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VarianceProxy","declaration:BanditRLProof.Concentration.varianceProxy_le_finiteArmVarianceProxy","declaration:BanditRLProof.Concentration.finiteArmVarianceProxy_pos_of_exists","declaration:BanditRLProof.Concentration.finiteArmPositiveVarianceProxy","declaration:BanditRLProof.Concentration.varianceProxy_le_finiteArmPositiveVarianceProxy","declaration:BanditRLProof.Concentration.finiteArmPositiveVarianceProxy_pos","declaration:BanditRLProof.RewardKernel.contextIndependentCenteredRewardKernelLaw_of_hasSubgaussianMGF","declaration:BanditRLProof.RewardKernel.contextIndependentBoundedCenteredRewardKernelLaw","declaration:BanditRLProof.RewardKernel.contextIndependentArmwiseBoundedCenteredRewardKernelLaw","declaration:BanditRLProof.Concentration.finiteContextArmVarianceProxy","declaration:BanditRLProof.Concentration.varianceProxy_le_finiteContextArmVarianceProxy","declaration:BanditRLProof.Concentration.finiteContextArmVarianceProxy_pos_of_exists","declaration:BanditRLProof.Concentration.finiteContextArmPositiveVarianceProxy","declaration:BanditRLProof.Concentration.varianceProxy_le_finiteContextArmPositiveVarianceProxy","declaration:BanditRLProof.Concentration.finiteContextArmPositiveVarianceProxy_pos","declaration:BanditRLProof.History.FiniteActionHistory","declaration:BanditRLProof.History.FiniteRewardHistory","declaration:BanditRLProof.History.FinitePairHistory","declaration:BanditRLProof.History.FiniteHistory","declaration:BanditRLProof.History.finiteActionHistoryOfTrace","declaration:BanditRLProof.History.finiteRewardHistoryOfTrace","declaration:BanditRLProof.History.completeRewardTrace","declaration:BanditRLProof.History.finiteHistoryOfTrace","declaration:BanditRLProof.History.finitePairHistoryOfTrace","declaration:BanditRLProof.History.extendPairHistorySucc","declaration:BanditRLProof.History.finiteActionHistoryOfTrace_apply","declaration:BanditRLProof.History.finiteRewardHistoryOfTrace_apply","declaration:BanditRLProof.History.completeRewardTrace_finiteRewardHistoryOfTrace_apply_of_le","declaration:BanditRLProof.History.finiteHistoryOfTrace_fst","declaration:BanditRLProof.History.finiteHistoryOfTrace_snd","declaration:BanditRLProof.History.finitePairHistoryOfTrace_apply","declaration:BanditRLProof.History.extendPairHistorySucc_apply_of_le","declaration:BanditRLProof.History.extendPairHistorySucc_apply_succ","declaration:BanditRLProof.History.finitePairHistoryOfTrace_succ","declaration:BanditRLProof.History.pairHistoryRewardProjection","declaration:BanditRLProof.History.pairHistoryRewardProjection_apply","declaration:BanditRLProof.History.pairHistoryRewardProjection_finitePairHistoryOfTrace","declaration:BanditRLProof.History.measurable_finiteActionHistory_eval","declaration:BanditRLProof.History.measurable_finiteRewardHistory_eval","declaration:BanditRLProof.History.measurable_finiteHistory_action_eval","declaration:BanditRLProof.History.measurable_finiteHistory_reward_eval","declaration:BanditRLProof.History.measurable_pairHistoryRewardProjection","declaration:BanditRLProof.History.measurable_finitePairHistoryOfTrace","declaration:BanditRLProof.History.measurable_extendPairHistorySucc","declaration:BanditRLProof.History.measurable_finiteActionHistoryOfTrace","declaration:BanditRLProof.History.measurable_finiteRewardHistoryOfTrace","declaration:BanditRLProof.History.measurable_finiteHistoryOfTrace","declaration:BanditRLProof.History.historyGenerators","declaration:BanditRLProof.History.historyGenerators_mono","declaration:BanditRLProof.History.historyMeasurableSpace","declaration:BanditRLProof.History.historyMeasurableSpace_mono","declaration:BanditRLProof.History.historyMeasurableSpace_le","declaration:BanditRLProof.History.historyFiltration","declaration:BanditRLProof.History.historyFiltration_apply","declaration:BanditRLProof.History.historyFiltrationSucc","declaration:BanditRLProof.History.historyFiltrationSucc_apply","declaration:BanditRLProof.History.measurableSet_action_mem_historyFiltration","declaration:BanditRLProof.History.measurable_action_mem_historyFiltration_of_lt","declaration:BanditRLProof.History.measurableSet_reward_mem_historyFiltration","declaration:BanditRLProof.History.measurable_reward_mem_historyFiltration_of_lt","declaration:BanditRLProof.History.measurable_finitePairHistoryOfTrace_mem_historyFiltration_of_lt","declaration:BanditRLProof.History.historyFiltration_succ_eq_comap_finitePairHistoryOfTrace","declaration:BanditRLProof.History.historyFiltrationSucc_eq_comap_finitePairHistoryOfTrace","declaration:BanditRLProof.History.historyFiltrationSucc_eq_of_action_eq_on_prefix","declaration:BanditRLProof.IndependenceFoundation.iIndepFun_infinitePi_coord","declaration:BanditRLProof.IndependenceFoundation.iIndepFun_rewardTrace_infinitePi","declaration:BanditRLProof.IntegrabilitySums.integrable_finset_sum","declaration:BanditRLProof.IntegrabilitySums.integrable_univ_sum","declaration:BanditRLProof.Measure.compProd_restrict_prod","declaration:BanditRLProof.Measure.compProd_restrict_eq_of_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oRegret_exploreArm_explorationPulls_mul_K_le_sum_gap_mul_explorationPulls","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