{"paper":{"title":"Scalable Bi-causal Optimal Transport via KL Relaxation and Policy Gradients","license":"http://creativecommons.org/licenses/by/4.0/","headline":"KL relaxation turns bi-causal optimal transport into a policy-gradient problem","cross_cats":["cs.LG"],"primary_cat":"math.OC","authors_text":"Haoyang Cao, Jesse Hoekstra, Renyuan Xu, Ruixun Zhang, Yumin Xu","submitted_at":"2026-05-17T05:41:01Z","abstract_excerpt":"Bi-causal optimal transport (OT) is a natural framework for comparing and coupling stochastic processes under nonanticipative information constraints, with important applications in robust finance, sequential uncertainty quantification, and multistage stochastic optimization. In particular, a learned bi-causal coupling naturally serves as a simulator for generating joint sample paths that respect both prescribed marginal laws and the underlying information flow. Its practical use, however, is limited by the computational difficulty of enforcing bi-causal coupling constraints over path space, e"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish dynamic programming principles for both the original and relaxed formulations, prove that the relaxed problem converges to the original bi-causal OT problem as the penalty grows, and derive explicit policy-gradient representations for the relaxed objective. Building on these results, we propose a practical policy-gradient algorithm with unbiased mini-batch estimators, variance reduction, and nonasymptotic regret guarantees.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The framework assumes that the KL relaxation preserves the recursive structure of the bi-causal problem sufficiently for dynamic programming and policy gradient methods to apply directly, and that marginal laws can be sampled to enable the stochastic optimization procedure described.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A KL-relaxed formulation of bi-causal optimal transport is solved via policy gradients with proven convergence to the original problem and nonasymptotic regret guarantees for the resulting algorithm.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"KL relaxation turns bi-causal optimal transport into a policy-gradient problem","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9dee8f7222e87e2119b5d5502b43a9a10ff7b3a725594bf29f7b23b5349d46fe"},"source":{"id":"2605.17271","kind":"arxiv","version":1},"verdict":{"id":"a35e1a80-85ed-45cb-9e7b-cbefb76d5d55","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:06:42.062019Z","strongest_claim":"We establish dynamic programming principles for both the original and relaxed formulations, prove that the relaxed problem converges to the original bi-causal OT problem as the penalty grows, and derive explicit policy-gradient representations for the relaxed objective. Building on these results, we propose a practical policy-gradient algorithm with unbiased mini-batch estimators, variance reduction, and nonasymptotic regret guarantees.","one_line_summary":"A KL-relaxed formulation of bi-causal optimal transport is solved via policy gradients with proven convergence to the original problem and nonasymptotic regret guarantees for the resulting algorithm.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The framework assumes that the KL relaxation preserves the recursive structure of the bi-causal problem sufficiently for dynamic programming and policy gradient methods to apply directly, and that marginal laws can be sampled to enable the stochastic optimization procedure described.","pith_extraction_headline":"KL relaxation turns bi-causal optimal transport into a policy-gradient problem"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17271/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.216522Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:13:21.007701Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.836529Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.778072Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"533ace73c6130e27e4071f3b93844809ddf63219741c0afb53d5e879dd24b21c"},"references":{"count":23,"sample":[{"doi":"","year":null,"title":"Here, the first inequality follows fromJπ n+1 ≥V n+1 and the monotonicity of operatorQπ n+1, and the last inequality follows from (15)","work_id":"95dd3457-e2d9-4af2-b89c-d36057606821","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"NX n=0 cn(Yn, Y ′ n) # =E π","work_id":"92e2157a-694f-45d2-9eac-bed3823b4161","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"43 The definition ofWbc in (5) as the infimum overπ∈ΠΠΠbc yieldsW bc(µ, µ′)≥inf π0∈Π(µ0,µ′ 0) Eπ0[V0]","work_id":"c18fb885-1533-4079-8484-d7398c1e9c12","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Notice that (A.3) holds for any initial coupling inΠ(µ0, µ′ 0)","work_id":"5b93d9f2-82ae-4ce9-ae32-b5e2e90bcdeb","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"To establish the second equivalent representation, notice thatγ ϵ ∈ M bc(µ, µ′)⊂Π ΠΠbc and Wbc(µ, µ′) = inf π0∈Π(µ0,µ′","work_id":"64f761fb-aff7-4675-ac60-d1ad02cfc366","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"62ca0d2ec3fc558ce01777bcf5f3d855e632f47dc06d0e8c47267ec71f3bb2bd","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"0d510d0506d2ab9762e536e8083e2e38ddeb5ba1fd4029059a5cf1789c2ca1b9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}