{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:2RWIEOJCIU3NHICHM35RGGCSP5","short_pith_number":"pith:2RWIEOJC","schema_version":"1.0","canonical_sha256":"d46c8239224536d3a04766fb1318527f647514268d87abfb484c3633af8accdf","source":{"kind":"arxiv","id":"2202.08658","version":2},"attestation_state":"computed","paper":{"title":"The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","stat.ML"],"primary_cat":"cs.LG","authors_text":"Emmanuel Abbe, Enric Boix-Adsera, Theodor Misiakiewicz","submitted_at":"2022-02-17T13:43:06Z","abstract_excerpt":"It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parameterizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest (non-linear but regular networks) no tight characterization has yet been achieved, despite significant developments.\n  We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small"},"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":"2202.08658","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2022-02-17T13:43:06Z","cross_cats_sorted":["cs.DS","stat.ML"],"title_canon_sha256":"2fc5e83f72092494dd79b90e9d01d927d824d307d01c7da66f39702829d5f45b","abstract_canon_sha256":"10738478d8fb04f5482612843fd0d7a01d90d95988d037bc57f57ef9e07c9159"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:59:21.857743Z","signature_b64":"eO22imgbQlsoI1BTfZtWJNM6UVVqJ3VnZJq9uTKSk4KdMbABhtT8qgQeQze7gOYPIf5Wb22MLmBDM426mJJMAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d46c8239224536d3a04766fb1318527f647514268d87abfb484c3633af8accdf","last_reissued_at":"2026-07-05T08:59:21.857340Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:59:21.857340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","stat.ML"],"primary_cat":"cs.LG","authors_text":"Emmanuel Abbe, Enric Boix-Adsera, Theodor Misiakiewicz","submitted_at":"2022-02-17T13:43:06Z","abstract_excerpt":"It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parameterizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest (non-linear but regular networks) no tight characterization has yet been achieved, despite significant developments.\n  We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.08658","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/2202.08658/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":"2202.08658","created_at":"2026-07-05T08:59:21.857403+00:00"},{"alias_kind":"arxiv_version","alias_value":"2202.08658v2","created_at":"2026-07-05T08:59:21.857403+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2202.08658","created_at":"2026-07-05T08:59:21.857403+00:00"},{"alias_kind":"pith_short_12","alias_value":"2RWIEOJCIU3N","created_at":"2026-07-05T08:59:21.857403+00:00"},{"alias_kind":"pith_short_16","alias_value":"2RWIEOJCIU3NHICH","created_at":"2026-07-05T08:59:21.857403+00:00"},{"alias_kind":"pith_short_8","alias_value":"2RWIEOJC","created_at":"2026-07-05T08:59:21.857403+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.01288","citing_title":"A Theory of Saddle Escape in Deep Nonlinear Networks","ref_index":2,"is_internal_anchor":false},{"citing_arxiv_id":"2605.01288","citing_title":"A Theory of Saddle Escape in Deep Nonlinear Networks","ref_index":2,"is_internal_anchor":false},{"citing_arxiv_id":"2605.01288","citing_title":"A Theory of Saddle Escape in Deep Nonlinear Networks","ref_index":2,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5","json":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5.json","graph_json":"https://pith.science/api/pith-number/2RWIEOJCIU3NHICHM35RGGCSP5/graph.json","events_json":"https://pith.science/api/pith-number/2RWIEOJCIU3NHICHM35RGGCSP5/events.json","paper":"https://pith.science/paper/2RWIEOJC"},"agent_actions":{"view_html":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5","download_json":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5.json","view_paper":"https://pith.science/paper/2RWIEOJC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2202.08658&json=true","fetch_graph":"https://pith.science/api/pith-number/2RWIEOJCIU3NHICHM35RGGCSP5/graph.json","fetch_events":"https://pith.science/api/pith-number/2RWIEOJCIU3NHICHM35RGGCSP5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5/action/storage_attestation","attest_author":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5/action/author_attestation","sign_citation":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5/action/citation_signature","submit_replication":"https://pith.science/pith/2RWIEOJCIU3NHICHM35RGGCSP5/action/replication_record"}},"created_at":"2026-07-05T08:59:21.857403+00:00","updated_at":"2026-07-05T08:59:21.857403+00:00"}