  We consider the problem of estimating an unknown $\theta\in {\mathbb{R}}^n$
from noisy observations under the constraint that $\theta$ belongs to certain
convex polyhedral cones in ${\mathbb{R}}^n$. Under this setting, we prove
bounds for the risk of the least squares estimator (LSE). The obtained risk
bound behaves differently depending on the true sequence $\theta$ which
highlights the adaptive behavior of $\theta$. As special cases of our general
result, we derive risk bounds for the LSE in univariate isotonic and convex
regression. We study the risk bound in isotonic regression in greater detail:
we show that the isotonic LSE converges at a whole range of rates from $\log
n/n$ (when $\theta$ is constant) to $n^{-2/3}$ (when $\theta$ is uniformly
increasing in a certain sense). We argue that the bound presents a benchmark
for the risk of any estimator in isotonic regression by proving nonasymptotic
local minimax lower bounds. We prove an analogue of our bound for model
misspecification where the true $\theta$ is not necessarily nondecreasing.
