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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; 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 $L_\\tau$ and the hard objective $L_0$, including an $O(\\tau^\\alpha)$ uniform comparison under a margin-tail condition, and yield $\\Gamma$-convergence of the sof","title":"Soft-to-Hard Routing in Sparse Mixture-of-Experts Models","url":"https://arxiv.org/abs/2605.02124","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.02124v2 Announce Type: replace-cross \nAbstract: 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; 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 $L_\\tau$ and the hard objective $L_0$, including an $O(\\tau^\\alpha)$ uniform comparison under a margin-tail condition, and yield $\\Gamma$-convergence of the sof","title":"Soft-to-Hard Routing in Sparse Mixture-of-Experts Models","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-26T04:43:39Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.02124"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:91165f729ff792c91a218cadf07ddddae8f4438efeb31e28c56c257e3f88752fa9a72fd44fced8d54cc1d6adeeffdd58b8ae11241a6eec69174285a4af1b5005","signer":"crovia.substrate","subject":{"observed_at":"2026-05-26T04:43:39Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.02124"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"34dbda07fda00a31429f1f7777f7ebb60e841536699005e219f18b27a2f7a1a0","leaf_index":151993,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"acfba842f2a5ff92389613beaac08925ec3026248a01de07fbf87c82ea08987d","side":"left"},{"sibling":"596d33d460d4d91929f7929664f55a0698a6e0ae907f26c8e82406bbebe80d98","side":"right"},{"sibling":"a43715409342c805e946b3d0ce7e47f6073ca27a4afcbe1c9fa60569a6c1f239","side":"right"},{"sibling":"5ee5d9404763f3a2da30594cd4277992e12b572453592eb5d3e709319009aa36","side":"left"},{"sibling":"c62cff0d3d3aed32cb04366f02a2a9ee67d16c26cd1b3385f11c28abe1034d44","side":"left"},{"sibling":"357631f93ff0924e1cbee5b13c28023fef9d7239eda735855e1fc3d83dde4741","side":"left"},{"sibling":"6f1153ad830bc8202e6e1954981665e478138b53ce7dc91b62be904bade1d7fe","side":"right"},{"sibling":"8b0755e3ba26ea95e61e8d0e6362ce6cbe9991903d18e148b0cba214724147ec","side":"left"},{"sibling":"6c8b632bc887ad8683d2d08a66a66babfd1e8e43de4b1b760ed3f3d35d5b02e1","side":"left"},{"sibling":"879666fab72e779ab55d0564eaabd64b00534fd6bba7f18412c7f31f61ffd09f","side":"right"},{"sibling":"f40ccedd90c323817e961adc0a2e2db82b8aabe192b6c9d5a373ff988987b207","side":"right"},{"sibling":"b85ea61ae405eed84392a7b6b1eee5536f5a38d6b04070638e23ec7b71e3443a","side":"right"},{"sibling":"d415e6939aee710631f5062799379b547d2c3e3d9a68f263bbb5a693285ab2ca","side":"left"},{"sibling":"e5893793e3591ed7f5e58ca94ffcfba46bb30f69fb1c25d5ba8ef49eb99f9126","side":"right"},{"sibling":"35ca36cee447f0ef7064a25d55f59357c66901e31427729c4c1d8b14aa8adb6c","side":"left"},{"sibling":"e3eecaf996dbe7229a7bb1d234c97aea97a252c5f7c89f8547b6d091db0f0e40","side":"right"},{"sibling":"55bcbd4da3e20d93931f7e58673f10232e81a5b1514d7396cb4b71e8f95788d0","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":152106,"merkle_root":"ac5182c6f3dd09931f2a689df5f4be36df7b55e55bcf106e195671f5ed55fd8f","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260526T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-26T05:37:34Z","sig_algorithm":"ed25519","signature":"a401243fbd2c077d29a623c6ef616c78fbd5c8ce1930afa7af7165b2386e5a8cf15d5094983a1c962e71b27b911a3f3ce09c8cfab9393be2ce5ce8ee6513da06","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_16c92cf898e53c7340fbeda6cac883f180ae5fd839d963911818fde60747c35e"}}