_le_sum_gap_mul_explorationPulls","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_add_le_sum_gap_mul_suffix_count_budget","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_add_le_sum_gap_mul_explorationPulls_add_suffix","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_add_eq_add_suffix_gap","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_add_eq_of_commitArm_eq_bestArm","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_add_le_sum_gap_mul_explorationPulls_of_commitArm_eq_bestArm","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_explorationPulls_mul_K_add_le_sum_gap_mul_explorationPulls_add_suffix_gap","declaration:BanditRLProof.ETC.centeredPairwiseRewardDiff","declaration:BanditRLProof.ETC.centeredPairwiseGapThreshold","declaration:BanditRLProof.ETC.selectedSubMean_sum_eq_sumRewards_sub_pullCount_mul","declaration:BanditRLProof.ETC.meanSubSelected_sum_eq_pullCount_mul_sub_sumRewards","declaration:BanditRLProof.ETC.sumRewards_le_imp_centered_pairwise_sum_ge","declaration:BanditRLProof.ETC.empMeanAtExploration_ge_best_event_subset_centered_pairwise_sum_event","declaration:BanditRLProof.ETC.actionWithCommit","declaration:BanditRLProof.ETC.actionWithCommit_eq_exploreArm_of_lt","declaration:BanditRLProof.ETC.actionWithCommit_eq_commitArm_of_ge","declaration:BanditRLProof.ETC.actionWithCommit_eq_bestArm_of_commitArm_eq_bestArm_of_explorationPulls_mul_K_le","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_eq_pullCount_exploreArm_of_le","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_explorationPulls_mul_K_eq","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_explorationPulls_mul_K_pos","declaration:BanditRLProof.ETC.ratCast_pullCount_actionWithCommit_explorationPulls_mul_K_pos","declaration:BanditRLProof.ETC.ratCast_pullCount_actionWithCommit_explorationPulls_mul_K_ne_zero","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_succ_eq_add_if_commitArm_of_ge","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_explorationPulls_mul_K_add_eq","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_explorationPulls_mul_K_add_eq_of_ne","declaration:BanditRLProof.ETC.pullCount_actionWithCommit_explorationPulls_mul_K_add_eq_commitArm","declaration:BanditRLProof.ETC.prob_argmaxCommitOracle_ne_bestArm_le_filtered_sum_centeredDiffSubGaussianTail","declaration:BanditRLProof.ETC.pseudoRegret_actionWithCommit_choice_le_sum_gap_mul_explorationPulls_add_suffix_badGap"]},{"id":"teaching:ucb","title":"4. 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andardHighProbabilityPseudoRegretBound_outerBudget_eq_telescoping","declaration:BanditRLProof.OFUL.telescopingStandardScalarAllRoundGapBound_eq_telescopingHighProbabilityPseudoRegretBound","declaration:BanditRLProof.OFUL.telescopingCanonicalExplicitHighProbabilityPseudoRegretAllHorizonViolationSet","declaration:BanditRLProof.OFUL.telescopingCanonicalExplicitHighProbabilityPseudoRegretAllHorizonViolationSet_eq_standard","declaration:BanditRLProof.OFUL.telescopingCanonicalStandardHighProbabilityPseudoRegret_nonneg_and_allHorizon_tail_le_explicitBound_of_linearSubgaussianEnvironment_of_featureBound_le_regularization","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeRadius","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeOptimisticScore","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeOptimisticScore_eq","declaration:BanditRLProof.OFUL.measurable_finiteHistoryScheduledScalarRidgeOptimisticScore","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeOptimisticAction","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeOptimisticAction_eq","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeOptimisticAction_score_max","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeOptimisticAlgorithm","declaration:BanditRLProof.OFUL.finiteHistoryTelescopingScalarRidgeOptimisticAlgorithm","declaration:BanditRLProof.OFUL.finiteHistoryTelescopingScalarRidgeOptimisticAlgorithm_initialAction","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidge_historyStepKernel_map_snd","declaration:BanditRLProof.OFUL.finiteHistoryScheduledScalarRidgeSelectedFeature","declaration:BanditRLProof.OFUL.canonicalHistoryTrajectory_observedFeature_succ_ae_eq_scheduledSelectedFeature","declaration:BanditRLProof.OFUL.measurable_finiteHistoryScheduledScalarRidgeOptimisticAction","declaration:BanditRLProof.OFUL.scheduledCanonicalHistoryTrajectoryPredictableFeature","declaration:BanditRLProof.OFUL.scheduledCanonicalHistoryTrajectoryPredictableFeature_stronglyMeasurable","declaration:BanditRLProof.OFUL.scheduledCanonicalHistoryTrajectory_action_zero_ae_eq_initialArm","declaration:BanditRLProof.OFUL.canonicalHistoryTrajectoryFeature_ae_eq_scheduledPredictableFeature","declaration:BanditRLProof.OFUL.canonicalHistoryTrajectoryFeature_ae_eq_scheduledPredictableFeature_all","declaration:BanditRLProof.OFUL.scheduledCanonicalHistoryTrajectoryPredictableResidual","declaration:BanditRLProof.OFUL.scheduledCanonicalHistoryTrajectoryPredictableResidual_stronglyAdapted","declaration:BanditRLProof.OFUL.scheduledPredictableFeature_projection_le_finiteActionProjectionBound","declaration:BanditRLProof.OFUL.CanonicalScheduledPredictableScalarRidgeResidualLaw","declaration:BanditRLProof.OFUL.canonicalScheduledPredictableScalarRidgeResidualLaw_of_linearSubgaussianEnvironment","declaration:BanditRLProof.OFUL.mem_allTimeTelescopingScalarRidgeConfidenceFailureSet_iff_of_feature_eq","declaration:BanditRLProof.OFUL.measure_telescopingCanonicalHistoryTrajectory_allTimeConfidenceFailureSet_le","declaration:BanditRLProof.OFUL.telescopingCanonicalHistoryTrajectory_action_succ_ae_eq_scheduledAction","declaration:BanditRLProof.OFUL.telescopingCanonicalHistoryTrajectory_action_succ_gap_le_of_not_mem_confidenceFailure","declaration:BanditRLProof.OFUL.telescopingCanonicalHistoryTrajectoryAllTimeSuccGapViolationSet","declaration:BanditRLProof.OFUL.telescopingCanonicalHistoryTrajectoryAllTimeSuccGapViolationSet_subset_confidenceFailure_ae","declaration:BanditRLProof.OFUL.measure_telescopingCanonicalHistoryTrajectoryAllTimeSuccGapViolationSet_le","declaration:BanditRLProof.OFUL.measure_telescopingCanonicalHistoryTrajectory_allTimeConfidenceFailureSet_le_of_linearSubgaussianEnvironment","declaration:BanditRLProof.OFUL.measure_telescopingCanonicalHistoryTrajectoryAllTimeSuccGapViolationSet_le_of_linearSubgaussianEnvironment","declaration:BanditRLProof.OFUL.blockStartForcedIndexSet","declaration:BanditRLProof.OFUL.card_blockStartForcedIndexSet_le_div_add_one","declaration:BanditRLProof.OFUL.blockStartForcedActionSuccessorPseudoRegret_le_card_mul","declaration:BanditRLProof.OFUL.blockStartForcedActionSuccessorPseudoRegret_le_div_add_one_mul","declaration:BanditRLProof.OFUL.blockStartForcedActionSuccessorPseudoRegret_le_div_add_one_mul_two_mul_parameterFeatureBound","declaration:BanditRLProof.OFUL.blockStartForcedCanonicalStandardHighProbabilityPseudoRegretScalarAllHorizonViolationSet","declaration:BanditRLProof.OFUL.blockStartForcedCanonicalStandardHighProbabilityPseudoRegretScalarAllHorizonViolationSet_subset","declaration:BanditRLProof.OFUL.blockStartForcedCanonicalStandardHighProbabilityPseudoRegret_nonneg_and_scalarAllHorizon_tail_le_explicitBound_of_linearSubgaussianEnvironment_of_featureBound_le_regularization","declaration:BanditRLProof.OFUL.deterministicHistoryCanonicalPredictableFeature","declaration:BanditRLProof.OFUL.deterministicHistoryCanonicalPredictableFeature_stronglyMeas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These mapped results do not complete the topic. Separately sourced Genalti2024 fixed-horizon supremum obstruction and actual-law adapter are additional reviewed results.","source":{"title":"Bandits with Heavy Tail","authors":"Bubeck, Cesa-Bianchi and Lugosi","year":2013,"url":"https://www.microsoft.com/en-us/research/wp-content/uploads/2017/01/BCL13.pdf","sha256":"df94efa3708dab85063d6c7a04e0812f264c1c6f13a284017eb2efeed5077ef3"},"qualifications":["The actual source-radius4 policy has an explicitly corrected regret guarantee; the printed coefficient is refuted, not accepted.","Clipping transfer is an estimator adaptation on one action trace, not a corruption-robust regret theorem.","The finite counterexample uses a fixed permissible deterministic tie rule; no universal randomized-tie certificate is claimed.","Recent-source audit and all-topic ICLR evidence remain open; the reviewed mappings do not establish topic completion.","Reading membership is not a proof dependency; the current page banner separately reports whether Lean was rerun for this build.","The primary source block above covers the BCL results. 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Bubeck, R. Munos, G. Stoltz and C. Szepesvari","year":2011,"url":"https://jmlr.org/papers/volume12/bubeck11a/bubeck11a.pdf","sha256":"dcbbc42ae1ddc5a21153594a542d593d85b4f18913d9776a16221e2e31be68ef"},"source_commit":"18d30b25e878894ffecf883d3c02d7ebaaa4af03","scope":"Reviewed full HOO chain and reward-family prototype snapshot; production port preserves proof bodies. For each real exponent above the actual dimension, the fixed causal algorithm and one infinite reward law give a positive gamma independent of horizon for all N>=1. Current build validation is reported separately.","qualifications":["General asymmetric dissimilarity, contained-ball packing and no attained optimum/compactness premise.","Root initialization, chronological indexing, log(max(N,2)) and extended logzero repairs explicit; representatives and left ties are source-permitted fixed choices.","Arbitrary arm-indexed probability reward family needs no global-law measurability; countable representative-node kernel preserves actual trajectory.","Cantor model has infinite arms and non-Dirac distinct-mean rewards, proved dimension<=2; d=3 rate is conservative, not exact dimension or sharpness.","Reading references are not dependency edges. Recent comparison acceptance and all-topic ICLR evidence remain open; topic incomplete."],"review_receipts":[{"path":"runs/extended-topics-20260919/hoo-production-review.json","sha256":"016ddb95c9a85b21c4fb46749f98a54005edafb649ac808be645008102e4d414"}],"hash_normalization_records":[{"path":"runs/extended-topics-20260921/topic-receipt-hash-normalization.json","sha256":"e43bfd99b7ce3f878498839a403decf0d493acb1e16dfa6bf0d1832da6fbba4f"}],"declarations":[{"name":"BanditRLProof.HOO.RegularCovering","statement_sha256":"e7de29dc5d7ed84878755d1dc69c1858ca1ca2522c866342b173044677fd6427","role":"model","source_locator":"A1/A2, Algorithm1, Definitions4-5 