Learning convex sets under uniform distribution

Abstract

In order to learn a convex set C, an algorithm is given a random sample of points and the information which of the points belong to C. From this sample a set C′ is constructed which is supposed to be a good approximation of C. The algorithm may have a small probability of failing. We measure the quality of the approximation by minimizing the probability that a random test point selected under the same distribution as the sample points is classified correctly. That minimum is taken over a set of distributions associated with C.

Learnability depends on the choice of these distributions of the sample points. If we allow too many distributions, then convex sets are not learnable. We set up a model for learning from equidistributed samples. Let 1/m be the error probability for classifying a random test point. We show that for learning d-dimensional convex sets a sample of size Ω(m ^(d+1)/2) is sufficient and necessary. The upper bound is obtained by analysing the sample size needed for the convex hull algorithm.

Research supported by DFG grant RE 672/1