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However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. We first propose ProbMoE Exact-$k$ routing, which samples $k$-expert subsets in the forward pass, and the backward pass uses gradients through each expert's exact marginal probability as a tractable surrogate for the true gradient. ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert al","title":"ProbMoE: Differentiable Probabilistic Routing for Mixture-of-Experts","url":"https://arxiv.org/abs/2606.01509","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.01509v1 Announce Type: cross \nAbstract: Mixture-of-Experts (MoE) models scale by activating only a small subset of experts per token. However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. 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ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert al","title":"ProbMoE: Differentiable Probabilistic Routing for Mixture-of-Experts","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-02T04:43:38Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.01509"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:e968e9b1c49e38583b75104587a3686130d1d8b531c9c7d7f0cdb4d576e233c43d4b4426d97d77fc3bdd17f990855e1ae710d507ca0049e182148f8e22e4f505","signer":"crovia.substrate","subject":{"observed_at":"2026-06-02T04:43:38Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.01509"},"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":"dc39976e3af0c528d4f34c220aef488dfcaa251d1ecef242923853aaca35f232","leaf_index":205622,"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":"f5f1cc5ec7efb339b1559b1e418a9b4bb4a518922265c46f928c2600b025975f","side":"right"},{"sibling":"fc174f98930968693eb91211a8160fa4bc9e0a3a6cfdae39b1e3a74cbf005c2b","side":"left"},{"sibling":"42102b0dfc26e2633cb1464379545bab0e49c6d7c486d5ac26ee462a17a18b83","side":"left"},{"sibling":"ebd67083cb0a8b3a39677753aa0aca084400b6baad779983ede181ad5337eea3","side":"right"},{"sibling":"6ecdd783142cf3bf726d7720408aec9cf14de71209d4178208dcf89e5b411974","side":"left"},{"sibling":"a8b18e6d0ec4a0f130fc8061b2313189b351327ff818e979c489def7e296d8a3","side":"left"},{"sibling":"1e82068bd784779754cf4c7ebad995756b8568c4c3050604d6f3f530fe4825b9","side":"right"},{"sibling":"41924649fb2045a8eb1c76bcccd1d2cd8c7e0ab0bba7b07a79bf25a0c1677263","side":"right"},{"sibling":"21332ee1c3955bf1803955bf945ab0966ce5e0aa48d8642320cdbc044a72b935","side":"left"},{"sibling":"6620a5008acf732cd3e57b0f2d1437293e88366c2e5b175ebeb7d796fc1e0c62","side":"left"},{"sibling":"a82575bfb494af7afcc13aaae718afa6f74030f09d71b02819ea25efd4186fc4","side":"right"},{"sibling":"e6adead8216db4cae92f0a036d53baebf30eed95a99c0d10758aa75bb7780f2f","side":"right"},{"sibling":"1acc2b7ff453ffd8c97b80ae4db5358780f0c6796874fd75403791dbe99f8cd7","side":"right"},{"sibling":"24d1bb4b13e0e46131b27b70a48e65fcf4e2e14b95e3bb83ade821e9df530f6b","side":"left"},{"sibling":"5f86f58c28b1a86ae06dfff4666bb9fba8866021a81fd4f1d200aa9af4722dfb","side":"right"},{"sibling":"f6cc6f94f6944ae21390afc65ac9e91dc31f84ee6e060681bba5ae08058294bd","side":"right"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"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":206226,"merkle_root":"d2a6d32b13cbf343fb143b21a756d0533864ae6577a376ee84ba867b949207ec","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260602T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-02T05:37:46Z","sig_algorithm":"ed25519","signature":"abd9956cfb19dd1fb8142c46a220bac2514848c6abb0e79b8b0940206cc3ebb00894d4daaf9f786427f82a7cc12482e7fda79054ebb06bceaa9b4b97e23fb30e","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_d6c7d34d3f2d6a698bd33dfcf713ed0e7d5524f66f9a1743e781dfea3287714b"}}