and actual countable reward construction","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.action","statement_sha256":"dc571d3c496958feadb2d250727b06b261339b50749767c56edfebefa44250d6","role":"algorithm","source_locator":"A1/A2, Algorithm1, Definitions4-5 and actual countable reward construction","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.trajectory","statement_sha256":"8768f673a5432a7377e18e91ce9166a373f3aa66f1759fc2e3088f7f0e753826","role":"algorithm","source_locator":"A1/A2, Algorithm1, Definitions4-5 and actual countable reward construction","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.RegularCovering.nearOptimalNodes_power_bound","statement_sha256":"f59b2af8c9c4d50c1ac840c7ab960aa9eb08ff04ce9cee84d9ce8a1af3ee362e","role":"producer","source_locator":"A1/A2, Algorithm1, Definitions4-5 and actual countable reward construction","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.RegularCovering.expected_pseudoRegret_rate_family","statement_sha256":"1f1cc9fb4840a7ac60ac417c0be0df1c6aa718d616d73536a029db014c54f747","role":"endpoint","source_locator":"Theorem6 repaired rate / actual expected reward identity","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.RegularCovering.expected_actual_eq_pseudoRegret_family","statement_sha256":"8cde329136a000983c73dd9187af16ca739150c93332223b7b1a9f129089bd9e","role":"endpoint","source_locator":"Theorem6 repaired rate / actual expected reward identity","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.RegularCovering.expected_actualRegret_rate_family","statement_sha256":"66b47c005d41151d4da043f542ea6f99550afd242ad61576fe381ec473f071c1","role":"endpoint","source_locator":"Theorem6 repaired rate / actual expected reward identity","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.Covering.familyNodeLaw","statement_sha256":"1e416b55b17c9b0b006a06155534406a43144ff590d9530c6ee3f22091861baf","role":"algorithm","source_locator":"A1/A2, Algorithm1, Definitions4-5 and actual countable reward construction","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"},{"name":"BanditRLProof.HOO.CantorModel.expected_actual_rate","statement_sha256":"dad79179bf439c8202b6618c1c2147bee8bca87574193b322102d1972e59865f","role":"endpoint","source_locator":"Theorem6 repaired rate / actual expected reward identity","review_receipt":"runs/extended-topics-20260919/hoo-production-review.json"}],"mathematical_reader":["A1 supplies a measurable binary covering, geometric diameter bounds and pairwise disjoint contained balls; A2 is weak Lipschitz smoothness for a general possibly asymmetric dissimilarity. Rewards are stationary probability laws M_x on [0,1] with mean f(x). The best value is sup f; attainment and compactness are not assumed.","For each real d strictly above the actual (4 nu1/nu2)-near-optimality dimension, there exists gamma>0, independent of N, such that expected actual and pseudo-regret are equal and at most gamma N^((d+1)/(d+2)) log(max(N,2))^(1/(d+2)) for every N>=1. This is an expected-regret statement, not a pathwise bound.","The proof first derives adaptive concentration from the true reward trajectory, then bounds poor-region visits. A1/A2 and the limsup dimension yield a horizon-independent bound on near-optimal node counts. The actual three-way tree partition bounds shallow, near-optimal and poor-branch regret. A geometric sum and an integer depth choice give the displayed rate. Bounded integrability and the actual conditional reward law identify realized and pseudo-regret expectations.","For arbitrary arm-indexed laws define Q(v)=M_(a_v), where a_v is the fixed representative of binary-word node v. The node space is countable, so Q is a measurable Markov kernel without global measurability of x -> M_x. Internally giving X the full sigma algebra lets the existing theorem apply, while preserving all regions, geometry, representatives, dimension and actions. Reducing definitions returns precisely the trajectory driven by Q. For an existing global kernel, the node-law equality is proved by reflexivity.","The family adapter reuses the existing pseudo-regret theorem, actual/pseudo expectation identity and actual-regret theorem. Its public-root canary uses the infinite binary-arm model with non-Dirac rewards and distinct means; dimension<=2 is proved, and d=3 yields the conservative N^(4/5) log(max(N,2))^(1/5) rate. Exact statements and proofs are available through the canonical declaration links below."]