Suppose that we first apply the Lasso to a design matrix, and then update one of its columns. In general, the signs of the Lasso coefficients may change, and there is no closed-form expression for updating the Lasso solution exactly. In this work, we propose an approximate formula for updating a debiased Lasso coefficient. We provide general nonasymptotic error bounds in terms of the norms and correlations of a given design matrix's columns, and then prove asymptotic convergence results for the case of a random design matrix with i.i.d.\ sub-Gaussian row vectors and i.i.d.\ Gaussian noise. Notably, the approximate formula is asymptotically correct for most coordinates in the proportional growth regime, under the mild assumption that each row of the design matrix is sub-Gaussian with a covariance matrix having a bounded condition number. Our proof only requires certain concentration and anti-concentration properties to control various error terms and the number of sign changes. In contrast, rigorously establishing distributional limit properties (e.g.\ Gaussian limits for the debiased Lasso) under similarly general assumptions has been considered open problem in the universality theory. As applications, we show that the approximate formula allows us to reduce the computation complexity of variable selection algorithms that require solving multiple Lasso problems, such as the conditional randomization test and a variant of the knockoff filter.

给定的研究论文提出了适用于Lasso的近似公式，通过该公式可以更新去偏Lasso的系数。论文证明了该近似公式在具有有界条件数协方差矩阵的亚高斯型设计矩阵情况下渐进收敛，并给出了一些具体应用案例。

广义去偏Lasso方法的稳定性及在基于重采样的变量选择中的应用