{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:ERTVQSU4UEYSWIOKMI3MD4SGUX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"47ac04ab56cac7634661dd5a1d28f3ffd04ed865b015a61fa74ce2315cef1e6b","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-05-25T16:43:51Z","title_canon_sha256":"f2f4e9cef2198697e0e22f5c18f3ebef5b32e3b6d19691a6e686222c3faec435"},"schema_version":"1.0","source":{"id":"2505.19227","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.19227","created_at":"2026-07-05T11:09:26Z"},{"alias_kind":"arxiv_version","alias_value":"2505.19227v1","created_at":"2026-07-05T11:09:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.19227","created_at":"2026-07-05T11:09:26Z"},{"alias_kind":"pith_short_12","alias_value":"ERTVQSU4UEYS","created_at":"2026-07-05T11:09:26Z"},{"alias_kind":"pith_short_16","alias_value":"ERTVQSU4UEYSWIOK","created_at":"2026-07-05T11:09:26Z"},{"alias_kind":"pith_short_8","alias_value":"ERTVQSU4","created_at":"2026-07-05T11:09:26Z"}],"graph_snapshots":[{"event_id":"sha256:9fe99e7afcece7ae93cafe28567b8c226d0f4fbd683149e64fb11c1f3061dbd0","target":"graph","created_at":"2026-07-05T11:09:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2505.19227/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Recent works have highlighted optimization difficulties faced by gradient descent in training the first and last layers of transformer-based language models, which are overcome by optimizers such as Adam. These works suggest that the difficulty is linked to the heavy-tailed distribution of words in text data, where the frequency of the $k$th most frequent word $\\pi_k$ is proportional to $1/k$, following Zipf's law. To better understand the impact of the data distribution on training performance, we study a linear bigram model for next-token prediction when the tokens follow a power law $\\pi_k ","authors_text":"Francis Bach, Frederik Kunstner","cross_cats":["math.OC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-05-25T16:43:51Z","title":"Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.19227","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f7ec1bc345dfe6c3c7e34139da17b09a847903ea6062b0f86a2c34809fbf5cb1","target":"record","created_at":"2026-07-05T11:09:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"47ac04ab56cac7634661dd5a1d28f3ffd04ed865b015a61fa74ce2315cef1e6b","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-05-25T16:43:51Z","title_canon_sha256":"f2f4e9cef2198697e0e22f5c18f3ebef5b32e3b6d19691a6e686222c3faec435"},"schema_version":"1.0","source":{"id":"2505.19227","kind":"arxiv","version":1}},"canonical_sha256":"2467584a9ca1312b21ca6236c1f246a5c27df86b7791ffb4854780c84b5b9294","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2467584a9ca1312b21ca6236c1f246a5c27df86b7791ffb4854780c84b5b9294","first_computed_at":"2026-07-05T11:09:26.728726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:09:26.728726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rQtS54YpUGKRFyukpar9cFCkc+aC2ZCPbit9cfOEYmWUIuTBG2gLCis2kfN5IyurUrCVHijlVg9vJeuWjX/8BA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:09:26.729223Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.19227","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7ec1bc345dfe6c3c7e34139da17b09a847903ea6062b0f86a2c34809fbf5cb1","sha256:9fe99e7afcece7ae93cafe28567b8c226d0f4fbd683149e64fb11c1f3061dbd0"],"state_sha256":"263a0d34477fff0b71c484e75648da5f9665f1f0d5ead4394f1f13d647446340"}