Mixture-of-Experts (MoE) has become a cornerstone in recent state-of-the-art large language models (LLMs). Traditionally, MoE relies on \\(\\mathrm{Softmax}\\) as the router score function to aggregate expert output, a designed choice that has persisted from the earliest MoE models to modern LLMs, and is now widely regarded as standard practice. However, the necessity of using \\(\\mathrm{Softmax}\\) to project router weights into a probability simplex remains an unchallenged assumption rather than a principled design choice. In this work, we first revisit the classical Nadaraya-Watson regression and observe that MoE shares the same mathematical formulation as Nadaraya-Watson regression. Furthermore, we show that both feed-forward neural network (FFN) and MoE can be interpreted as a special case of Nadaraya-Watson regression, where the kernel function corresponds to the input neurons of the output layer. Motivated by these insights, we propose the \\(\\textbf{zero-additional-cost}\\) Kernel Inspired Router with Normalization (KERN), an FFN-style router function, as an alternative to \\(\\mathrm{Softmax}\\). We demonstrate that this router generalizes both \\(\\mathrm{Sigmoid}\\)– and \\(\\mathrm{Softmax}\\)-based routers. Based on empirical observations and established practices in FFN implementation, we recommend the use of \\(\\mathrm{ReLU}\\) activation and \\(\\ell_2\\)-normalization in \\(\\mathrm{KERN}\\) router function. Comprehensive experiments in MoE and LLM validate the effectiveness of the proposed FFN-style router function \\methodNorm.<\/p>\n","protected":false},"excerpt":{"rendered":"
Mixture-of-Experts (MoE) has become a cornerstone in recent state-of-the-art large language models (LLMs). Traditionally, MoE relies on as the router score function to aggregate expert output, a designed choice that has persisted from the earliest MoE models to modern LLMs, and is now widely regarded as standard practice. 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