On the Convergence Rate for Stochastic Approximation in the Nonsmooth Setting

We consider a stochastic approximation (SA) method for finding the minimizer of a function f, which is convex but nondifferentiable at the minimizer. Due to the nondifferentiability at the minimizer, f is allowed to increase at a positive rate in a neighborhood of the minimizer. From this property, we show that the nth estimate for the minimizer generated by the SA procedure converges at a rate of 1/n in the mean, which is significantly faster than the classical convergence rates for differentiable functions f. We also discuss an example from an inventory control system that exhibits a convex but nondifferentiable cost function.