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Stochastic Gradient Descent (SGD) is one of the simplest and most popularstochastic optimization methods. While it has already been theoreticallystudied for decades, the classical analysis usually required non-trivialsmoothness assumptions, which do not apply to many modern applications of SGDwith non-smooth objective functions such as support vector machines. In thispaper, we investigate the performance of SGD without such smoothnessassumptions, as well as a running average scheme to convert the SGD iterates toa solution with optimal optimization accuracy. In this framework, we prove thatafter T rounds, the suboptimality of the last SGD iterate scales asO(log(T)/\sqrt{T}) 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 arethe first bounds of this kind, and almost match the minimax-optimal ratesobtainable by appropriate averaging schemes. We also propose a new and simpleaveraging scheme, which not only attains optimal rates, but can also be easilycomputed on-the-fly (in contrast, the suffix averaging scheme proposed inRakhlin et al. (2011) is not as simple to implement). Finally, we provide someexperimental illustrations.

Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes