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Many contemporary large-scale applications involve building interpretablemodels linking a large set of potential covariates to a response in a nonlinearfashion, such as when the response is binary. Although this modeling problemhas been extensively studied, it remains unclear how to effectively control thefraction of false discoveries even in high-dimensional logistic regression, notto mention general high-dimensional nonlinear models. To address such apractical problem, we propose a new framework of $model$-$X$ knockoffs, whichreads from a different perspective the knockoff procedure (Barber and Cand\`es,2015) originally designed for controlling the false discovery rate in linearmodels. Whereas the knockoffs procedure is constrained to homoscedastic linearmodels with $n\ge p$, the key innovation here is that model-X knockoffs providevalid inference from finite samples in settings in which the conditionaldistribution of the response is arbitrary and completely unknown. Furthermore,this holds no matter the number of covariates. Correct inference in such abroad setting is achieved by constructing knockoff variables probabilisticallyinstead of geometrically. To do this, our approach requires the covariates berandom (independent and identically distributed rows) with a distribution thatis known, although we provide preliminary experimental evidence that ourprocedure is robust to unknown/estimated distributions. To our knowledge, noother procedure solves the $controlled$ variable selection problem in suchgenerality, but in the restricted settings where competitors exist, wedemonstrate the superior power of knockoffs through simulations. Finally, weapply our procedure to data from a case-control study of Crohn's disease in theUnited Kingdom, making twice as many discoveries as the original analysis ofthe same data.

Panning for Gold: Model-X Knockoffs for High-dimensional Controlled Variable Selection