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Estimating the Lasso's Effective Noise

Johannes Lederer, Michael Vogt; 22(276):1−32, 2021.

Abstract

Much of the theory for the lasso in the linear model $Y = \boldsymbol{X} \beta^* + \varepsilon$ hinges on the quantity $2\| \boldsymbol{X}^\top \varepsilon \|_\infty / n$, which we call the lasso's effective noise. Among other things, the effective noise plays an important role in finite-sample bounds for the lasso, the calibration of the lasso's tuning parameter, and inference on the parameter vector $\beta^*$. In this paper, we develop a bootstrap-based estimator of the quantiles of the effective noise. The estimator is fully data-driven, that is, does not require any additional tuning parameters. We equip our estimator with finite-sample guarantees and apply it to tuning parameter calibration for the lasso and to high-dimensional inference on the parameter vector $\beta^*$.

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