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Accuracy Assessment for High-dimensional Linear Regression

This paper considers point and interval estimation of the $\ell_q$ loss of an estimator in high-dimensional linear regression with random design. We establish the minimax rate for estimating the $\ell_{q}$ loss and the minimax expected length of confidence intervals for the $\ell_{q}$ loss of rate-optimal estimators of the regression vector, including commonly used estimators such as Lasso, scaled Lasso, square-root Lasso and Dantzig Selector. Adaptivity of the confidence intervals for the $\ell_{q}$ loss is also studied. Both the setting of known identity design covariance matrix and known noise level and the setting of unknown design covariance matrix and unknown noise level are studied. The results reveal interesting and significant differences between estimating the $\ell_2$ loss and $\ell_q$ loss with $1\le q <2$ as well as between the two settings. New technical tools are developed to establish rate sharp lower bounds for the minimax estimation error and the expected length of minimax and adaptive confidence intervals for the $\ell_q$ loss. A significant difference between loss estimation and the traditional parameter estimation is that for loss estimation the constraint is on the performance of the estimator of the regression vector, but the lower bounds are on the difficulty of estimating its $\ell_q$ loss. The technical tools developed in this paper can also be of independent interest.