Distribution-dependent concentration inequalities for tighter generalization bounds

We prove several distribution-dependent extensions of Hoeffding and McDiarmid's inequalities with (difference-) unbounded and hierarchically (difference-) bounded functions. For this purpose, several assumptions about the probabilistic boundedness and bounded differences are introduced. Our approaches improve the previous concentration inequalities' bounds, and achieve tight bounds in some exceptional cases where the original inequalities cannot hold. Furthermore, we discuss the potential applications of our extensions in VC dimension and Rademacher complexity. Then we obtain generalization bounds for (difference-) unbounded loss functions and tighten the existing generalization bounds.