{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:V3ZUVYL4SZ3K5NJTLTPMLCII66","short_pith_number":"pith:V3ZUVYL4","canonical_record":{"source":{"id":"2402.05576","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-02-08T11:23:11Z","cross_cats_sorted":[],"title_canon_sha256":"2219a57ced1ca304467d72744139ac436a93fd85059ee4147e985fd09970587f","abstract_canon_sha256":"1d21d0ff0ec5d4981241c58d617f18a9643f6cf8ddef72bf480238f179b19640"},"schema_version":"1.0"},"canonical_sha256":"aef34ae17c9676aeb5335cdec58908f7a677bbb6d241bc0ef060f641b12c6409","source":{"kind":"arxiv","id":"2402.05576","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.05576","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"arxiv_version","alias_value":"2402.05576v4","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.05576","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"pith_short_12","alias_value":"V3ZUVYL4SZ3K","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"V3ZUVYL4SZ3K5NJT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"V3ZUVYL4","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:V3ZUVYL4SZ3K5NJTLTPMLCII66","target":"record","payload":{"canonical_record":{"source":{"id":"2402.05576","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-02-08T11:23:11Z","cross_cats_sorted":[],"title_canon_sha256":"2219a57ced1ca304467d72744139ac436a93fd85059ee4147e985fd09970587f","abstract_canon_sha256":"1d21d0ff0ec5d4981241c58d617f18a9643f6cf8ddef72bf480238f179b19640"},"schema_version":"1.0"},"canonical_sha256":"aef34ae17c9676aeb5335cdec58908f7a677bbb6d241bc0ef060f641b12c6409","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:47.019503Z","signature_b64":"IaiNRVoNrIabNOBVvwsedI9epKnQBEY2/8gEstOVJC0oLllXc/tnjfizrD1y6sHbMzNZWzQO8Ai5r4SbTGQHAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aef34ae17c9676aeb5335cdec58908f7a677bbb6d241bc0ef060f641b12c6409","last_reissued_at":"2026-05-18T03:09:47.018745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:47.018745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.05576","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"chQuoz3WDNTGI51esFbZ12cwx1cHeWjt9jwOWbcs4zq53NhILyPmZMMMCmNY+rpoijzBp+ET8mfhwckpq7IfCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:51:20.644160Z"},"content_sha256":"9709ca304850328dd0197c8e17c502b14122686e853fa4642215d8fa75112200","schema_version":"1.0","event_id":"sha256:9709ca304850328dd0197c8e17c502b14122686e853fa4642215d8fa75112200"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:V3ZUVYL4SZ3K5NJTLTPMLCII66","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"A. Martina Neuman, Anastasis Kratsios, Gudmund Pammer","submitted_at":"2024-02-08T11:23:11Z","abstract_excerpt":"Machine learning models with inputs in a Euclidean space $\\mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the constant $c>0$ obtained through classical methods can be large in terms of the ambient dimension $d$ and machine precision, posing a challenge when $N$ is small to realistically large. In this paper, we derive a family of generalization bounds $\\{c_m/N^{1/(2\\vee m)}\\}_{m=1}^{\\infty}$ tailored for learning models on digital computers, which adapt to both the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.05576","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T03:09:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yedoI7s+guBueQD4rQdbvGZpCvGR64DPTQuOOlTz0Q5k9oVfFYb4ATImDbjyXlgwIeZOxRiNzVjrkZYwf0MSCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:51:20.644733Z"},"content_sha256":"c3a3edd062ea328826e59c10c22dff83858f2464b7187e4ebef0cd832ae5c4ab","schema_version":"1.0","event_id":"sha256:c3a3edd062ea328826e59c10c22dff83858f2464b7187e4ebef0cd832ae5c4ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66/bundle.json","state_url":"https://pith.science/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66/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":"2026-05-27T05:51:20Z","links":{"resolver":"https://pith.science/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66","bundle":"https://pith.science/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66/bundle.json","state":"https://pith.science/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V3ZUVYL4SZ3K5NJTLTPMLCII66/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:V3ZUVYL4SZ3K5NJTLTPMLCII66","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":"1d21d0ff0ec5d4981241c58d617f18a9643f6cf8ddef72bf480238f179b19640","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-02-08T11:23:11Z","title_canon_sha256":"2219a57ced1ca304467d72744139ac436a93fd85059ee4147e985fd09970587f"},"schema_version":"1.0","source":{"id":"2402.05576","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.05576","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"arxiv_version","alias_value":"2402.05576v4","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.05576","created_at":"2026-05-18T03:09:47Z"},{"alias_kind":"pith_short_12","alias_value":"V3ZUVYL4SZ3K","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"V3ZUVYL4SZ3K5NJT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"V3ZUVYL4","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:c3a3edd062ea328826e59c10c22dff83858f2464b7187e4ebef0cd832ae5c4ab","target":"graph","created_at":"2026-05-18T03:09:47Z","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"},"paper":{"abstract_excerpt":"Machine learning models with inputs in a Euclidean space $\\mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the constant $c>0$ obtained through classical methods can be large in terms of the ambient dimension $d$ and machine precision, posing a challenge when $N$ is small to realistically large. In this paper, we derive a family of generalization bounds $\\{c_m/N^{1/(2\\vee m)}\\}_{m=1}^{\\infty}$ tailored for learning models on digital computers, which adapt to both the ","authors_text":"A. Martina Neuman, Anastasis Kratsios, Gudmund Pammer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-02-08T11:23:11Z","title":"Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.05576","kind":"arxiv","version":4},"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":"sha256:9709ca304850328dd0197c8e17c502b14122686e853fa4642215d8fa75112200","target":"record","created_at":"2026-05-18T03:09:47Z","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":"1d21d0ff0ec5d4981241c58d617f18a9643f6cf8ddef72bf480238f179b19640","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-02-08T11:23:11Z","title_canon_sha256":"2219a57ced1ca304467d72744139ac436a93fd85059ee4147e985fd09970587f"},"schema_version":"1.0","source":{"id":"2402.05576","kind":"arxiv","version":4}},"canonical_sha256":"aef34ae17c9676aeb5335cdec58908f7a677bbb6d241bc0ef060f641b12c6409","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aef34ae17c9676aeb5335cdec58908f7a677bbb6d241bc0ef060f641b12c6409","first_computed_at":"2026-05-18T03:09:47.018745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:47.018745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IaiNRVoNrIabNOBVvwsedI9epKnQBEY2/8gEstOVJC0oLllXc/tnjfizrD1y6sHbMzNZWzQO8Ai5r4SbTGQHAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:47.019503Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.05576","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9709ca304850328dd0197c8e17c502b14122686e853fa4642215d8fa75112200","sha256:c3a3edd062ea328826e59c10c22dff83858f2464b7187e4ebef0cd832ae5c4ab"],"state_sha256":"56741865bdf18a8d100ab5f348a45ad8bf418f54d09978553d794a5920f719fd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DDxwOzpmvvE+FXx0lQrf0SyasZ8Y2HRPe6QB3OwKh/k/ph3qKufWpPQNPTG9kdT22QEQ8wb6XlKX0dmCajwUBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:51:20.647435Z","bundle_sha256":"c10f872d406a90892e600fc62bd7edb37ad49c2094eefb001fc7f863f06f00b8"}}