I am aware that convergence in stochastic gradient problems is very sensitive to the variance of your gradient estimator. One issue I'm running into is that the gradient is a random vector and so would require a different metric to comparing the variance of univariate estimators.
I've seen a paper (https://arxiv.org/pdf/1705.07880.pdf) where variance reduction techniques are applied to minimise the trace of the estimator covariance matrix, however there are usually dependencies between gradient components which is not accounted for when taking the trace.
I have heard that it is standard in the optimisation community to compare the relative efficiency of two gradient estimators by using the expected value of the squared norm of the gradient (apparently convergence is dependent on this). Is this correct? If so, can anyone share a reference?