I've found things such as the Robbins-Monroe conditions for the learning rate, as well as a proof from Robbins, Siegmund, 1971 which gives convergence to a local minima provided that the expectation and variance of the gradient are both bounded.
I'm looking for a reference that talks about the relationship between the variance of the gradient and convergence of SGD. Specifically, I know that lower variance is good, but given that the gradient is a vector, how is "lower" defined, and what are the guaranteed benefits of reducing the variance?