC learning? (I am most interested in this direction of implication.)

Please provide references if you have them.