{"paper":{"title":"Propagation of Chaos in Contextual Flow Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Finite-context models converge to infinite-context versions uniformly in depth and training steps at optimal Wasserstein rates.","cross_cats":["math.AP","math.OC","math.PR","math.ST","stat.TH"],"primary_cat":"cs.LG","authors_text":"Kaizhao Liu, Philippe Rigollet, Shi Chen, Zhengjiang Lin","submitted_at":"2026-05-16T02:03:20Z","abstract_excerpt":"We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length $n$ becomes a statistical resource. Exploiting the McKean--Vlasov structure of the dynamics and the classical machinery of p"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate n^{-1/d} for general CFMs and parametric rate n^{-1/2} for a restricted class of CFMs that includes transformers as a special case.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The dynamics of the attention blocks admit a McKean-Vlasov structure so that the finite-context empirical measure converges to a population measure in the large-n limit; this is invoked to apply classical propagation-of-chaos machinery (abstract, paragraph on McKean-Vlasov structure).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Derives forward and backward propagation-of-chaos bounds for finite vs. infinite-context transformers modeled as contextual flow maps, achieving Wasserstein rate n^{-1/d} generally and n^{-1/2} for transformer-like cases.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite-context models converge to infinite-context versions uniformly in depth and training steps at optimal Wasserstein rates.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4b1696ea3d5175fd5b9a23981ff54ee7bd313e6513acf15eac02af4d313263bd"},"source":{"id":"2605.16747","kind":"arxiv","version":1},"verdict":{"id":"9491591e-e120-4485-9bee-b853e3fc8d04","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:15:44.139730Z","strongest_claim":"We establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate n^{-1/d} for general CFMs and parametric rate n^{-1/2} for a restricted class of CFMs that includes transformers as a special case.","one_line_summary":"Derives forward and backward propagation-of-chaos bounds for finite vs. infinite-context transformers modeled as contextual flow maps, achieving Wasserstein rate n^{-1/d} generally and n^{-1/2} for transformer-like cases.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The dynamics of the attention blocks admit a McKean-Vlasov structure so that the finite-context empirical measure converges to a population measure in the large-n limit; this is invoked to apply classical propagation-of-chaos machinery (abstract, paragraph on McKean-Vlasov structure).","pith_extraction_headline":"Finite-context models converge to infinite-context versions uniformly in depth and training steps at optimal Wasserstein rates."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16747/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.163510Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:22:11.512694Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:01:56.329523Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.459780Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"6303bd8a460a220a3ff93483a6cd5fc147a16daacf37cb5c4b5df01fe11acdd7"},"references":{"count":21,"sample":[{"doi":"","year":null,"title":"[ÁLGRB26] Antonio Álvarez-López, Borjan Geshkovski, and Domènec Ruiz-Balet","work_id":"04dcc830-26c9-4d53-a03a-9610389db975","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Perceptrons and localization of attention’s mean-field landscape","work_id":"9fa6df2f-3410-4bf8-9e33-0ba750fb8576","ref_index":2,"cited_arxiv_id":"2601.21366","is_internal_anchor":true},{"doi":"","year":null,"title":"[BCL+26] Giuseppe Bruno, Shi Chen, Zhengjiang Lin, Yury Polyanskiy, and Philippe Rigollet","work_id":"dd1dccbe-d10b-4755-a09d-154d6b7a49b0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Scaling Limits of Long-Context Transformers","work_id":"a43dad01-bd2f-4ec0-a7fb-6fe560c35c34","ref_index":4,"cited_arxiv_id":"2605.08505","is_internal_anchor":true},{"doi":"","year":null,"title":"[BO20] Tom B","work_id":"990fb2ff-4f53-40a4-8805-541835999e17","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"4a8a7d006c96f7534755a2dd51241ede42cc0ed174d6806b03d5cf8fb7e4560f","internal_anchors":6},"formal_canon":{"evidence_count":1,"snapshot_sha256":"bcc4a03d1502d975e7a32e3bdd0bdc3e5d74aad94837da5ba10b65d479e9a36a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}