Soft-to-Hard Routing in Sparse Mixture-of-Experts Models
Softmax routing approaches hard top-1 routing as the temperature tends to zero, but the limiting passage is singular at router ties. This paper develops a boundary-layer calculus for this soft-to-hard limit in population squared-loss mixture-of-experts regression. For a router with logits $a_k(x;φ)$, the relevant local quantity is the top-two margin $Δ(x;φ)$, and the relevant global quantity is the boundary mass $\mathbb{P}(Δ(X;φ)\le w)$. Under smoothness and transversality assumptions, coarea and tubular-neighborhood estimates show how this mass scales with the slab width; in the binary case the leading coefficient is an explicit surface integral over the routing interface. These geometr...
| Authors | Reza Rastegar |
|---|---|
| Primary category | cs.LG |
| Published | 4 May 2026 |
| Last updated at source | 23 May 2026 |
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