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While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symm","title":"Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers","url":"https://arxiv.org/abs/2605.18106","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.18106v1 Announce Type: cross \nAbstract: A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. 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More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symm","title":"Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-19T04:43:36Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.18106"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:7449a0f414da8a31c30ed490feab8f7bf8fd62e9ea04bae10706d0f34e026e353a34671dde683f13bb98f672e4c9b7a5558e2fa3b743dfa82b342507df38e903","signer":"crovia.substrate","subject":{"observed_at":"2026-05-19T04:43:36Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.18106"},"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":"1403a5fa8d5d9500895ddadb1ae176da7c3e5a331f4d8d3613ca09c585cf0091","leaf_index":142829,"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":"2bd58989c04ae388af46b61ccb0009d80c63171743689d8a51bcb1b14ef38bca","side":"left"},{"sibling":"1c54144163dc65ad72abd61b1feea33a47eb5666b3ba55759960ab1d6055d2bb","side":"right"},{"sibling":"e5a69e6ea0b67f11b42748b66c1cbec97fd484abb94a9f6dfd81380349449eb7","side":"left"},{"sibling":"11aebd1813f1dd042a642e6220babc25ce71cad67eb7ab197948a368d0e74bc3","side":"left"},{"sibling":"123a34c927a0499526eca827616a1bbf60e8975c959e15a902c001a391f09bdf","side":"right"},{"sibling":"b20281a18efefd22a53865024a324dd7382dccc561660387151a9918c829a505","side":"left"},{"sibling":"323685a66e9243202fc51767d169b1aade6efd749dca8b6de8ac2c6fde5340ff","side":"left"},{"sibling":"bda8ed627eb0c8b21066bfa3a60b10b3fd1daf60f49295408a27bab24b140eb3","side":"left"},{"sibling":"f1c796d3bd453570426dcee9ab20072f19762202f84b7993afba2c7c0ee9ec3b","side":"left"},{"sibling":"2e0ce989d789c88e796991ef014ce7e4e1f96c0d4ede9da8d31afcf5ba6a8f46","side":"right"},{"sibling":"202f1bead178ef3785968d50d3d188264a95192a077654c331612e04a34cbfbe","side":"left"},{"sibling":"72249c8c8b068386e35d16f4bd0bbeb9ba820ca217ef0f0d28396c9fe493f5f0","side":"left"},{"sibling":"ea64599340f7ffdf17ad0cbc1d9401ef8870a347e3847bdc106d06b1673df09c","side":"right"},{"sibling":"8f4c0fbe56b6c010fbb8c782ebcd478079bb3f991d8704e2534209a075d9163c","side":"left"},{"sibling":"4db1f363729507e27a60851cf6ed334d7b9acdef194ed7d419aba4d2bd367a4a","side":"right"},{"sibling":"a86ee18c45e7fcc408b6007eaece05aa75b2d9ae30252e9e878462b4dffbef7b","side":"right"},{"sibling":"1d18e7663d43ccff0122ecc7ee12645bb16afb607b218e81b1ea2408f863cb78","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":143302,"merkle_root":"999156d40a7c61d9ddd52b7338f3cbda3e68f53bace070c7b616ea194e23b123","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260519T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-19T05:37:30Z","sig_algorithm":"ed25519","signature":"b1a252cc66ff32bed1d10dd88a6b2a200e3856d3dbcfcc4ee55e02e00f3d548e854ed9c544704b222bd5d315492c4a935ba2d90d727c585a67899b0ad602fc05","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_f24beea6019d65b466a61cf7babc6d448ede41f8be81a1bae345d91a1bce3c35"}}