{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:ULS4B4RQBISTBL7OVYFEKVO2Y2","short_pith_number":"pith:ULS4B4RQ","schema_version":"1.0","canonical_sha256":"a2e5c0f2300a2530afeeae0a4555dac69b64f086e2ffd4cdb81ead85a3fa6199","source":{"kind":"arxiv","id":"2106.01101","version":2},"attestation_state":"computed","paper":{"title":"Learning a Single Neuron with Bias Using Gradient Descent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Gal Vardi, Gilad Yehudai, Ohad Shamir","submitted_at":"2021-06-02T12:09:55Z","abstract_excerpt":"We theoretically study the fundamental problem of learning a single neuron with a bias term ($\\mathbf{x} \\mapsto \\sigma(<\\mathbf{w},\\mathbf{x}> + b)$) in the realizable setting with the ReLU activation, using gradient descent. Perhaps surprisingly, we show that this is a significantly different and more challenging problem than the bias-less case (which was the focus of previous works on single neurons), both in terms of the optimization geometry as well as the ability of gradient methods to succeed in some scenarios. We provide a detailed study of this problem, characterizing the critical poi"},"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":"2106.01101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-06-02T12:09:55Z","cross_cats_sorted":["cs.NE","stat.ML"],"title_canon_sha256":"73f1a903f6c4afe93254415097e00828aadfa541c0efe466e539ab9a8fc34944","abstract_canon_sha256":"7a4bfb2361dc9ffcaa6d34239be6f8438744b5bdc0f93a993afaa37d7ec42dfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:54:21.067100Z","signature_b64":"dfQfzbCgCJB4hozBkZRUjDlDZUzpoz5E3mM4E0dqDarLTKu5v08oNrpvtMWtk6HEhdICkNrn4v+I0ZuTB58MAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2e5c0f2300a2530afeeae0a4555dac69b64f086e2ffd4cdb81ead85a3fa6199","last_reissued_at":"2026-07-05T03:54:21.066599Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:54:21.066599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Learning a Single Neuron with Bias Using Gradient Descent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","stat.ML"],"primary_cat":"cs.LG","authors_text":"Gal Vardi, Gilad Yehudai, Ohad Shamir","submitted_at":"2021-06-02T12:09:55Z","abstract_excerpt":"We theoretically study the fundamental problem of learning a single neuron with a bias term ($\\mathbf{x} \\mapsto \\sigma(<\\mathbf{w},\\mathbf{x}> + b)$) in the realizable setting with the ReLU activation, using gradient descent. Perhaps surprisingly, we show that this is a significantly different and more challenging problem than the bias-less case (which was the focus of previous works on single neurons), both in terms of the optimization geometry as well as the ability of gradient methods to succeed in some scenarios. We provide a detailed study of this problem, characterizing the critical poi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.01101","kind":"arxiv","version":2},"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/2106.01101/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":"2106.01101","created_at":"2026-07-05T03:54:21.066649+00:00"},{"alias_kind":"arxiv_version","alias_value":"2106.01101v2","created_at":"2026-07-05T03:54:21.066649+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.01101","created_at":"2026-07-05T03:54:21.066649+00:00"},{"alias_kind":"pith_short_12","alias_value":"ULS4B4RQBIST","created_at":"2026-07-05T03:54:21.066649+00:00"},{"alias_kind":"pith_short_16","alias_value":"ULS4B4RQBISTBL7O","created_at":"2026-07-05T03:54:21.066649+00:00"},{"alias_kind":"pith_short_8","alias_value":"ULS4B4RQ","created_at":"2026-07-05T03:54:21.066649+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/ULS4B4RQBISTBL7OVYFEKVO2Y2","json":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2.json","graph_json":"https://pith.science/api/pith-number/ULS4B4RQBISTBL7OVYFEKVO2Y2/graph.json","events_json":"https://pith.science/api/pith-number/ULS4B4RQBISTBL7OVYFEKVO2Y2/events.json","paper":"https://pith.science/paper/ULS4B4RQ"},"agent_actions":{"view_html":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2","download_json":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2.json","view_paper":"https://pith.science/paper/ULS4B4RQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2106.01101&json=true","fetch_graph":"https://pith.science/api/pith-number/ULS4B4RQBISTBL7OVYFEKVO2Y2/graph.json","fetch_events":"https://pith.science/api/pith-number/ULS4B4RQBISTBL7OVYFEKVO2Y2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2/action/storage_attestation","attest_author":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2/action/author_attestation","sign_citation":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2/action/citation_signature","submit_replication":"https://pith.science/pith/ULS4B4RQBISTBL7OVYFEKVO2Y2/action/replication_record"}},"created_at":"2026-07-05T03:54:21.066649+00:00","updated_at":"2026-07-05T03:54:21.066649+00:00"}