Global risk bounds and adaptation in univariate convex regression

Understand

We consider the problem of nonparametric estimation of a convex regression function $\phi_0$.

• We study the risk of the least squares estimator (LSE) under the natural squared error loss.
• We show that the risk is always bounded from above by $n^{-4/5}$ modulo logarithmic factors while being much smaller when $\phi_0$ is well-approximable by a piecewise affine convex function with not too many affine pieces (in which case, the risk is at most $1/n$ up to logarithmic factors).
• On the other hand, when $\phi_0$ has curvature, we show that no estimator can have risk smaller than a constant multiple of $n^{-4/5}$ in a very strong sense by proving a "local" minimax lower bound.