  We consider $L_2$Boosting, a special case of Friedman's generic boosting
algorithm applied to linear regression under $L_2$-loss. We study $L_2$Boosting
for an arbitrary regularization parameter and derive an exact closed form
expression for the number of steps taken along a fixed coordinate direction.
This relationship is used to describe $L_2$Boosting's solution path, to
describe new tools for studying its path, and to characterize some of the
algorithm's unique properties, including active set cycling, a property where
the algorithm spends lengthy periods of time cycling between the same
coordinates when the regularization parameter is arbitrarily small. Our fixed
descent analysis also reveals a repressible condition that limits the
effectiveness of $L_2$Boosting in correlated problems by preventing desirable
variables from entering the solution path. As a simple remedy, a data
augmentation method similar to that used for the elastic net is used to
introduce $L_2$-penalization and is shown, in combination with decorrelation,
to reverse the repressible condition and circumvents $L_2$Boosting's
deficiencies in correlated problems. In itself, this presents a new explanation
for why the elastic net is successful in correlated problems and why methods
like LAR and lasso can perform poorly in such settings.
