An Improved Global Risk Bound in Concave Regression

Understand

A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator.

• The best known risk bound, as had appeared in \citet{GSvex}, scaled like $\log(en) n^{-4/5}$ under the mean squared error loss, up to a constant factor.
• The authors in \cite{GSvex} 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 $n^{-4/5}$ up to constant factors.