  In this paper, we address the challenging problem of selecting tuning
parameters for high-dimensional sparse regression. We propose a simple and
computationally efficient method, called path thresholding (PaTh), that
transforms any tuning parameter-dependent sparse regression algorithm into an
asymptotically tuning-free sparse regression algorithm. More specifically, we
prove that, as the problem size becomes large (in the number of variables and
in the number of observations), PaTh performs accurate sparse regression, under
appropriate conditions, without specifying a tuning parameter. In
finite-dimensional settings, we demonstrate that PaTh can alleviate the
computational burden of model selection algorithms by significantly reducing
the search space of tuning parameters.
