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Under mild assumptions stochastic gradient methods asymptotically achieve anoptimal rate of convergence if the arithmetic mean of all iterates is returnedas an approximate optimal solution. However, in the absence of stochasticnoise, the arithmetic mean of all iterates converges considerably slower to theoptimal solution than the iterates themselves. And also in the presence ofnoise, when a finite termination of the stochastic gradient method isconsidered, the arithmetic mean is not necessarily the best possibleapproximation to the unknown optimal solution. This paper aims at identifyingoptimal strategies in a particularly simple case, the minimization of astrongly convex function with i. i. d. noise terms and finite termination.Explicit formulas for the stochastic error and the optimization error arederived in dependence of certain parameters of the SGD method. The aim was tochoose parameters such that both stochastic error and optimization error arereduced compared to arithmetic averaging. This aim could not be achieved;however, by allowing a slight increase of the stochastic error it was possibleto select the parameters such that a significant reduction of the optimizationerror could be achieved. This reduction of the optimization error has a strongeffect on the approximate solution generated by the stochastic gradient methodin case that only a moderate number of iterations is used or when the initialerror is large. The numerical examples confirm the theoretical results andsuggest that a generalization to non-quadratic objective functions may bepossible.

Optimized convergence of stochastic gradient descent by weighted averaging