  Confidence sets play a fundamental role in statistical inference. In this
paper, we consider confidence intervals for high dimensional linear regression
with random design. We first establish the convergence rates of the minimax
expected length for confidence intervals in the oracle setting where the
sparsity parameter is given. The focus is then on the problem of adaptation to
sparsity for the construction of confidence intervals. Ideally, an adaptive
confidence interval should have its length automatically adjusted to the
sparsity of the unknown regression vector, while maintaining a prespecified
coverage probability. It is shown that such a goal is in general not
attainable, except when the sparsity parameter is restricted to a small region
over which the confidence intervals have the optimal length of the usual
parametric rate. It is further demonstrated that the lack of adaptivity is not
due to the conservativeness of the minimax framework, but is fundamentally
caused by the difficulty of learning the bias accurately.
