{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:UEDMRKQ4HRY4VIH4HI37WQTDXZ","short_pith_number":"pith:UEDMRKQ4","canonical_record":{"source":{"id":"2006.09313","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2020-06-16T16:57:12Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"ab172f93cbdb2dd6b6edc176dd535488f7f8ba37f21b63bd209a5be7db9ad8a8","abstract_canon_sha256":"128d889a22b6a80f6e44f8dd74a1b8a4c66c1f3fef8df8fa210562ef61410c89"},"schema_version":"1.0"},"canonical_sha256":"a106c8aa1c3c71caa0fc3a37fb4263be71dbad4f94397424050c2cc8ea25f2e9","source":{"kind":"arxiv","id":"2006.09313","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.09313","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"arxiv_version","alias_value":"2006.09313v3","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.09313","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"pith_short_12","alias_value":"UEDMRKQ4HRY4","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"pith_short_16","alias_value":"UEDMRKQ4HRY4VIH4","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"pith_short_8","alias_value":"UEDMRKQ4","created_at":"2026-07-05T03:47:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:UEDMRKQ4HRY4VIH4HI37WQTDXZ","target":"record","payload":{"canonical_record":{"source":{"id":"2006.09313","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2020-06-16T16:57:12Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"ab172f93cbdb2dd6b6edc176dd535488f7f8ba37f21b63bd209a5be7db9ad8a8","abstract_canon_sha256":"128d889a22b6a80f6e44f8dd74a1b8a4c66c1f3fef8df8fa210562ef61410c89"},"schema_version":"1.0"},"canonical_sha256":"a106c8aa1c3c71caa0fc3a37fb4263be71dbad4f94397424050c2cc8ea25f2e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:47:12.599452Z","signature_b64":"aW1egDfYeqNWmuKcAYsLxlspr/EEUbRZNc6oXDiSBzzeGvLYWinxJsuprDF8tLb9oxxnHZB0Le2HcvlnqdrQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a106c8aa1c3c71caa0fc3a37fb4263be71dbad4f94397424050c2cc8ea25f2e9","last_reissued_at":"2026-07-05T03:47:12.598987Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:47:12.598987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.09313","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:47:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cUtsPpqbhIV4zTv1WbqH2iZmYnEk5VJxNkcAJWjb+9b7gsQIens7gj5Lv3TLUhsGPF+B6mcbvclf6H2ovGTVCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T04:56:36.124721Z"},"content_sha256":"5aabce0bc371d4b85ed6e259799e209fb7a9e4314fe8e64e70baf5120e0b2fef","schema_version":"1.0","event_id":"sha256:5aabce0bc371d4b85ed6e259799e209fb7a9e4314fe8e64e70baf5120e0b2fef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:UEDMRKQ4HRY4VIH4HI37WQTDXZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"George Deligiannidis, Murat A. Erdogdu, Ozan Sener, Umut \\c{S}im\\c{s}ekli","submitted_at":"2020-06-16T16:57:12Z","abstract_excerpt":"Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.09313","kind":"arxiv","version":3},"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/2006.09313/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:47:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g9VxBfV3d2+e1JtKEQYQ1EQDjk0yUeSAVVSnA6hNjSC6kGFWTpvhllP1vlil/PSSCPJO4wq65QfzbWwB9Tj4Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T04:56:36.125115Z"},"content_sha256":"bb0ce03dc9037ae45139848d680c451f4b8b1610a43b300bba60809efeac3d32","schema_version":"1.0","event_id":"sha256:bb0ce03dc9037ae45139848d680c451f4b8b1610a43b300bba60809efeac3d32"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ/bundle.json","state_url":"https://pith.science/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-22T04:56:36Z","links":{"resolver":"https://pith.science/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ","bundle":"https://pith.science/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ/bundle.json","state":"https://pith.science/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UEDMRKQ4HRY4VIH4HI37WQTDXZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:UEDMRKQ4HRY4VIH4HI37WQTDXZ","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":"128d889a22b6a80f6e44f8dd74a1b8a4c66c1f3fef8df8fa210562ef61410c89","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2020-06-16T16:57:12Z","title_canon_sha256":"ab172f93cbdb2dd6b6edc176dd535488f7f8ba37f21b63bd209a5be7db9ad8a8"},"schema_version":"1.0","source":{"id":"2006.09313","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.09313","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"arxiv_version","alias_value":"2006.09313v3","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.09313","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"pith_short_12","alias_value":"UEDMRKQ4HRY4","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"pith_short_16","alias_value":"UEDMRKQ4HRY4VIH4","created_at":"2026-07-05T03:47:12Z"},{"alias_kind":"pith_short_8","alias_value":"UEDMRKQ4","created_at":"2026-07-05T03:47:12Z"}],"graph_snapshots":[{"event_id":"sha256:bb0ce03dc9037ae45139848d680c451f4b8b1610a43b300bba60809efeac3d32","target":"graph","created_at":"2026-07-05T03:47:12Z","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/2006.09313/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under ","authors_text":"George Deligiannidis, Murat A. Erdogdu, Ozan Sener, Umut \\c{S}im\\c{s}ekli","cross_cats":["cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2020-06-16T16:57:12Z","title":"Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.09313","kind":"arxiv","version":3},"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":"sha256:5aabce0bc371d4b85ed6e259799e209fb7a9e4314fe8e64e70baf5120e0b2fef","target":"record","created_at":"2026-07-05T03:47:12Z","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":"128d889a22b6a80f6e44f8dd74a1b8a4c66c1f3fef8df8fa210562ef61410c89","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2020-06-16T16:57:12Z","title_canon_sha256":"ab172f93cbdb2dd6b6edc176dd535488f7f8ba37f21b63bd209a5be7db9ad8a8"},"schema_version":"1.0","source":{"id":"2006.09313","kind":"arxiv","version":3}},"canonical_sha256":"a106c8aa1c3c71caa0fc3a37fb4263be71dbad4f94397424050c2cc8ea25f2e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a106c8aa1c3c71caa0fc3a37fb4263be71dbad4f94397424050c2cc8ea25f2e9","first_computed_at":"2026-07-05T03:47:12.598987Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:47:12.598987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aW1egDfYeqNWmuKcAYsLxlspr/EEUbRZNc6oXDiSBzzeGvLYWinxJsuprDF8tLb9oxxnHZB0Le2HcvlnqdrQBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:47:12.599452Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.09313","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5aabce0bc371d4b85ed6e259799e209fb7a9e4314fe8e64e70baf5120e0b2fef","sha256:bb0ce03dc9037ae45139848d680c451f4b8b1610a43b300bba60809efeac3d32"],"state_sha256":"8c069b41582b218ee8acdfa8497991d28cf8ed2dfe13de466f5d24db8c8d8684"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NNdMu9J5/DHVr3Krv1eTwtnsg2cUglo35aNAMHQKmJ+6+ffCjlwyrfKd1DdjhrjgtWWu/KNaJAanQlBuIR/IAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T04:56:36.127637Z","bundle_sha256":"b52229f2d3ba20be4e03346077f8bb905577ccfca0845c274177e9aa19457486"}}