  In the setting of high-dimensional linear models with Gaussian noise, we
investigate the possibility of confidence statements connected to model
selection. Although there exist numerous procedures for adaptive point
estimation, the construction of adaptive confidence regions is severely limited
(cf. Li, 1989). The present paper sheds new light on this gap. We develop exact
and adaptive confidence sets for the best approximating model in terms of risk.
One of our constructions is based on a multiscale procedure and a particular
coupling argument. Utilizing exponential inequalities for noncentral
chi-squared distributions, we show that the risk and quadratic loss of all
models within our confidence region are uniformly bounded by the minimal risk
times a factor close to one.
