A Power Analysis for Knockoffs with the Lasso Coefficient-Difference Statistic Asaf Weinstein ∗ Weijie J. Su † Ma lgorzata Bogdan ‡ Rina F. Barber § Emmanuel J. Cand` es ¶ Abstract In a linear model with possibly many predictors, we consider variable selection procedures given by {1 ≤ j ≤ p : |  β j (λ)| >t}, where  β(λ) is the Lasso estimate of the regression coefficients, and where λ and t may be data dependent. Ordinary Lasso selection is captured by using t = 0, thus allowing to control only λ, whereas thresholded-Lasso selection allows to control both λ and t. The potential advantages of the latter over the former in terms of power—figuratively, opening up the possibility to look further down the Lasso path—have been quantified recently leveraging advances in approxi- mate message-passing (AMP) theory, but the implications are actionable only when assuming substantial knowledge of the underlying signal. In this work we study theoretically the power of a knockoffs-calibrated counterpart of thresholded-Lasso that enables us to control FDR in the realistic situation where no prior in- formation about the signal is available. Although the basic AMP framework remains the same, our analysis requires a significant technical extension of existing theory in order to handle the pairing between original variables and their knockoffs. Relying on this extension we obtain exact asymptotic predictions for the true positive proportion achievable at a prescribed type I error level. In particular, we show that the knockoffs version of thresholded-Lasso can perform much better than ordinary Lasso selection if λ is chosen by cross-validation on the augmented matrix. 1 Introduction Suppose that we observe a matrix X ∈ R n×p of predictor measurements and a response vector Y ∈ R n , and assume that Y = Xβ + ξ, ξ ∼N n (0,σ 2 I ), (1.1) where β =(β 1 , ..., β p ) ⊤ and σ 2 are unknown. In many modern applications where the linear model is appropriate, p is large and we may have a reason to believe a priori that β j is small in magnitude for most j =1, ..., p. For example, in genetics X ij might encode the state (presence or absence) ∗ School of Computer Science and Engineering, Hebrew University of Jerusalem, Israel † Department of Statistics, University of Pennsylvania ‡ Department of Mathematics, University of Wroclaw, Poland, and Department of Statistics, Lund University, Sweden § Department of Statistics, University of Chicago ¶ Department of Statistics and Department of Mathematics, Stanford University 1 arXiv:2007.15346v1 [math.ST] 30 Jul 2020