Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

Abstract

AbstractIn this paper, we prove multilevel concentration inequalities for bounded functionals $$f = f(X_1, \ldots , X_n)$$f=f(X1,…,Xn) of random variables $$X_1, \ldots , X_n$$X1,…,Xn that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes $$f(X) = \sup _{g \in {\mathcal {F}}} {|g(X)|}$$f(X)=supg∈F|g(X)| and suprema of homogeneous chaos in bounded random variables in the Banach space case $$f(X) = \sup _{t} {\Vert \sum _{i_1 \ne \ldots \ne i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots X_{i_d}\Vert }_{{\mathcal {B}}}$$f(X)=supt‖∑i1≠…≠idti1…idXi1⋯Xid‖B. The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U-statistics with bounded kernels h and for the number of triangles in an exponential random graph model.