The algorithm we are using is "compute the streaming average of cluster assignment". AFAIK MB k-means was not derived as an interpretation of some second order gradient descent of the k-means objective function but as a streaming approximation of the batch averaging done by batch k-means. I don't know if there is a theory paper for the interpretation of MB k-means as a gradient descent optimizer.

Anyway theoretical convergence analysis of non-convex objective functions without making further assumptions on the data is pretty pointless in my opinion.