Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes
; Proceedings of the 30th International Conference on Machine Learning, PMLR 28(1):71-79, 2013.
Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD \emphwithout such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the \emphlast SGD iterate scales as O(\log(T)/\sqrtT) for non-smooth convex objective functions, and O(\log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in \citetRakhShaSri12arxiv is not as simple to implement). Finally, we provide some experimental illustrations.