{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:KK2VDR3II7WTTAAEIARKMPLOVN","short_pith_number":"pith:KK2VDR3I","schema_version":"1.0","canonical_sha256":"52b551c76847ed3980044022a63d6eab60d001db0dccca821fa0906d656d32bb","source":{"kind":"arxiv","id":"1912.07242","version":1},"attestation_state":"computed","paper":{"title":"More Data Can Hurt for Linear Regression: Sample-wise Double Descent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NE","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Preetum Nakkiran","submitted_at":"2019-12-16T08:28:26Z","abstract_excerpt":"In this expository note we describe a surprising phenomenon in overparameterized linear regression, where the dimension exceeds the number of samples: there is a regime where the test risk of the estimator found by gradient descent increases with additional samples. In other words, more data actually hurts the estimator. This behavior is implicit in a recent line of theoretical works analyzing \"double-descent\" phenomenon in linear models. In this note, we isolate and understand this behavior in an extremely simple setting: linear regression with isotropic Gaussian covariates. In particular, th"},"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":"1912.07242","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2019-12-16T08:28:26Z","cross_cats_sorted":["cs.LG","cs.NE","math.ST","stat.TH"],"title_canon_sha256":"c962ef7a2c6953adc48191a86acce1f557a9271dae75e56f6fabd56d90877731","abstract_canon_sha256":"87a490ec76f2bad8971e515c38ccabff603d90a852602f2a97ae1f3afba1558a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:26:19.912932Z","signature_b64":"yLmKS2WZiseT1p75WFmiKpp7FsRpPBVbHFrdANTBogZHrEn0xTUwdXufcvf5dghT3jwqxboC+QzHTlY6aQJ0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52b551c76847ed3980044022a63d6eab60d001db0dccca821fa0906d656d32bb","last_reissued_at":"2026-07-05T00:26:19.912601Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:26:19.912601Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"More Data Can Hurt for Linear Regression: Sample-wise Double Descent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NE","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Preetum Nakkiran","submitted_at":"2019-12-16T08:28:26Z","abstract_excerpt":"In this expository note we describe a surprising phenomenon in overparameterized linear regression, where the dimension exceeds the number of samples: there is a regime where the test risk of the estimator found by gradient descent increases with additional samples. In other words, more data actually hurts the estimator. This behavior is implicit in a recent line of theoretical works analyzing \"double-descent\" phenomenon in linear models. In this note, we isolate and understand this behavior in an extremely simple setting: linear regression with isotropic Gaussian covariates. In particular, th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.07242","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/1912.07242/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":"1912.07242","created_at":"2026-07-05T00:26:19.912657+00:00"},{"alias_kind":"arxiv_version","alias_value":"1912.07242v1","created_at":"2026-07-05T00:26:19.912657+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1912.07242","created_at":"2026-07-05T00:26:19.912657+00:00"},{"alias_kind":"pith_short_12","alias_value":"KK2VDR3II7WT","created_at":"2026-07-05T00:26:19.912657+00:00"},{"alias_kind":"pith_short_16","alias_value":"KK2VDR3II7WTTAAE","created_at":"2026-07-05T00:26:19.912657+00:00"},{"alias_kind":"pith_short_8","alias_value":"KK2VDR3I","created_at":"2026-07-05T00:26:19.912657+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":6,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.27142","citing_title":"Estimation of High Dimensional Bounded Discrete Graphical Models via Regularized Generalized Score Matching","ref_index":161,"is_internal_anchor":false},{"citing_arxiv_id":"2606.24903","citing_title":"The Geometry of Saturation: Effective Rank Predicts When Labels Stop Helping in Few-Shot Classification","ref_index":28,"is_internal_anchor":false},{"citing_arxiv_id":"2606.10089","citing_title":"A Theory on Flow Matching with Neural Networks","ref_index":31,"is_internal_anchor":false},{"citing_arxiv_id":"2605.29548","citing_title":"Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention","ref_index":117,"is_internal_anchor":false},{"citing_arxiv_id":"2405.00592","citing_title":"Scaling and renormalization in high-dimensional regression","ref_index":17,"is_internal_anchor":false},{"citing_arxiv_id":"2605.14200","citing_title":"How to Scale Mixture-of-Experts: From muP to the Maximally Scale-Stable Parameterization","ref_index":290,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN","json":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN.json","graph_json":"https://pith.science/api/pith-number/KK2VDR3II7WTTAAEIARKMPLOVN/graph.json","events_json":"https://pith.science/api/pith-number/KK2VDR3II7WTTAAEIARKMPLOVN/events.json","paper":"https://pith.science/paper/KK2VDR3I"},"agent_actions":{"view_html":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN","download_json":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN.json","view_paper":"https://pith.science/paper/KK2VDR3I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1912.07242&json=true","fetch_graph":"https://pith.science/api/pith-number/KK2VDR3II7WTTAAEIARKMPLOVN/graph.json","fetch_events":"https://pith.science/api/pith-number/KK2VDR3II7WTTAAEIARKMPLOVN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN/action/storage_attestation","attest_author":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN/action/author_attestation","sign_citation":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN/action/citation_signature","submit_replication":"https://pith.science/pith/KK2VDR3II7WTTAAEIARKMPLOVN/action/replication_record"}},"created_at":"2026-07-05T00:26:19.912657+00:00","updated_at":"2026-07-05T00:26:19.912657+00:00"}