Doubly robust and computationally efficient high-dimensional variable selection

The variable selection problem is to discover which of a large set of predictors is associated with an outcome of interest, conditionally on the other predictors. This problem has been widely studied, but existing approaches lack either power against complex alternatives, robustness to model misspecification, computational efficiency, or quantification of evidence against individual hypotheses. We present tower PCM (tPCM), a statistically and computationally efficient solution to the variable selection problem that does not suffer from these shortcomings. tPCM adapts the best aspects of two existing procedures that are based on similar functionals: the holdout randomization test (HRT) and the projected covariance measure (PCM). The former is a model-X test that utilizes many resamples and few machine learning fits, while the latter is an asymptotic doubly-robust style test for a single hypothesis that requires no resamples and many machine learning fits. Theoretically, we demonstrate the validity of tPCM, and perhaps surprisingly, the asymptotic equivalence of HRT, PCM, and tPCM. In so doing, we clarify the relationship between two methods from two separate literatures. An extensive simulation study verifies that tPCM can have significant computational savings compared to HRT and PCM, while maintaining nearly identical power.