}},{"id":"combinatorial","title":"Combinatorial bandits","kind":"setting","url":"banditrlwiki/topics/combinatorial/index.html","case_refs":[],"node_ids":["declaration:BanditRLProof.CUCB.FeedbackModel","declaration:BanditRLProof.CUCB.FeedbackModel.charged_observation_tail","declaration:BanditRLProof.CUCB.FeedbackModel.nice_event_probability","declaration:BanditRLProof.CUCB.FiniteExample.deterministic_regret","declaration:BanditRLProof.CUCB.FiniteExample.no_bad_regret","declaration:BanditRLProof.CUCB.FiniteExample.noisy_each_arm","declaration:BanditRLProof.CUCB.FiniteExample.probabilistic_regret","declaration:BanditRLProof.CUCB.FiniteExample.randomizedSource","declaration:BanditRLProof.CUCB.FiniteExample.randomized_probabilistic_regret","declaration:BanditRLProof.CUCB.FiniteExample.refined_regret","declaration:BanditRLProof.CUCB.FiniteExample.regret_one_positive","declaration:BanditRLProof.CUCB.FiniteExample.source","declaration:BanditRLProof.CUCB.SourceModel","declaration:BanditRLProof.CUCB.SourceModel.approximationRegret_eq_gap_sum","declaration:BanditRLProof.CUCB.SourceModel.approximationRegret_le_underSampled_add_source_tail","declaration:BanditRLProof.CUCB.SourceModel.chargeData_sufficient","declaration:BanditRLProof.CUCB.SourceModel.polynomial_threshold_integral_probabilistic","declaration:BanditRLProof.CUCB.SourceModel.theorem_one_refined_regret","declaration:BanditRLProof.CUCB.SourceModel.theorem_two_deterministic","declaration:BanditRLProof.CUCB.SourceModel.theorem_two_probabilistic","declaration:BanditRLProof.CUCB.SourceModel.underChargeCount_power_sum_le","declaration:BanditRLProof.CUCB.cucbTrajectory","declaration:BanditRLProof.Thompson.uniformActionMeasure"],"status":"mapped-reviewed-partial","formalization":{"schema_version":1,"title":"CUCB: reviewed full source-qualified regret chain","semantic_status":"accepted-with-explicit-delta","source":{"title":"Combinatorial Multi-Armed Bandit and Its Extension to Probabilistically Triggered Arms","authors":"Wei Chen, Yajun Wang, Yang Yuan and Qinshi Wang","year":2016,"url":"https://jmlr.org/papers/volume17/14-298/14-298.pdf","sha256":"6a29fc188cd1490c53864eb4f839e572551c171388fc27ba256a7e61424e856c"},"source_commit":"604e4b7011c51d9a65726051bb531ccf3b105df4","scope":"Algorithm 1 and both complete performance theorems under CUCB-CONTRACT, with actual primitive feedback, nonlinear scores, fresh oracle and concrete noisy consumers. Independent blind/source/repair acceptance is bound separately from current build and publication validation.","qualifications":["Observed marginal compatibility is required: arbitrary outcome-dependent censoring is not covered. Fresh measurable oracle and inverse range on needed positive gaps are explicit.","Distinct selected subsets, nonempty possible-trigger sets and removal of never-triggerable arms; parameterized duplicate-subset application extensions are outside this contract.","Normalized analysis-only charging repairs mixed p=1/p<1 thresholds; conditional mixing, weighted counting and zero/small-horizon boundaries are explicit. The learner is unchanged.","The signed benchmark is H alpha beta OPT minus expected actual reward, not ordinary OPT regret. Both Theorem 2 trigger branches retain exact constants and residuals.","Finite concavity is separately proved for actual counts, not an actual dependency of the current common-cutoff endpoint proof.","The primary noisy input-dependent oracle model has R(1)=1/4; the supplementary uniform randomized oracle ignores input and is not a learning-improvement example.","Reading references are not proof edges. Recent-source complete disposition, ICLR evidence and whole-topic acceptance remain incomplete."],"declarations":[{"name":"BanditRLProof.CUCB.FeedbackModel","role":"model","source_locator":"Section 2: primitive triggered feedback; observed-marginal compatibility made explicit","statement_sha256":"bf000aa46bdabd57ab4b095b77482499c239df24b8c4cdf84c8a996fdf040f8c","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel","role":"model","source_locator":"Section 2: monotone smooth reward and approximation oracle; inverse-range clarification","statement_sha256":"eb44a6c03195d204fa477f45d7f79b8eeaff6e271b80336c6da89173cee70c49","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.cucbTrajectory","role":"algorithm","source_locator":"Algorithm 1: one causal CUCB trajectory, zero initial counts and initial indices one","statement_sha256":"5b0b02ebb00fa59bbfdf552706034708b40d13a548d63cf073f9dfeebfba87fb","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.chargeData_sufficient","role":"producer","source_locator":"Analysis-counter repair: normalized charge choice handles mixed trigger probabilities","statement_sha256":"8066b653e4dce2b0c23f2d40934dc9d50ff290527664b7eafab78a05a72d93a1","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FeedbackModel.nice_event_probability","role":"producer","source_locator":"Proof of confidence control from actual primitive laws","statement_sha256":"3ce5c15748c5666730221c2cd561ef2712385ba3318bda29a76d16be72ba0990","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FeedbackModel.charged_observation_tail","role":"producer","source_locator":"Proof of charged-trigger concentration under adaptive action selection","statement_sha256":"f69390ac17fdf2d6dfe54cb1e28e1104e598c1e2ebe5cc56ccd3c7050de1bba1","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.approximationRegret_eq_gap_sum","role":"producer","source_locator":"Original signed approximation regret and actual-reward identity","statement_sha256":"85c317ad23446b4bc234c84aab6e6ed2a7f66227028f7f6b13a44669aec49c70","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.approximationRegret_le_underSampled_add_source_tail","role":"producer","source_locator":"Oracle failure credit cancellation and source tail