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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.02124 |
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| _version_ | 1866913159100497920 |
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| author | Rastegar, Reza |
| author_facet | Rastegar, Reza |
| contents | 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 geometric estimates give quantitative bounds between the soft objective $L_τ$ and the hard objective $L_0$, including an $O(τ^α)$ uniform comparison under a margin-tail condition, and yield $Γ$-convergence of the soft objectives on compact parameter spaces. The main conclusion is that the zero-temperature approximation is controlled by the probability carried by an $O(τ)$ neighborhood of the routing interfaces, not by temperature alone. After isolating this boundary-layer part of the problem, we record a conditional landscape-transfer theorem from hard to small-temperature soft routing and a reduced two-expert Gaussian calculation illustrating local symmetry breaking. Synthetic diagnostics are included only as controlled checks of the boundary-layer predictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02124 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Soft-to-Hard Routing in Sparse Mixture-of-Experts Models Rastegar, Reza Machine Learning Artificial Intelligence Probability 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 geometric estimates give quantitative bounds between the soft objective $L_τ$ and the hard objective $L_0$, including an $O(τ^α)$ uniform comparison under a margin-tail condition, and yield $Γ$-convergence of the soft objectives on compact parameter spaces. The main conclusion is that the zero-temperature approximation is controlled by the probability carried by an $O(τ)$ neighborhood of the routing interfaces, not by temperature alone. After isolating this boundary-layer part of the problem, we record a conditional landscape-transfer theorem from hard to small-temperature soft routing and a reduced two-expert Gaussian calculation illustrating local symmetry breaking. Synthetic diagnostics are included only as controlled checks of the boundary-layer predictions. |
| title | Soft-to-Hard Routing in Sparse Mixture-of-Experts Models |
| topic | Machine Learning Artificial Intelligence Probability |
| url | https://arxiv.org/abs/2605.02124 |