Risk estimation for high-dimensional lasso regression

In high-dimensional estimation, analysts are faced with more parameters $p$ than available observations $n$, and asymptotic analysis of performance allows the ratio $p/n\rightarrow \infty$. This situation makes regularization both necessary and desirable in order for estimators to possess theoretical guarantees. However, the amount of regularization, often determined by one or more tuning parameters, is integral to achieving good performance. In practice, choosing the tuning parameter is done through resampling methods (e.g. cross-validation), generalized information criteria, or reformulating the optimization problem (e.g. square-root lasso or scaled sparse regression). Each of these techniques comes with varying levels of theoretical guarantee for the low- or high-dimensional regimes. However, there are some notable deficiencies in the literature. The theory, and sometimes practice, of many methods relies on either the knowledge or estimation of the variance parameter, which is difficult to estimate in high dimensions. In this paper, we provide theoretical intuition suggesting that some previously proposed approaches based on information criteria work poorly in high dimensions. We introduce a suite of new risk estimators leveraging the burgeoning literature on high-dimensional variance estimation. Finally, we compare our proposal to many existing methods for choosing the tuning parameters for lasso regression by providing an extensive simulation to examine their finite sample performance. We find that our new estimators perform quite well, often better than the existing approaches across a wide range of simulation conditions and evaluation criteria.