constants","statement_sha256":"6fb1511608b48a9d29399666d26830d104e8b318e2d769493f0fc881ed309f77","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.theorem_one_refined_regret","role":"endpoint","source_locator":"Theorem 1: refined gap-dependent integral bound, with disclosed analysis repair","statement_sha256":"2d10a204bb9ce6f5e6e33b05befee8408900068ecf23502011ab7d6d57df1818","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.polynomial_threshold_integral_probabilistic","role":"producer","source_locator":"Theorem 2: mixed-trigger polynomial envelope integrated on actual gap interval","statement_sha256":"358b339edc16f8dd1c63d86056844e57b44546a7ad72952f1bb67b3f715d2bfc","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.underChargeCount_power_sum_le","role":"producer","source_locator":"Finite concavity obligation on actual counts; separately proved, not used by cutoff endpoint route","statement_sha256":"b65e024c7c5937944b2f8ce954b5518974d622e10d890469c838b293e71ab44c","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.theorem_two_deterministic","role":"endpoint","source_locator":"Theorem 2: p*=1 branch, H>=1 including H=1","statement_sha256":"89389da4f54d28fe1ae2300743282f66260391891a63ac14ccc1cca760594637","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.SourceModel.theorem_two_probabilistic","role":"endpoint","source_locator":"Theorem 2: p*<1 branch, H>=1 including H=1","statement_sha256":"8eddde83425293c8eaf63c1fef280790fb572647164f392b4bcc138a4e94ed80","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.source","role":"canary","source_locator":"Constructed 64-atom nonlinear noisy model; not a source-paper example","statement_sha256":"f6d5ef25437eb87dd4ac61afc65c5f37911e235575fe62358308350db9667678","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.noisy_each_arm","role":"canary","source_locator":"Each actual arm law has probability of one strictly between zero and one","statement_sha256":"6a577c1125aa66d9610f88f9168f98fd7ee5ba54789038254ebbb38ae041dd91","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.regret_one_positive","role":"canary","source_locator":"Actual learning-trajectory R(1)=1/4","statement_sha256":"a6f1831d635afdaee39a5abf08dd38df74a84edb7a9ea1d65a723d4ee96ba642","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.refined_regret","role":"canary","source_locator":"Concrete full Theorem 1 instance","statement_sha256":"3632f7dfd4828ddb8081dc9b1fa00a92e475a70f69e5a842b2cdfc4ded87f63b","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.probabilistic_regret","role":"canary","source_locator":"Concrete random-trigger Theorem 2 instance","statement_sha256":"966d02d3a3e151b14835aa8fe3b1d0ccbc3673917fb0799cbc9bebcb96bcc988","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.randomizedSource","role":"canary","source_locator":"Supplementary uniform action oracle with beta=1/2; ignores input","statement_sha256":"53246c4c8675d311a799e1cc28986d8bbe7013e3c1ae2a1d067d6d05fd9b5375","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.randomized_probabilistic_regret","role":"canary","source_locator":"Actual signed-regret theorem for supplementary randomized oracle","statement_sha256":"75918a8f106179e46e395ae8bc4e197a74971c5b77b5445182bc89c2ae238dfa","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.deterministic_regret","role":"canary","source_locator":"Noisy full-observation recovery of Theorem 2","statement_sha256":"9f7df4c377cbf39340675c05a4c54e592d259053c7b9df1ab762b8321bcf9c94","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.CUCB.FiniteExample.no_bad_regret","role":"canary","source_locator":"Alpha=1/3 no-bad-action boundary, all horizons","statement_sha256":"0f1dff63726b36ab60e52f09f2102003c45faf6b859b922359555d021feb57c9","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"},{"name":"BanditRLProof.Thompson.uniformActionMeasure","role":"reuse","source_locator":"Existing shared finite uniform measure used by the concrete sample law and randomized oracle; not an efficiency experiment","statement_sha256":"677d33f78c8167529cb23a8ce7e9a281806b66a3cc72fd11d78049364bc793d1","review_receipt":"runs/extended-topics-20260919/cucb-mapping-review-input.json"}],"topic_complete":false,"review_receipts":[{"path":"runs/extended-topics-20260919/cucb-mapping-review-input.json","sha256":"954c0cb17dda23aa1cf2ba7e06ce6960c4d0763b77a38fb6d7103e343ff59bd0"}],"hash_normalization_records":[{"path":"runs/extended-topics-20260921/topic-receipt-hash-normalization.json","sha256":"e43bfd99b7ce3f878498839a403decf0d493acb1e16dfa6bf0d1832da6fbba4f"}],"mathematical_reader":["Finite distinct feasible subsets act on m bounded base arms. The environment produces a mask, outcomes and a nonnegative integrable total reward. For every action a and arm i, law(observe i and value_i in B)=p_i^a D_i(B). This supports adaptive concentration while permitting dependence among arms. Minimum trigger probabilities are derived from actual laws.","At prefix length n, CUCB uses index one for zero observations, otherwise min(empirical mean + sqrt(3 log(n+1)/(2 count)),1). A fixed measurable approximation-oracle kernel draws the action before fresh environment feedback. One infinite causal trajectory serves every horizon; analysis counters and unknown true