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In this work, we show that discrete Gradient Descent (GD) with a large step size tells a different story. We prove that single-path solutions are sharp minima, whereas distributing signals across pathways reduces sharpness by a factor that decreases with both the number of pathways and depth. Consequently, while early training reproduces the depth-driven symmetry breaking predicted by GF, oscillations at the Edge of Stability subsequently override this tendency and drive the network into a re-balancing phase, where signals redistribute across pathways. Together, these results clarify how depth shapes pathway competition and explain why large-step GD favors shared representations rather than persistent single-pathway dominance.","title":"Gradient Descent with Large Step Size Restores Symmetry in Deep Linear Networks with Multi-Pathway","url":"https://arxiv.org/abs/2606.05219","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.05219v1 Announce Type: cross \nAbstract: Recent analyses of multi-pathway Deep Linear Networks use Gradient Flow to predict a \"winner-takes-all\" specialization in which path symmetry breaks and each feature concentrates in a single pathway. In this work, we show that discrete Gradient Descent (GD) with a large step size tells a different story. We prove that single-path solutions are sharp minima, whereas distributing signals across pathways reduces sharpness by a factor that decreases with both the number of pathways and depth. Consequently, while early training reproduces the depth-driven symmetry breaking predicted by GF, oscillations at the Edge of Stability subsequently override this tendency and drive the network into a re-balancing phase, where signals redistribute across pathways. Together, these results clarify how depth shapes pathway competition and explain why large-step GD favors shared representations rather than persistent single-pathway dominance.","title":"Gradient Descent with Large Step Size Restores Symmetry in Deep Linear Networks with Multi-Pathway","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-06T04:43:19Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.05219"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:5cb9d8c6ee7703b59661b7d6c33aceef0c5e121289f58d17b2546e8f5a3db2097aec21d4a276ee11009c3e40b3aacbc04aa6520d79e06aaaa71293fb560bc30c","signer":"crovia.substrate","subject":{"observed_at":"2026-06-06T04:43:19Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.05219"},"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":"b25a2f8deef606b3fc5cd6bdf85ce6cf705f930bc5409e0f2e65b833906045d1","leaf_index":219263,"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":"b6a941bc112ae1dd358839cdb6f2e74bd7dd5ce659c90087aa27bea0336bc199","side":"left"},{"sibling":"5132a7a5db34cac5d4059df7e4505b5b07ab9d9f6da798e119a4370494083753","side":"left"},{"sibling":"a380ccc580b7f968ea78eccffa8bb7ae4f72bc272ce6f4d7c332b3972a152828","side":"left"},{"sibling":"9a6907dbadc11f2c9b87922997aae80737fda5cd1a4d0266f57e3abde2b5d057","side":"left"},{"sibling":"1067976a95094a3b641898a2637fb226be05e899d82eff1a55d7ce9ea1c1d1c8","side":"left"},{"sibling":"55a9984d36ea19e3b3022744f50bc60dd27b253cce112377e2e9c25cd3a1ab3c","side":"left"},{"sibling":"63213cfb37562a3ef2d70c7ab846d4203323a4985a9908a7078e73d8d4ab2f65","side":"left"},{"sibling":"8f35ffd1b994bec37a9cc24e0047790f92b64b006d85bef48f319d6d28a40687","side":"right"},{"sibling":"bd0dfa76bed61a2c5e95135a7304a2b8c4fd064958dbf768664232880d0abaa5","side":"right"},{"sibling":"84d2509eab51047589142ed6da8c496305d2fbcbe148e0e6755163db2c7a4bc4","side":"right"},{"sibling":"9e3ea17e834fab022f2eabcfedb8ea0ac95c1f9fb57edc5004dded68522d3c9e","side":"right"},{"sibling":"41d58fea95a95071715ee23ef8bcd15f5867a3639da28e62a0641bc95eb83094","side":"left"},{"sibling":"27ad9d6a9ab792d708709017242a61b9ca519da4e035f87a342811aae221d000","side":"left"},{"sibling":"5f303e2a7840c60038ff2d035b1cd911feefb0fba880de2d737c6671ace594d4","side":"right"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"1a61eadbf0217d063ab78291ccafdc0c92907f7d6ccdc3357534ef89f07d78ae","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":219672,"merkle_root":"d3e32d3a61ca02ce6b1f0b2db86721107770b250e8a5bf762a2c225d2f03c870","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260606T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-06T05:38:35Z","sig_algorithm":"ed25519","signature":"dab214c2d4d857f01383c8e93a521a774b1aba60eaee5677d4e43e4074f0342b2c6a9b9bfcff0eba74f7ac81fcb490dd0e43727979c1a7e8979c7e11547fb101","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_20a1bd677e9e6d5c8386b09852401a8968898a17d0e75144ec702e9c0d7f2d6c"}}