An Improved Global Risk Bound in Concave Regression

A new risk bound is presented in the problem of convex/concave function estimation, for the least squares estimator. The best known risk bound, as had appeared in Guntuboyina and Sen(2013), scaled like $\log(en) n^{-4/5}$ under the mean squared error loss, upto a constant factor. The authors in Guntuboyina and Sen(2013) had conjectured that the logarithmic term may be an artifact of their proof. We show that indeed the logarithmic term is unnecessary and prove a risk bound which scales like $n^{-4/5}$ upto constant factors. Our risk bound holds in expectation as well as with high probability and also extends to the case of model misspecification.