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

Candès, Emmanuel J.; Fan, Yingying; Janson, Lucas; Lv, Jinchi

doi:10.48550/arxiv.1610.02351

Cited by 26 publications

(79 citation statements)

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“…Classical methods of FDR control depend on the assumptions on how the features and the responses are related Hochberg, 1995, Gavrilov et al, 2009]. Barber and Candes in their seminal paper [Candes et al, 2016], proposed a novel FDR control approach, called the Model-X knockoff that can be used as a statistical wrapper around any machine learning method that can select features. Model-X knockoff does not rely on the nature of the relationship between the features and responses and therefore is model-free.…”

Section: Introductionmentioning

confidence: 99%

“…Existing methods of knockoff generation either (i) assume the distribution of the features, or (ii) incorporate a generative model to learn the feature distribution from data. Second-order knockoff [Candes et al, 2016] assumes that the distribution of the features is jointly Gaussian. Another knockoff generation 2 Overview of the knockoff filter Given that X = (X 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…Otherwise, it will be considered as a relevant or important variable. Under these assumptions, the primary goal of the Model-X Knockoff filter [Candes et al, 2016] is to discover as many relevant variables as possible while keeping the FDR under control. FDR is defined as following,…”

Section: Introductionmentioning

confidence: 99%

“…To control FDR, Model-X knockoff [Candes et al, 2016] generates knockoff, Xa random vector that satisfies the following properties: Exchangeability property ensures that for any subset B ⊂ {1, 2, . .…”

Section: Introductionmentioning

confidence: 99%

“…A naive approach called Second-order knockoff [Candes et al, 2016] only approximates first two moments to satisfy the exchangeability condition assuming that the covariates follow a multivariate Gaussian distribution. However, Second-order knockoffs provide insufficient guarantees for FDR control if the assumption does not hold.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling

Masud¹,

Werenski²,

Murphy³

et al. 2021

Preprint

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We consider the problem of generating valid knockoffs for knockoff filtering which is a statistical method that provides provable false discovery rate guarantees for any model selection procedure. To this end, we are motivated by recent advances in multivariate distribution-free goodness-of-fit tests namely, the rank energy (RE), that is derived using theoretical results characterizing the optimal maps in the Monge's Optimal Transport (OT) problem. However, direct use of use RE for learning generative models is not feasible because of its high computational and sample complexity, saturation under large support discrepancy between distributions, and non-differentiability in generative parameters. To alleviate these, we begin by proposing a variant of the RE, dubbed as soft rank energy (sRE), and its kernel variant called as soft rank maximum mean discrepancy (sRMMD) using entropic regularization of Monge's OT problem. We then use sRMMD to generate deep knockoffs and show via extensive evaluation that it is a novel and effective method to produce valid knockoffs, achieving comparable, or in some cases improved tradeoffs between detection power Vs false discoveries.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling

Masud¹,

Werenski²,

Murphy³

et al. 2021

Preprint

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The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled Chi-square

Sur

Chen

Candès

2019

Probab. Theory Relat. Fields

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Logistic regression is used thousands of times a day to fit data, predict future outcomes, and assess the statistical significance of explanatory variables. When used for the purpose of statistical inference, logistic models produce p-values for the regression coefficients by using an approximation to the distribution of the likelihood-ratio test. Indeed, Wilks' theorem asserts that whenever we have a fixed number p of variables, twice the log-likelihood ratio (LLR) 2Λ is distributed as a χ 2 k variable in the limit of large sample sizes n; here, χ 2 k is a chi-square with k degrees of freedom and k the number of variables being tested. In this paper, we prove that when p is not negligible compared to n, Wilks' theorem does not hold and that the chi-square approximation is grossly incorrect; in fact, this approximation produces p-values that are far too small (under the null hypothesis).Assume that n and p grow large in such a way that p/n → κ for some constant κ < 1/2. We prove that for a class of logistic models, the LLR converges to a rescaled chi-square, namely, 2Λ d → α(κ)χ 2 k , where the scaling factor α(κ) is greater than one as soon as the dimensionality ratio κ is positive. Hence, the LLR is larger than classically assumed. For instance, when κ = 0.3, α(κ) ≈ 1.5. In general, we show how to compute the scaling factor by solving a nonlinear system of two equations with two unknowns. Our mathematical arguments are involved and use techniques from approximate message passing theory, from non-asymptotic random matrix theory and from convex geometry. We also complement our mathematical study by showing that the new limiting distribution is accurate for finite sample sizes.Finally, all the results from this paper extend to some other regression models such as the probit regression model.

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A modern maximum-likelihood theory for high-dimensional logistic regression

Sur

Candès

2019

Proc. Natl. Acad. Sci. U.S.A.

196

185

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Students in statistics or data science usually learn early on that when the sample size n is large relative to the number of variables p, fitting a logistic model by the method of maximum likelihood produces estimates that are consistent and that there are well-known formulas that quantify the variability of these estimates which are used for the purpose of statistical inference. We are often told that these calculations are approximately valid if we have 5 to 10 observations per unknown parameter. This paper shows that this is far from the case, and consequently, inferences produced by common software packages are often unreliable. Consider a logistic model with independent features in which n and p become increasingly large in a fixed ratio. We prove that (i) the maximum-likelihood estimate (MLE) is biased, (ii) the variability of the MLE is far greater than classically estimated, and (iii) the likelihood-ratio test (LRT) is not distributed as a χ2. The bias of the MLE yields wrong predictions for the probability of a case based on observed values of the covariates. We present a theory, which provides explicit expressions for the asymptotic bias and variance of the MLE and the asymptotic distribution of the LRT. We empirically demonstrate that these results are accurate in finite samples. Our results depend only on a single measure of signal strength, which leads to concrete proposals for obtaining accurate inference in finite samples through the estimate of this measure.

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

Cited by 26 publications

References 31 publications

Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling

Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling

The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled Chi-square

A modern maximum-likelihood theory for high-dimensional logistic regression

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