Accelerate Stochastic Subgradient Method by Leveraging Local Error Bound

In this paper, we propose two accelerated stochastic {\bf subgradient} methods for stochastic non-strongly convex optimization problems by leveraging a generic local error bound condition. The novelty of the proposed methods lies at smartly leveraging the recent historical solution to tackle the variance in the stochastic subgradient. The key idea of both methods is to iteratively solve the original problem approximately in a local region around a recent historical solution with size of the local region gradually decreasing as the solution approaches the optimal set. The difference of the two methods lies at how to construct the local region. The first method uses an explicit ball constraint and the second method uses an implicit regularization approach. For both methods, we establish the improved iteration complexity in a high probability for achieving an $\epsilon$-optimal solution. Besides the improved order of iteration complexity with a high probability, the proposed algorithms also enjoy a logarithmic dependence on the distance of the initial solution to the optimal set. When applied to the $\ell_1$ regularized polyhedral loss minimization (e.g., hinge loss, absolute loss), the proposed stochastic methods have a logarithmic iteration complexity.