Pith Number

pith:T35VLSWO

pith:2026:T35VLSWONCUXZ5INWIT47RJRK5

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Soft-to-Hard Routing in Sparse Mixture-of-Experts Models

Reza Rastegar

The zero-temperature limit of softmax-routed mixture-of-experts is governed by a thin geometric layer around routing interfaces rather than the full input space.

arxiv:2605.02124 v2 · 2026-05-04 · cs.LG · cs.AI · math.PR

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Claims

C1strongest claim

Under smoothness and transversality assumptions on the router and input law, we prove coarea/tube estimates showing that this mass is linear in the slab width, with leading constant given by a surface integral over the routing interface in the binary case. These estimates yield quantitative soft-to-hard risk bounds and, under compactness and uniform margin control, Γ-convergence of the soft objectives to the hard-routing objective. The main conclusion is that the zero-temperature limit is controlled by a thin geometric layer around routing interfaces, not by the full input space.

C2weakest assumption

Smoothness and transversality assumptions on the router and input law (invoked to obtain the coarea/tube estimates for boundary mass).

C3one line summary

Boundary mass in MoE is linear in slab width under smoothness and transversality, so the zero-temperature limit is governed by a thin geometric layer around routing interfaces rather than the full input space.

Cited by

1 paper in Pith

A Minimal Bifurcation Model of Load Imbalance in a Softmax Mixture-of-Experts Router

Receipt and verification

First computed	2026-05-26T01:03:31.888653Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

9efb55cace68a97cf50db227cfc53157698e20cf20547a1d20201b260c92060b

Aliases

arxiv: 2605.02124 · arxiv_version: 2605.02124v2 · doi: 10.48550/arxiv.2605.02124 · pith_short_12: T35VLSWONCUX · pith_short_16: T35VLSWONCUXZ5IN · pith_short_8: T35VLSWO

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/T35VLSWONCUXZ5INWIT47RJRK5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 9efb55cace68a97cf50db227cfc53157698e20cf20547a1d20201b260c92060b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e06f1a663987cc5f2fab70f5cb827c7b97435ca3e64b83812d3a55df72556871",
    "cross_cats_sorted": [
      "cs.AI",
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-04T01:07:21Z",
    "title_canon_sha256": "53d4eed950168c374c94cd2d0bdae311b41a0eb57e64b4fedbaaad0767a1fa3f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.02124",
    "kind": "arxiv",
    "version": 2
  }
}