  In large-scale modern data analysis, first-order optimization methods are
usually favored to obtain sparse estimators in high dimensions. This paper
performs theoretical analysis of a class of iterative thresholding based
estimators defined in this way. Oracle inequalities are built to show the
nearly minimax rate optimality of such estimators under a new type of
regularity conditions. Moreover, the sequence of iterates is found to be able
to approach the statistical truth within the best statistical accuracy
geometrically fast. Our results also reveal different benefits brought by
convex and nonconvex types of shrinkage.
