  We study the estimation error of constrained M-estimators, and derive
explicit upper bounds on the expected estimation error determined by the
Gaussian width of the constraint set. Both of the cases where the true
parameter is on the boundary of the constraint set (matched constraint), and
where the true parameter is strictly in the constraint set (mismatched
constraint) are considered. For both cases, we derive novel universal
estimation error bounds for regression in a generalized linear model with the
canonical link function. Our error bound for the mismatched constraint case is
minimax optimal in terms of its dependence on the sample size, for Gaussian
linear regression by the Lasso.
