Distribution-dependent concentration inequalities for tighter generalization bounds

TL;DR

This paper develops distribution-dependent concentration inequalities that provide tighter generalization bounds for learning models, especially in cases where traditional bounds are too loose, by relaxing conditions and extending existing inequalities.

Contribution

It introduces four new conditions for probabilistic boundedness and bounded differences, deriving extended Hoeffding's and McDiarmid's inequalities that are more adaptable to specific distributions.

Findings

Derived distribution-dependent extensions of Hoeffding's and McDiarmid's inequalities.

Provided tighter bounds for certain functions and distribution scenarios.

Discussed applications to generalization bounds in learning theory.

Abstract

Concentration inequalities are indispensable tools for studying the generalization capacity of learning models. Hoeffding's and McDiarmid's inequalities are commonly used, giving bounds independent of the data distribution. Although this makes them widely applicable, a drawback is that the bounds can be too loose in some specific cases. Although efforts have been devoted to improving the bounds, we find that the bounds can be further tightened in some distribution-dependent scenarios and conditions for the inequalities can be relaxed. In particular, we propose four types of conditions for probabilistic boundedness and bounded differences, and derive several distribution-dependent extensions of Hoeffding's and McDiarmid's inequalities. These extensions provide bounds for functions not satisfying the conditions of the existing inequalities, and in some special cases, tighter bounds. Furthermore, we obtain generalization bounds for unbounded and hierarchy-bounded loss functions. Finally we discuss the potential applications of our extensions to learning theory.