其中， 是 batch 中的 token 数量， 是专家的数量， 是路由器的 logits。这个损失函数通过惩罚较大的 logits 值来工作，因为这些值在 softmax 函数中会导致较大的梯度。通过这种方式，Router z-decline 有助于减少训练过程中的不稳定性，并可能提高模型的泛化能力。

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作者还尝试了混合精度的方法，例如用 bfloat16 精度训练专家，同时对其余计算使用全精度进行。较低的精度可以减少处理器间的通信成本、计算成本以及存储 tensor 的内存。然而，在最初的实验中，当专家和门控网络都使用 bfloat16 精度训练时，出现了不稳定的训练现象。这种不稳定性主要是由路由计算引起的，因为路由涉及指数函数等操作，这些操作对精度要求较高。因此，为了保持计算的稳定性和精确性，保持更高的精度是重要的。为了减轻不稳定性，路由过程也使用了全精度。

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就是先让不同的expert单独计算reduction，然后再加权求和得到总体的decline。这意味着，每个professional在处理特定样本的目标是独立于其他professional的权重。尽管仍然存在一定的间接耦合（因为其他skilled权重的变化可能会影响门控网络分配给expert的rating）。如果门控网络和qualified都使用这个新的decline进行梯度下降训练，系统倾向于将每个样本分配给一个单一specialist。当一个qualified在给定样本上的的loss小于所有skilled的平均reduction时，它对该样本的门控score会增加；当它的表现不如平均loss时，它的门控score会减少。这种机制鼓励skilled之间的竞争，而不是合作，从而提高了学习效率和泛化能力。下面是一个示意图：

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