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

Adaptive Confidence Sets for the Optimal Approximating Model