gaps do not enter the learner.","Let d(a)=alpha OPT-r(a), Delta=max_a max(0,d(a)), and R(H)=H alpha beta OPT-E sum reward. The actual reward identity gives R(H)=E sum d(A_t)-H alpha(1-beta)OPT. This signed credit cancels the oracle-failure charge after taking expectations.","Write ell_H(d,p)=log(H)c(d,p), with u=f^{-1}(d), c(d,1)=6/u^2 and c(d,p)=max(12/(u^2 p),24/p) for p<1. The repaired analysis charges argmin N_i/c(d,p_i). If that arm exceeds its threshold, all possible arms do. Charge is known after the action but before feedback; integrating the actual action mixture produces the trigger tail, with no extra number-of-actions factor.","Theorem 1 gives R(H)<=sum_i [d_i,min ell_H(d_i,min,p_i)+integral from d_i,min to d_i,max of ell_H(x,p_i) dx]+[1+(2+1{p*<1})pi^2/6]m Delta for H>=1. Empty bad-action families contribute zero. Distinct old integer counters and finite layer-cake integration retain the zero-counter residual. Confidence and trigger failures are derived and summed, not assumed.","For f(u)=gamma u^omega, gamma>0 and 0<omega<=1, Theorem 2 gives 2gamma/(2-omega)(6m log H)^(omega/2)H^(1-omega/2)+(1+pi^2/3)m Delta when p*=1. For p*<1 replace 6 by 12/p*, pi^2/3 by pi^2/2, and add sum_i(24 log H/p_i)Delta. A common cutoff and power-tail integration give both rates; H=1 is handled separately.","The concrete 64-atom model has three noisy arms and two distinct two-arm actions, product rewards, actual minimum trigger probabilities (1/2,1,1/2), and positive first-round regret 1/4. Other actual consumers cover a randomized beta=1/2 oracle, full-observation deterministic triggering and a no-bad-action boundary. Canonical links below expose exact statements and proofs."]}},{"id":"causal","title":"Causal bandits","kind":"setting","url":"banditrlwiki/topics/causal/index.html","case_refs":[],"node_ids":["declaration:BanditRLProof.Causal.GraphModel.expected_simpleRegret_bounds","declaration:BanditRLProof.Causal.GraphModel.expected_simpleRegret_explicit_rate","declaration:BanditRLProof.Causal.GraphModel.expected_simpleRegret_optimal","declaration:BanditRLProof.Causal.GraphModel.expected_simpleRegret_source_bound","declaration:BanditRLProof.Causal.GraphModel.expected_simpleRegret_uniform","declaration:BanditRLProof.Causal.GraphModel.rewardMean_eq_integral","declaration:BanditRLProof.Causal.GraphModel.roundLaw","declaration:BanditRLProof.Causal.GraphModel.sampleEstimate","declaration:BanditRLProof.Causal.GraphModel.sampleEstimate_source_confidence","declaration:BanditRLProof.Causal.GraphModel.sampleLaw","declaration:BanditRLProof.Causal.GraphModel.sampleRecommendation","declaration:BanditRLProof.Causal.GraphModel.sampleWeightedBit","declaration:BanditRLProof.Causal.NodeCodec.joint_encodeTables","declaration:BanditRLProof.Causal.NodeGraphModel.designCost_encoded","declaration:BanditRLProof.Causal.NodeGraphModel.expected_simpleRegret_encoded","declaration:BanditRLProof.Causal.NodeGraphModel.expected_simpleRegret_explicit_rate","declaration:BanditRLProof.Causal.NodeGraphModel.expected_simpleRegret_optimal","declaration:BanditRLProof.Causal.NodeGraphModel.expected_simpleRegret_source_bound","declaration:BanditRLProof.Causal.NodeGraphModel.expected_simpleRegret_uniform","declaration:BanditRLProof.Causal.NodeGraphModel.sampleEstimate","declaration:BanditRLProof.Causal.NodeGraphModel.sampleLaw","declaration:BanditRLProof.Causal.NodeGraphModel.sampleLaw_encoded","declaration:BanditRLProof.Causal.NodeGraphModel.sampleRecommendation","declaration:BanditRLProof.Causal.NodeGraphModel.sampleWeightedBit","declaration:BanditRLProof.Causal.ParallelParameters.allocated_mass_domination","declaration:BanditRLProof.Causal.ParallelParameters.allocation","declaration:BanditRLProof.Causal.ParallelParameters.allocationWeight_sum","declaration:BanditRLProof.Causal.ParallelParameters.allocation_cost_le","declaration:BanditRLProof.Causal.ParallelParameters.allocation_covers","declaration:BanditRLProof.Causal.ParallelParameters.expected_simpleRegret_parallel","declaration:BanditRLProof.Causal.ParallelParameters.expected_simpleRegret_parallel_optimal","declaration:BanditRLProof.Causal.ParallelParameters.graph","declaration:BanditRLProof.Causal.ParallelParameters.graph_designCost","declaration:BanditRLProof.Causal.ParallelParameters.graph_parentLaw","declaration:BanditRLProof.Causal.ParallelParameters.optimal_cost_le","declaration:BanditRLProof.Causal.ParallelParameters.rareActions_card_le","declaration:BanditRLProof.Causal.ParallelParameters.rarity","declaration:BanditRLProof.Causal.ParallelParameters.rarity_minimal","declaration:BanditRLProof.Causal.ParallelParameters.rarity_spec","declaration:BanditRLProof.Causal.ParallelParameters.rootLaw","declaration:BanditRLProof.Causal.ParallelParameters.rootLaw_mass","declaration:BanditRLProof.Causal.nodeIntervention_factorization","declaration:BanditRLProof.Causal.nodeJoint","declaration:BanditRLProof.Causal.productNodeCodec"],"status":"mapped-reviewed-partial","formalization":{"schema_version":1,"title":"Reviewed native heterogeneous causal sampling and actual expected regret","semantic_status":"accepted-with-explicit-delta","topic_complete":false,"source":{"title":"Causal Bandits: Learning Good Interventions via Causal