{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:XORUO7RHA7YZO6BECWZIZS6PEI","short_pith_number":"pith:XORUO7RH","canonical_record":{"source":{"id":"2404.04931","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.LG","submitted_at":"2024-04-07T12:07:33Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"881067d502118d9ae5fe3b3e832a55ad817bd7c15ec4bd84d5a157395c399289","abstract_canon_sha256":"bf47c04afcbc8b4d2cd8a4afd20acdd8d86bf272725f2e696a064307f528680e"},"schema_version":"1.0"},"canonical_sha256":"bba3477e2707f197782415b28ccbcf222421da5b0cca1db4f718face4aeb3ca7","source":{"kind":"arxiv","id":"2404.04931","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.04931","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"arxiv_version","alias_value":"2404.04931v2","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.04931","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"pith_short_12","alias_value":"XORUO7RHA7YZ","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"pith_short_16","alias_value":"XORUO7RHA7YZO6BE","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"pith_short_8","alias_value":"XORUO7RH","created_at":"2026-07-05T08:06:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:XORUO7RHA7YZO6BECWZIZS6PEI","target":"record","payload":{"canonical_record":{"source":{"id":"2404.04931","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.LG","submitted_at":"2024-04-07T12:07:33Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"881067d502118d9ae5fe3b3e832a55ad817bd7c15ec4bd84d5a157395c399289","abstract_canon_sha256":"bf47c04afcbc8b4d2cd8a4afd20acdd8d86bf272725f2e696a064307f528680e"},"schema_version":"1.0"},"canonical_sha256":"bba3477e2707f197782415b28ccbcf222421da5b0cca1db4f718face4aeb3ca7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:06:48.312588Z","signature_b64":"PsjrFSJq0q4JTUT9of9t4ghyPPN6xh6qiKxKXUzz9aDm121Jxb9m0WfSFr7Cvd7M1IU/FmD1Ta4v5i6b1A6eDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bba3477e2707f197782415b28ccbcf222421da5b0cca1db4f718face4aeb3ca7","last_reissued_at":"2026-07-05T08:06:48.311992Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:06:48.311992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.04931","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:06:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QNOcgm1N5tNs4ipwWVfKv4DT/DNSvIsPmsaJqpAUBDwstLZl+B+2AmnqXybWdWuLHF3bpkrF7nrISfXOfklsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T12:37:23.460055Z"},"content_sha256":"67ecb4cab823b33ac8d4235189935fb7fbb0754c9190f6e2b13a62ab7c457135","schema_version":"1.0","event_id":"sha256:67ecb4cab823b33ac8d4235189935fb7fbb0754c9190f6e2b13a62ab7c457135"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:XORUO7RHA7YZO6BECWZIZS6PEI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Sample Complexity of Gradient Descent in Stochastic Convex Optimization","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Roi Livni","submitted_at":"2024-04-07T12:07:33Z","abstract_excerpt":"We analyze the sample complexity of full-batch Gradient Descent (GD) in the setup of non-smooth Stochastic Convex Optimization. We show that the generalization error of GD, with common choice of hyper-parameters, can be $\\tilde \\Theta(d/m + 1/\\sqrt{m})$, where $d$ is the dimension and $m$ is the sample size. This matches the sample complexity of \\emph{worst-case} empirical risk minimizers. That means that, in contrast with other algorithms, GD has no advantage over naive ERMs. Our bound follows from a new generalization bound that depends on both the dimension as well as the learning rate and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.04931","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/2404.04931/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":"2026-07-05T08:06:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z1eJIE+sQc3njadzz141pc7VekzY8HD2qmXNOzF04BarMcZcfd99rj7J2xoan7hqp2wzR/53c7iCmg+Pxj4KAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T12:37:23.460353Z"},"content_sha256":"78675cb8b58e8b8d17f2522e53fd7350128cd07ae7bb754213237fc4ca4beefb","schema_version":"1.0","event_id":"sha256:78675cb8b58e8b8d17f2522e53fd7350128cd07ae7bb754213237fc4ca4beefb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XORUO7RHA7YZO6BECWZIZS6PEI/bundle.json","state_url":"https://pith.science/pith/XORUO7RHA7YZO6BECWZIZS6PEI