Estimating the Lasso's Effective Noise

Much of the theory for the lasso in the linear model Y = X β^* + varepsilon hinges on the quantity 2 | X^top varepsilon |∞ / 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 β^*. 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 β^*.