We study the complexity of the entire regularization path for least squares regression with 1-norm penalty, known as the Lasso. Every regression parameter in the Lasso changes linearly as a function of the regularization value. The number of changes is regarded as the Lasso's complexity. Experimental results using exact path following exhibit polynomial complexity of the Lasso in the problem size. Alas, the path complexity of the Lasso on artificially designed regression problems is exponential We use smoothed analysis as a mechanism for bridging the gap between worst case settings and the de facto low complexity. Our analysis assumes that the observed data has a tiny amount of intrinsic noise. We then prove that the Lasso's complexity is polynomial in the problem size.