The maximum a-posteriori (MAP) perturbation framework has emerged as a useful approach for inference and learning in high dimensional complex models. By maximizing a randomly perturbed potential function, MAP perturbations generate unbiased samples from the Gibbs distribution. Unfortunately, the computational cost of generating so many high-dimensional random variables can be prohibitive. More efficient algorithms use sequential sampling strategies based on the expected value of low dimensional MAP perturbations. This paper develops new measure concentration inequalities that bound the number of samples needed to estimate such expected values. Applying the general result to MAP perturbations can yield a more efficient algorithm to approximate sampling from the Gibbs distribution. The measure concentration result is of general interest and may be applicable to other areas involving expected estimations.
There are several results on measure concentration for Lipschitz functions of random variables (c.f. Maurey and ). In this work we use logarithmic Sobolev inequalities @cite_0 and prove a new measure concentration result for random variables. To do this we generalize a classic result of on Poincar ' e inequalities to non-strongly log-concave distributions, and also recover the concentration result of for functions of Laplace random variables.
@cite_0
2167969281 1873349594 1972615020 2165651967
Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are 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')| 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 paper we prove 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 why concentration should hold. Indeed, given an event that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where occurs. The point is that the resulting typical changes c_k are often much smaller than the worst case ones. To illustrate its application we consider 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 Bollob 'as and Erd o s. In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of random symmetric matrices. We show that our concentration inequality is stronger than the well-known concentration inequality due to Kim and Vu [29]. The main advantage of our method in comparison with the existing ones is a wide range of random variables we can handle and bounds for previously intractable regimes of high degree polynomials and small expectations. On the negative side we show that even for boolean random variables each term in our concentration inequality is tight. We prove concentration inequalities for some classes of Markov chains and (F-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures of Talagrand. The method is based on information inequalities put forward by Marton in case of contracting Markov chains. Using a simple duality argument on entropy, our results also include the family of logarithmic Sobolev inequalities for convex functions. Applications to bounds on supremum of dependent empirical processes complete this work. The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincar ' e -type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.