  The statistics literature of the past 15 years has established many favorable
properties for sparse diminishing-bias regularization: techniques which can
roughly be understood as providing estimation under penalty functions spanning
the range of concavity between $L_0$ and $L_1$ norms. However, lasso
$L_1$-regularized estimation remains the standard tool for industrial `Big
Data' applications because of its minimal computational cost and the presence
of easy-to-apply rules for penalty selection. In response, this article
proposes a simple new algorithm framework that requires no more computation
than a lasso path: the path of one-step estimators (POSE) does $L_1$ penalized
regression estimation on a grid of decreasing penalties, but adapts
coefficient-specific weights to decrease as a function of the coefficient
estimated in the previous path step. This provides sparse diminishing-bias
regularization at no extra cost over the fastest lasso algorithms. Moreover,
our `gamma lasso' implementation of POSE is accompanied by a reliable heuristic
for the fit degrees of freedom, so that standard information criteria can be
applied in penalty selection. We also provide novel results on the distance
between weighted-$L_1$ and $L_0$ penalized predictors; this allows us to build
intuition about POSE and other diminishing-bias regularization schemes. The
methods and results are illustrated in extensive simulations and in application
of logistic regression to evaluating the performance of hockey players.
