Oracle Inequalities for High-dimensional Prediction
This provides foundational theoretical support for practitioners using high-dimensional regression methods, though it is incremental as it extends existing oracle inequality frameworks.
The paper tackles the problem of establishing theoretical guarantees for prediction accuracy in high-dimensional linear regression using penalized estimators like the lasso, by deriving a general oracle inequality that holds for any design matrix and demonstrates consistent prediction.
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. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bound 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.