On Adaptive confidence Ellipsoids for sparse high dimensional linear models (2310.15659v1)

Abstract: In high-dimensional linear models the problem of constructing adaptive confidence sets for the full parameter is known to be generally impossible. We propose re-weighted loss functions under which constructing fully adaptive confidence sets for the parameter is shown to be possible. We give necessary and sufficient conditions on the weights for adaptive confidence sets to exist, and exhibit a concrete rate-optimal procedure in the feasible regime.