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Identifying the relevant variables for a classification model with correctconfidence levels is a central but difficult task in high-dimension. Despitethe core role of sparse logistic regression in statistics and machine learning,it still lacks a good solution for accurate inference in the regime where thenumber of features $p$ is as large as or larger than the number of samples $n$.Here, we tackle this problem by improving the Conditional Randomization Test(CRT). The original CRT algorithm shows promise as a way to output p-valueswhile making few assumptions on the distribution of the test statistics. As itcomes with a prohibitive computational cost even in mildly high-dimensionalproblems, faster solutions based on distillation have been proposed. Yet, theyrely on unrealistic hypotheses and result in low-power solutions. To improvethis, we propose \emph{CRT-logit}, an algorithm that combines avariable-distillation step and a decorrelation step that takes into account thegeometry of $\ell_1$-penalized logistic regression problem. We provide atheoretical analysis of this procedure, and demonstrate its effectiveness onsimulations, along with experiments on large-scale brain-imaging and genomicsdatasets.

A Conditional Randomization Test for Sparse Logistic Regression in High-Dimension