Finding significant combinations of features in the presence of categorical covariates
In high-dimensional settings, where the number of features p is much larger than the number of samples n, methods that systematically examine arbitrary combinations of features have only recently begun to be explored. However, none of the current methods is able to assess the association between feature combinations and a target variable while conditioning on a categorical covariate. As a result, many false discoveries might occur due to unaccounted confounding effects. We propose the Fast Automatic Conditional Search (FACS) algorithm, a significant discriminative itemset mining method which conditions on categorical covariates and only scales as O(k log k), where k is the number of states of the categorical covariate. Based on the Cochran-Mantel-Haenszel Test, FACS demonstrates superior speed and statistical power on simulated and real-world datasets compared to the state of the art, opening the door to numerous applications in biomedicine.
This paper proposed a discriminative itemset mining algorithm named FACS that deals with categorical co-founding effect in statistical analysis with specific applications in biology.  FACS corrects for a confounding covariate in multiple hypothesis testing. The main contribution of FACS is to allow applying Tarone's testability criterion to the CMH test. At the same time, FACS is very fast and scalable.  
The paper is well written and clearly organized, especially where it motivates the problem and the goals of the paper. It is a novel point of view to aim to find all feature subsets for which a statistical association test rejects the null hypothesis, and thus allowing to correct for a confounding categorical covariate. The authors results in the scope of the paper demonstrates that FACS keeps the computational efficiency, statistical power and the ability to correct for multiple hypothesis testing of existing method. The introduction of the branch-and-bound algorithm for conditional statistical association tests is novel and well-explained. Along with the results, the authors provide good intuition behind the methods and the key aspects, such as the appropriate testability criterion for the CMH test and the pruning criterion for the CMH test. So I think this work has provides a lot of new insights to field of machine learning in very high-dimensional settings.   The paper has some minor shortcomings: 1) The proposed technique FACS is limited to applications with binary features with binary labels and cofounding factors. This setting is restricting its wide applications, although the extension of continuous setting is not trivial. 2) The experimental part is weak. In simulations, I would not agree with the authors' conclusion that Bonf-CMH is worse than FACS. Bonf-CMH actually performs well in terms of speed and false positive detection with good precision (Figure 1C). More insights could be put into the distinction between the two methods. Further, in real-world applications, FACS is only tested on small size GWAS dataset. I would strongly recommend authors to apply FACS on large scale GWAS data, where we have many cases and controls and large number of genes and report the effectiveness. On such a small datasets, it is really hard to validate the proposed algorithm.