I have written these notes to accompany my first two tutorial lectures on concentration of measure given at IPAM during March 20–23, 2018. I intend the lectures to be quite basic, so they will miss out many large and important topics related to concentration. These notes are a guide to my lectures and to further reading, so only a few proofs are given in full here. I am grateful to a number of ...

Concentration bounds for non-product, non-Haar measures are fairly recent: the first such result was obtained for contracting Markov chains by Marton in 1996. Since then, several other such results have been proved; with few exceptions, these rely on coupling techniques. Though coupling is of unquestionable utility as a theoretical tool, it appears to have some limitations. Coupling has yet to ...

We prove an apparently novel concentration of measure result for Markov tree processes. The bound we derive reduces to the known bounds for Markov processes when the tree is a chain, thus strictly generalizing the known Markov process concentration results. We employ several techniques of potential independent interest, especially for obtaining similar results for more general directed acyclic ...

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Stability conditions can be thought of as a way of controlling the variance of the learning process. Strong stability conditions additionally imply concentration of certain quantities around their expected values. It was shown recently that stability of learning algorithms is closely related to their generalization and consistency. In this paper we examine stability conditions from this point o...

Suppose A is a finite set, let P be a discrete distribution on A, and let M be an arbitrary “mass” function on A. We give a precise characterization of the most efficient way in which An can be almost-covered using spheres of a fixed radius. An almost-covering is a subset Cn of An, such that the union of the spheres centered at the points of Cn has probability close to one with respect to the p...