A Power and Prediction Analysis for Knockoffs with Lasso Statistics

TL;DR

This paper analyzes the power of knockoff procedures with Lasso statistics, showing they nearly match an oracle's performance in controlling false negatives and achieving optimal prediction errors under Gaussian designs.

Contribution

It provides a theoretical analysis of the power of knockoff methods with Lasso, demonstrating near-optimal performance in false negative control and prediction accuracy for sparse signals.

Findings

Knockoffs asymptotically achieve near-optimal power.

Model selection via knockoff filtering yields nearly ideal prediction errors.

Analysis is based on i.i.d. Gaussian design assumptions.

Abstract

Knockoffs is a new framework for controlling the false discovery rate (FDR) in multiple hypothesis testing problems involving complex statistical models. While there has been great emphasis on Type-I error control, Type-II errors have been far less studied. In this paper we analyze the false negative rate or, equivalently, the power of a knockoff procedure associated with the Lasso solution path under an i.i.d. Gaussian design, and find that knockoffs asymptotically achieve close to optimal power with respect to an omniscient oracle. Furthermore, we demonstrate that for sparse signals, performing model selection via knockoff filtering achieves nearly ideal prediction errors as compared to a Lasso oracle equipped with full knowledge of the distribution of the unknown regression coefficients. The i.i.d. Gaussian design is adopted to leverage results concerning the empirical distribution of the Lasso estimates, which makes power calculation possible for both knockoff and oracle procedures.