Oracle Inequalities for High-dimensional Prediction
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The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. However, the common theoretical bounds for the predictive performance of these estimators hinge on strong, in practice unverifiable assumptions on the design. In this paper, we introduce a new set of oracle inequalities for prediction in high-dimensional linear regression. These bounds hold irrespective of the design matrix. Moreover, since the proofs rely only on convexity and continuity arguments, the bounds apply to a wide range of penalized estimators. Overall, the bounds demonstrate that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bounds can help to identify the potential of specific estimators, and they can help to get a sense of the prediction accuracy in a given application.
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