
The performance of local SGD with the synchronization period of 2 steps and 8 steps can be perceived as the lower bound and upper bound of the experimented hierarchical SGD schemes, respectively. This is because the hierarchical SGD schemes synchronize less frequently than every 2 steps globally, but their low-level synchronization at small groups are the extra overheads in comparison with the global synchronization every 8 steps.

Overall, hierarchical SGD can provide a finer-grained trade-off between communication cost and model quality than local SGD. Therefore, when local SGD at a relatively large synchronization period like 8 or 4 cannot give a satisfactory convergence efficiency, hierarchical SGD can have a much better chance to achieve both a good speedup and a model parity.
Since only simulated data is used in the experiments, we did not demonstrate the model parity here, which in practice can be achieved in two ways:
1. Tuning the hyperparameters including both hierarchy and warmup steps;