A sharp concentration inequality with applications

Abstract

We present a new general concentration-of-measure inequality and illustrate its power by applications in random combinatorics. The results nd direct applications in some problems of learning theory.

Model selection by bootstrap penalization for classification

“…This would entail improving the inequality given in Proposition 2 under propitious conditions, namely margin type conditions. Boucheron et al's (2000) concentration inequality seems to be the adequate tool, though it can not be directly applied because of the dependence between the weights involved in the bootstrap processes. Some refined Poissonization techniques may allow us to overcome this difficulty, and this may be the subject of a future work.…”

“…In this paper, we focus on random penalty functions. This subject has been tackled by Buescher and Kumar (1996), Lugosi and Nobel (1999) and Boucheron et al (2000) but the most interesting works for our approach are the ones due to Koltchinskii (2001) and Bartlett et al (2002). Let ξ denote the sample (X 1 , Y 1 ), .…”

“…This remark naturally leads to the idea of data-driven penalization. Buescher and Kumar (1996), Lugosi and Nobel (1999), Boucheron et al (2000) introduce some penalties involving the related empirical coverings or empirical Shatter coefficients. Inspired by the method of Rademacher symmetrization commonly used in the empirical processes theory (see for instance Van der Vaart and Wellner, 1996), Koltchinskii (2001) and Bartlett et al (2002) independently propose the so-called Rademacher penalties which are based on random variables of the form:…”

Concentration inequalities for functions of independent variables

“…One candidate for comparison is the result proved in [2], which has almost the same conclusion (slightly better with (2=3) t in the denominator in (6) instead of t), but requires the much stronger self-boundedness conditions…”

“…Originating in the work of Leonard Gross on logarithmic Sobolev inequalities for Gaussian measures [6], the method has been developed and re…ned by Ledoux, Bobkov, Massart, Boucheron, Lugosi, Rio, Bousquet and others ( see [8], [10], [11], [2], [3], etc) to become an important tool in the study of empirical processes and learning theory. In [2, Boucheron at al] a general theorem on con…guration functions is presented, which improves on the results obtained from the convex distance inequality.…”