{"paper":{"title":"Soft-to-Hard Routing in Sparse Mixture-of-Experts Models","license":"http://creativecommons.org/licenses/by/4.0/","headline":"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.","cross_cats":["cs.AI","math.PR"],"primary_cat":"cs.LG","authors_text":"Reza Rastegar","submitted_at":"2026-05-04T01:07:21Z","abstract_excerpt":"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 $a_k(x;\\phi)$, the relevant local quantity is the top-two margin $\\Delta(x;\\phi)$, and the relevant global quantity is the boundary mass $\\mathbb{P}(\\Delta(X;\\phi)\\le w)$. Under smoothness and transversality assumptions, coarea and tubular-neighborhood estimates show how this mass scales with the slab width; "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Smoothness and transversality assumptions on the router and input law (invoked to obtain the coarea/tube estimates for boundary mass).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9052a3f7b35e30765ff533e7f5be6ef1aff9b86e005e7e33ff43dcf98a1edfce"},"source":{"id":"2605.02124","kind":"arxiv","version":2},"verdict":{"id":"419aaaa3-02fc-4612-84e2-a55874e05822","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T16:45:21.115516Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Smoothness and transversality assumptions on the router and input law (invoked to obtain the coarea/tube estimates for boundary mass).","pith_extraction_headline":"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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.02124/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T16:37:35.121075Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T04:01:22.619875Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:39:19.837957Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5b6dbe82667f81e5a6750515c31bcd19ef6b3b34248c008de5aaf38ee2c2c9d7"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}