$($Distributed SGD$)$ Assume we$$re training a model with distributed SGD with m $=6$ machines. For n $=76,800$ data points and mini$-$batch size b $=1,536$ on the master node, how many distributed rounds do we need to finish $10$ epochs? Assume the time for processing a mini$-$batch by the workers is $4$s and each communication $($from master to workers or from workers to master$)$ takes $1\mathrm{.}5$s$,$ and every worker increases the aggregation time at the master by $0\mathrm{.}015$s$.$ Hint: Remember to include communication time from master to workers and workers to master!

$(1)$ How long does it take to train on the entire data for $10$ epochs?

$(2)$ After having how many workers the communication and aggregation time dominates $($is strictly greater than$)$ the gradient computation time?