Mathematics > Probability
[Submitted on 3 Dec 2018 (v1), last revised 19 May 2020 (this version, v5)]
Title:Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities
View PDFAbstract:In this paper we prove multilevel concentration inequalities for bounded functionals $f = f(X_1, \ldots, X_n)$ of random variables $X_1, \ldots, X_n$ 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 in both dependent and independent random variables. This includes deviation inequalities for empirical processes $f(X) = \sup_{g \in \mathcal{F}} \lvert g(X) \rvert$ and suprema of homogeneous chaos in bounded random variables in the Banach space case given by $f(X) = \sup_{t} \lVert \sum_{i_1 \neq \ldots \neq i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots X_{i_d}\rVert_{\mathcal{B}}$. The latter application is comparable to earlier results of Boucheron-Bousquet-Lugosi-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.
Submission history
From: Arthur Sinulis [view email][v1] Mon, 3 Dec 2018 21:52:52 UTC (39 KB)
[v2] Thu, 27 Dec 2018 16:40:46 UTC (27 KB)
[v3] Wed, 13 Mar 2019 15:19:30 UTC (34 KB)
[v4] Fri, 21 Jun 2019 09:32:20 UTC (38 KB)
[v5] Tue, 19 May 2020 17:56:13 UTC (46 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.