On the method of typical bounded differences

ACO Seminar
Tuesday, November 15, 2016 - 13:30
1 hour (actually 50 minutes)
Skiles 005
Cambridge University and Georgia Tech
Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that functions of random variables are typically near their means. Of particular importance is the case where f(X) is a function of independent random variables X=(X_1,...,X_n). Here the well-known bounded differences inequality (also called McDiarmid's or Hoeffding--Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X)-f(X')| \leq c_k whenever X,X' differ only in X_k. While this is easy to check, the main disadvantage is that it considers worst-case changes c_k, which often makes the resulting bounds too weak to be useful. In this talk we discuss a variant of the bounded differences inequality which can be used to establish concentration of functions f(X) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations as to why concentration should hold. Indeed, given a `good' event G that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where G occurs. The point is that the resulting typical changes c_k are often much smaller than the worst case ones. If time permits, we shall illustrate its application by considering the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdös.