When dealing with modern big data sets, a very common theme is reducing the set through a random process. These generally work by making “many simple estimates” of the full data set, and then judging them as a whole. Perhaps magically, these “many simple estimates” can provide a very accurate and small representation of the large data set. The key tool in showing how many of these simple estima...

When dealing with modern big data sets, a very common theme is reducing the set through a random process. These generally work by making “many simple estimates” of the full data set, and then judging them as a whole. Perhaps magically, these “many simple estimates” can provide a very accurate and small representation of the large data set. The key tool in showing how many of these simple estima...

A new line of work [6, 9, 15, 2] demonstrates how differential privacy [8] can be used as a mathematical tool for guaranteeing generalization in adaptive data analysis. Specifically, if a differentially private analysis is applied on a sample S of i.i.d. examples to select a lowsensitivity function f , then w.h.p. f (S) is close to its expectation, although f is being chosen based on the data. ...

1 Concentration Inequalities Lemma 1 (Markov’s inequality). Let X be a non-negative random variable. For all λ > 0, Pr[X > λ] ≤ E[X] λ Lemma 2 (Chebyshev Inequality). For all λ > 0, Pr[|X −E[X]| > λ] ≤ var[X] λ2 Lemma 3 (The Chernoff Bound: Upper bound). Let X1, X2, ..., Xn be independent random variables taking values in {0, 1} with E[Xi] = pi. Let X = ∑n i=1Xi, and μ = E[X]. Then the followin...

The problem of const.ructing shifts of work for busdrivers from given workblocks is shown to be a minimum cost matching problem. A branch and bound algorithm for this problem is derived which, while not having the guaranteed polynomial time complexity of the well known efficient labelling methods, is shown in extensive compu lational testing to be (1) extremely fast OIl the class of problem whe...