The way I know of to bound generalization error by Rademacher complexity is Theorem 2.4 in this lecture notes, http://ttic.uchicago.edu/~tewari/lectures/lecture9.pdf. Here the quantity on the LHS that Rademacher complexity is trying to upperbound is given as, $L_{\phi}(\hat{f}_{\phi}^*)-\min_{f \in F} L_{\phi}(f)$ where $F$ is some "hypothesis class" of functions mapping $ f :X \rightarrow D$, $\phi : D \times Y \rightarrow [0,1]$ is the "loss function", "$\phi-$"loss of any function $g : X \rightarrow D$ is defined as, $L_\phi(g) = \mathbb{E}[\phi(f(x),y)]$ - where the expectation is taken over some distribution over the points $(x,y) \in X \times Y$ and $\hat{f}_{\phi}^*$ is what the ERM returns over some $m$ samples i.e $\hat{f}_{\phi}^* = \mathrm{argmin}_{f \in F} \frac{1}{m} \sum_{i=1}^m \phi(f(x_i),y_i)$.
The above setting is called ``agnostic" because at no point was it assumed that, $\exists$ any ground-truth labelling function $L \in F$ such that $y = L(x)$ but rather the class $F$ is to be seen to be trying to learn via empirical risk minimization a distribution, say ${\cal D}$, over $X \times Y$.
My question is 3 fold , is there any analogue of this Theorem $2.4$ when,
(a) an existence of a $L$ is assumed with $L$ may or maynot be in $F$. (the later is I guess often called the ``realizable setting") (...I have seen some papers trying to bound generalization error of a specific algorithm in the realizable setting but I somehow dont see Rademacher complexity defined in those settings!..)
(b) the loss function $\phi$ is not assumed to be bounded above but only assumed to be bounded below.
(c) AND most importantly, say I have a class of labelling functions ${\cal L}$ mapping $X \rightarrow Y$ and I want to say the following, "Given a loss function $\phi$, irrespective of which member of ${\cal L}$ labels the data (maybe also irrespective of the distribution over $X$ used to measure $L_{\phi}$) the member of class $F$ obtained via ERM on the data, can never generalize well". Is there a version of Rademacher complexity which captures this?
