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Nesterov's accelerated gradient (AG) is a popular technique to optimizeobjective functions comprising two components: a convex loss and a penaltyfunction. While AG methods perform well for convex penalties, such as theLASSO, convergence issues may arise when it is applied to nonconvex penalties,such as SCAD. A recent proposal generalizes Nesterov's AG method to thenonconvex setting. The proposed algorithm requires specification of severalhyperparameters for its practical application. Aside from some generalconditions, there is no explicit rule for selecting the hyperparameters, andhow different selection can affect convergence of the algorithm. In thisarticle, we propose a hyperparameter setting based on the complexity upperbound to accelerate convergence, and consider the application of this nonconvexAG algorithm to high-dimensional linear and logistic sparse learning problems.We further establish the rate of convergence and present a simple and usefulbound to characterize our proposed optimal damping sequence. Simulation studiesshow that convergence can be made, on average, considerably faster than that ofthe conventional proximal gradient algorithm. Our experiments also show thatthe proposed method generally outperforms the current state-of-the-art methodsin terms of signal recovery.

Accelerated Gradient Methods for Sparse Statistical Learning with Nonconvex Penalties