In many federated learning (FL) models, a common strategy employed to ensure the progress in the training process, is to wait for at least $M$ clients out of the total $N$ clients to send back their local gradients based on a reporting deadline $T$, once the parameter server (PS) has broadcasted the global model. If enough clients do not report back within the deadline, the particular round is considered to be a failed round and the training round is restarted from scratch. If enough clients have responded back, the round is deemed successful and the local gradients of all the clients that responded back are used to update the global model. In either case, the clients that failed to report back an update within the deadline would have wasted their computational resources. Having a tighter deadline (small $T$) and waiting for a larger number of participating clients (large $M$) leads to a large number of failed rounds and therefore greater communication cost and computation resource wastage. However, having a larger $T$ leads to longer round durations whereas smaller $M$ may lead to noisy gradients. Therefore, there is a need to optimize the parameters $M$ and $T$ such that communication cost and the resource wastage is minimized while having an acceptable convergence rate. In this regard, we show that the average age of a client at the PS appears explicitly in the theoretical convergence bound, and therefore, can be used as a metric to quantify the convergence of the global model. We provide an analytical scheme to select the parameters $M$ and $T$ in this setting.

翻译：在许多联邦学习（FL）模型中，为确保训练进程推进，常采用的一种策略是：参数服务器（PS）广播全局模型后，设定一个回报截止时间 $T$，并等待至少 $M$ 个客户端（共 $N$ 个）基于该截止时间返回其本地梯度。若在截止时间内未收到足够客户端的回报，则该轮训练视为失败，需从头重启训练轮次。若收到足够客户端的响应，则该轮视为成功，所有响应客户端的本地梯度将被用于更新全局模型。无论哪种情况，未能在截止时间内回报更新的客户端均会浪费其计算资源。设定更紧的截止时间（较小的 $T$）并等待更多客户端参与（较大的 $M$）会导致大量失败轮次，从而增加通信成本与计算资源浪费。然而，较大的 $T$ 会导致单轮时长增加，而较小的 $M$ 则可能引入梯度噪声。因此，有必要优化参数 $M$ 与 $T$，以在保持可接受收敛速度的同时，最小化通信成本与资源浪费。对此，我们证明客户端在参数服务器处的平均时效（age）显式地出现在理论收敛界中，因而可作为量化全局模型收敛性的度量指标。我们在此设定下提出了一种用于选择参数 $M$ 与 $T$ 的解析方案。