{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:7OVOS26ECPV6T4NJC5NB6OPAQA","short_pith_number":"pith:7OVOS26E","schema_version":"1.0","canonical_sha256":"fbaae96bc413ebe9f1a9175a1f39e08011f70671f0b427dc873a2b1ca16676e6","source":{"kind":"arxiv","id":"2606.21581","version":1},"attestation_state":"computed","paper":{"title":"Convergence Analysis of Muon-type Methods with Inexact LMO in the Degenerate Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Peter Richt\\'arik, Xun Qian","submitted_at":"2026-06-19T16:36:53Z","abstract_excerpt":"Muon-type methods have demonstrated potentially superior performance over Adam and its variants, and have shown hyperparameter transferability across model sizes when specific norms are chosen for the LMO in deep architectures. However, while the LMO is solved approximately via iterative algorithms in practice, most convergence analyses consider the ideal case where the search direction is the exact solution to the LMO. Recently, the inexact Muon update was analyzed by Shulgin et al. [2025], which reveals a fundamental coupling between the inexactness and the optimal step size and momentum. Ho"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.21581","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-06-19T16:36:53Z","cross_cats_sorted":[],"title_canon_sha256":"319625040babad1127696e6192c27a45f14bfcef04aba6b702d9b4c5135fc166","abstract_canon_sha256":"519c3ea2acffdface3215ca57c2ea1fca7c15915d8769dbb2660d1fc3a0c08bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T01:13:15.308360Z","signature_b64":"wT+qjPViQyrF4t2r3laXVeFh72x8gooUddtiaKYTn1NCLgBz4tbZYAk6QLrP2azz4tAQU5qIjUJPULFxC3/NCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbaae96bc413ebe9f1a9175a1f39e08011f70671f0b427dc873a2b1ca16676e6","last_reissued_at":"2026-06-23T01:13:15.307845Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T01:13:15.307845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence Analysis of Muon-type Methods with Inexact LMO in the Degenerate Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Peter Richt\\'arik, Xun Qian","submitted_at":"2026-06-19T16:36:53Z","abstract_excerpt":"Muon-type methods have demonstrated potentially superior performance over Adam and its variants, and have shown hyperparameter transferability across model sizes when specific norms are chosen for the LMO in deep architectures. However, while the LMO is solved approximately via iterative algorithms in practice, most convergence analyses consider the ideal case where the search direction is the exact solution to the LMO. Recently, the inexact Muon update was analyzed by Shulgin et al. [2025], which reveals a fundamental coupling between the inexactness and the optimal step size and momentum. Ho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21581","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.21581/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.21581","created_at":"2026-06-23T01:13:15.307915+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.21581v1","created_at":"2026-06-23T01:13:15.307915+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.21581","created_at":"2026-06-23T01:13:15.307915+00:00"},{"alias_kind":"pith_short_12","alias_value":"7OVOS26ECPV6","created_at":"2026-06-23T01:13:15.307915+00:00"},{"alias_kind":"pith_short_16","alias_value":"7OVOS26ECPV6T4NJ","created_at":"2026-06-23T01:13:15.307915+00:00"},{"alias_kind":"pith_short_8","alias_value":"7OVOS26E","created_at":"2026-06-23T01:13:15.307915+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA","json":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA.json","graph_json":"https://pith.science/api/pith-number/7OVOS26ECPV6T4NJC5NB6OPAQA/graph.json","events_json":"https://pith.science/api/pith-number/7OVOS26ECPV6T4NJC5NB6OPAQA/events.json","paper":"https://pith.science/paper/7OVOS26E"},"agent_actions":{"view_html":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA","download_json":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA.json","view_paper":"https://pith.science/paper/7OVOS26E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.21581&json=true","fetch_graph":"https://pith.science/api/pith-number/7OVOS26ECPV6T4NJC5NB6OPAQA/graph.json","fetch_events":"https://pith.science/api/pith-number/7OVOS26ECPV6T4NJC5NB6OPAQA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA/action/storage_attestation","attest_author":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA/action/author_attestation","sign_citation":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA/action/citation_signature","submit_replication":"https://pith.science/pith/7OVOS26ECPV6T4NJC5NB6OPAQA/action/replication_record"}},"created_at":"2026-06-23T01:13:15.307915+00:00","updated_at":"2026-06-23T01:13:15.307915+00:00"}