An Inequality for Nearly Log-Concave Distributions With Applications to Learning

We prove that given a nearly log-concave distribution, in any partition of the space to two well separated sets, the measure of the points that do not belong to these sets is large. We apply this isoperimetric inequality to derive lower bounds on the generalization error in learning. We further consider regression problems and show that if the inputs and outputs are sampled from a nearly log-concave distribution, the measure of points for which the prediction is wrong by more than epsi₀ and less than epsi₁ is (roughly) linear in epsi₁-epsi₀, as long as epsi₀ is not too small, and epsi₁ not too large. We also show that when the data are sampled from a nearly log-concave distribution, the margin cannot be large in a strong probabilistic sense