We consider the problem of PAC-learning decision trees, i.e., learning a decision tree over the n-dimensional hypercube from independent random labeled examples. Despite significant effort, no polynomial-time algorithm is known for learning polynomial-sized decision trees (even trees of any super-constant size), even when examples are assumed to be drawn from the uniform distribution on 0,1 ^n. We give an algorithm that learns arbitrary polynomial-sized decision trees for most product distributions . In particular, consider a random product distribution where the bias of each bit is chosen independently and uniformly from, say, [.49,.51]. Then with high probability over the parameters of the product distribution and the random examples drawn from it, the algorithm will learn any tree. More generally, in the spirit of smoothed analysis, we consider an arbitrary product distribution whose parameters are specified only up to a [-c,c] accuracy (perturbation), for an arbitrarily small positive constant c.
@cite_9: We introduce the smoothed analysis of algorithms, which continuously interpolates between the worst-case and average-case analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has smoothed complexity polynomial in the input size and the standard deviation of Gaussian perturbations.
We consider a smoothed learning'' model inspired by Smoothed Analysis, which Spielman and Teng introduced to explain why the simplex method for linear programming (LP) usually runs in polynomial time @cite_9 . Roughly speaking, they show that if each parameter of an LP is perturbed by a small amount, then the simplex method will run in polynomial time with high probability (in fact, the expected run-time will be polynomial). For LP's arising from nature or business (as opposed to reduction from another computational problem), the parameters are measurements or estimates that have some inherent inaccuracy or uncertainty. Hence, the model is reasonable for a large class of interesting LP's.