Replacing mean by median over batch-size to lessen the impact of outliers

In the case of training a Neural Network on a regression task. Assuming the data has a significant amount of outliers. Provided that the error needs to be RMS and not MAE. Can it be better (as in less sensitive to the outliers) to replace the average over batch size in the weights update by a median over batch size computation?

For a batch size large enough, this should lessen the impact the contribution from the outliers. It does not seem to be common though, at least to current knowledge. What are the shortcomings of this approach?