This talk studies hypothesis testing and confidence interval construction in high-dimensional linear models where robustness is view in the context of whether the data is truly sparse. We will show new concepts of uniform and essentially uniform non-testability that allow the study of the limitations of tests across a broad set of alternatives. Uniform non-testability identifies an extensive collection of alternatives such that the power of any test, against any alternative in this group, is asymptotically at most equal to the nominal size, whereas minimaxity shows the existence of one, particularly "bad" alternative. Implications of the new constructions include new minimax testability results that in sharp contrast to existing results do not depend on the sparsity of the model parameters and are therefore robust. We identify new tradeoffs between testability and feature correlation. In particular, we show that in models with weak feature correlations minimax lower bound can be attained by a confidence interval whose width has the parametric rate regardless of the size of the model sparsity.

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