VC uniform convergence theorem (${\sup }_{h\in H}|{\hat{R}}_m(h)-R(h)|\le \eta$)

A finite VC dimension gives simultaneous control of empirical misclassification risk for every member of a hypothesis class. Sufficient independent sample size is $A{\eta }^{-2}(v\log (2/\eta )+\log (2/\delta ))$. The displayed event implies an empirical risk minimization rule has misclassification risk at most ${\inf }_{h}R(h)+2\eta$, including for unrealizable labels.^[ref]