  The extrapolation strategy raised by Nesterov, which can accelerate the
convergence rate of gradient descent methods by orders of magnitude when
dealing with smooth convex objective, has led to tremendous success in training
machine learning tasks. In this article, the convergence of individual iterates
of projected subgradient (PSG) methods for nonsmooth convex optimization
problems is theoretically studied based on Nesterov's extrapolation, which we
name individual convergence. We prove that Nesterov's extrapolation has the
strength to make the individual convergence of PSG optimal for nonsmooth
problems. In light of this consideration, a direct modification of the
subgradient evaluation suffices to achieve optimal individual convergence for
strongly convex problems, which can be regarded as making an interesting step
toward the open question about stochastic gradient descent (SGD) posed by
Shamir. Furthermore, we give an extension of the derived algorithms to solve
regularized learning tasks with nonsmooth losses in stochastic settings.
Compared with other state-of-the-art nonsmooth methods, the derived algorithms
can serve as an alternative to the basic SGD especially in coping with machine
learning problems, where an individual output is needed to guarantee the
regularization structure while keeping an optimal rate of convergence.
Typically, our method is applicable as an efficient tool for solving
large-scale $l$1-regularized hinge-loss learning problems. Several comparison
experiments demonstrate that our individual output not only achieves an optimal
convergence rate but also guarantees better sparsity than the averaged
solution.