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XORUO7RHA7YZO6BECWZIZS6PEI/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":"2026-08-05T12:37:23Z","links":{"resolver":"https://pith.science/pith/XORUO7RHA7YZO6BECWZIZS6PEI","bundle":"https://pith.science/pith/XORUO7RHA7YZO6BECWZIZS6PEI/bundle.json","state":"https://pith.science/pith/XORUO7RHA7YZO6BECWZIZS6PEI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XORUO7RHA7YZO6BECWZIZS6PEI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:XORUO7RHA7YZO6BECWZIZS6PEI","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":"bf47c04afcbc8b4d2cd8a4afd20acdd8d86bf272725f2e696a064307f528680e","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.LG","submitted_at":"2024-04-07T12:07:33Z","title_canon_sha256":"881067d502118d9ae5fe3b3e832a55ad817bd7c15ec4bd84d5a157395c399289"},"schema_version":"1.0","source":{"id":"2404.04931","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.04931","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"arxiv_version","alias_value":"2404.04931v2","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.04931","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"pith_short_12","alias_value":"XORUO7RHA7YZ","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"pith_short_16","alias_value":"XORUO7RHA7YZO6BE","created_at":"2026-07-05T08:06:48Z"},{"alias_kind":"pith_short_8","alias_value":"XORUO7RH","created_at":"2026-07-05T08:06:48Z"}],"graph_snapshots":[{"event_id":"sha256:78675cb8b58e8b8d17f2522e53fd7350128cd07ae7bb754213237fc4ca4beefb","target":"graph","created_at":"2026-07-05T08:06:48Z","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/2404.04931/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We analyze the sample complexity of full-batch Gradient Descent (GD) in the setup of non-smooth Stochastic Convex Optimization. We show that the generalization error of GD, with common choice of hyper-parameters, can be $\\tilde \\Theta(d/m + 1/\\sqrt{m})$, where $d$ is the dimension and $m$ is the sample size. This matches the sample complexity of \\emph{worst-case} empirical risk minimizers. That means that, in contrast with other algorithms, GD has no advantage over naive ERMs. Our bound follows from a new generalization bound that depends on both the dimension as well as the learning rate and ","authors_text":"Roi Livni","cross_cats":["math.OC"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.LG","submitted_at":"2024-04-07T12:07:33Z","title":"The Sample Complexity of Gradient Descent in Stochastic Convex Optimization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.04931","kind":"arxiv","version":2},"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":"sha256:67ecb4cab823b33ac8d4235189935fb7fbb0754c9190f6e2b13a62ab7c457135","target":"record","created_at":"2026-07-05T08:06:48Z","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":"bf47c04afcbc8b4d2cd8a4afd20acdd8d86bf272725f2e696a064307f528680e","cross_cats_sorted":["math.OC"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"cs.LG","submitted_at":"2024-04-07T12:07:33Z","title_canon_sha256":"881067d502118d9ae5fe3b3e832a55ad817bd7c15ec4bd84d5a157395c399289"},"schema_version":"1.0","source":{"id":"2404.04931","kind":"arxiv","version":2}},"canonical_sha256":"bba3477e2707f197782415b28ccbcf222421da5b0cca1db4f718face4aeb3ca7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bba3477e2707f197782415b28ccbcf222421da5b0cca1db4f718face4aeb3ca7","first_computed_at":"2026-07-05T08:06:48.311992Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:06:48.311992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PsjrFSJq0q4JTUT9of9t4ghyPPN6xh6qiKxKXUzz9aDm121Jxb9m0WfSFr7Cvd7M1IU/FmD1Ta4v5i6b1A6eDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:06:48.312588Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.04931","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67ecb4cab823b33ac8d4235189935fb7fbb0754c9190f6e2b13a62ab7c457135","sha256:78675cb8b58e8b8d17f2522e53fd7350128cd07ae7bb754213237fc4ca4beefb"],"state_sha256":"0d2955376703ef3a6ecea0c62415818c5f2bcaabe0de4d955423f21841f829b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BJ9rKpuwVwdYzZFc4swa81GPrZ3RmwKzgZKUEGt0oDNwlGmm8298Xy/gd4xWQ35owX9YN4GMUJCmZR8eq1ukCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T12:37:23.463267Z","bundle_sha256":"b635f7e2fa3141d641458e6d2f4b7daafdef79028e28bb6c314884f635ff7d1b"}}