Inference","authors":"F. Lattimore, T. Lattimore and M. D. Reid","year":2016,"url":"https://proceedings.neurips.cc/paper_files/paper/2016/file/b4288d9c0ec0a1841b3b3728321e7088-Paper.pdf","sha256":"99aa9427e02e31883510dfcd0b11ba4f45097dc30dd98a094cbf13068f93b747"},"source_commit":"6ac086dcd71dd6829c9859e13aefef4a9f04c282","scope":"Native heterogeneous finite-node actual-law expected regret; complete frozen noisy diagnostics; repaired parallel allocation, actual graph-law/cost bridge, true-cost regret consumers and both boundary witnesses. Remaining source screening and ICLR evidence are open.","qualifications":["Each node has its own nonempty finite type; final performance theorem constructs the finite product codec internally. Binary readout extends the literal binary source reward node.","Coverage is explicit; zero allocation weights are allowed only when Q covers every parent support.","Unchanged source truncation tuning and fixed-order ties; coefficient and failure direction repairs disclosed.","Uniform actual threshold uses m, with K substituted only in RHS. Optimal allocation minimizes cost, not a proved ordering of actual regrets.","Frozen noisy test now proves concentrated coverage, cost 8/3, B=2 biases, conditional versus interventional means, an uncovered design, actual tuned T=1 recommendation and exact expected regret 1/5, plus concentrated all-horizon bound. The B=2 bias test is distinct from the actual one-round tuning.","Parallel bridge and witnesses are reviewed with strict rarity, normalized observation mass 1-D, N>=2 and known q. Actual thresholds retain designCost. Remaining source audit and ICLR evidence remain required; no topic completion.","Heterogeneous Fin3/Fin2 canary proves native factorization, means 5/12, 1/4, 3/4 and uniform all-horizon rate; no exact diagnostic cost or random recommendation distribution is claimed.","Evidence erratum: three historical sampling file hashes are invalid. Fresh independent blind/source review binds five foundations and seven sampling modules plus canary to132a834; old records retained. Identity checks use explicit LF normalization and run in the harness. No Lean proof change or topic completion."],"review_receipts":[{"path":"runs/extended-topics-20260919/causal-sampling-review.json","sha256":"03caccaa6a4ceb29e726a9283580530b609485092c3fa24f163dc996d2b41812"},{"path":"runs/extended-topics-20260919/causal-heterogeneous-review.json","sha256":"768c9d7358ad92d878d70e6840c8187bd19705670e7634280bb16c0f624cf5ef"},{"path":"runs/extended-topics-20260919/causal-noisy-diagnostics-review.json","sha256":"1c82ee814f73077e600f6521276372aaa1e952e542b17087060e5cf1f4578888"},{"path":"runs/extended-topics-20260919/causal-parallel-review.json","sha256":"c624c990f7f00b8235c00f8d391b6ca59a1236af893fc32eec84ccfbd88d88f5"},{"path":"runs/extended-topics-20260920/causal-review-rebinding.json","sha256":"c2ef4c2e2fd78fba0a330ddcfc71088c7306afdedd64c4cde116468629f87e1e"}],"hash_normalization_records":[{"path":"runs/extended-topics-20260921/topic-receipt-hash-normalization.json","sha256":"e43bfd99b7ce3f878498839a403decf0d493acb1e16dfa6bf0d1832da6fbba4f"}],"declarations":[{"name":"BanditRLProof.Causal.GraphModel.roundLaw","statement_sha256":"8ecf39ef4e790aaa4d924af1c78a33471d050744afcfa6933e4990efcbafd3d7","role":"algorithm","source_locator":"Algorithm2/Theorem3 repaired common-alphabet chain; 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The finite product law and reward projection derive independent paired observations; no supplied tail or confidence premise.","The observable estimator averages Y(P_a/Q)1{P_a/Q<=B}; it uses known parent laws and observed bits. Its mean differs from true intervention reward by nonnegative truncation bias at most m/B. The true reward mean is proved equal to the intervention integral.","Bounded centered MGF at tilt 1/(2B), with B=sqrt(mT/L), L=log(2TK), gives simultaneous confidence radius sqrt(2mL/T)+3BL/T with failure at most 1/T.","The least empirical maximizer satisfies regret <=2epsilon+m/B on the good event and <=1 everywhere. Integration gives (2sqrt(2)+7)sqrt(mL/T)+1/T and the cap 1. Since L>=1/2, explicit constant 3sqrt(2)+7 suffices.","Existing attained optimal-design and uniform-coverage theorems instantiate the same actual learner. The original noisy graph has true means 1/2, 3/10, 7/10; the exact noisy diagnostics are now reviewed, and the parallel-design bridge and witnesses are accepted separately with explicit source corrections.","Native dependent node tables define the joint before encoding. Induction proves its pushforward under a constructed finite product codec; intervention and parent laws, coverage, exact design cost and full product sample law are preserved.","The estimator, fixed-order recommendation and actual regret agree pathwise; integral transport yields the native expected-regret endpoint without an external codec premise. The additional Fin3/Fin2 canary has exact means 5/12, 1/4 and 3/4.","The local noisy canary derives all parent masses from the actual DAG. Sampling only observation still covers every intervention and has exact cost 8/3. At T=1 the source threshold lies strictly between 1 and 4/3, estimates are (Y,0,0), fixed-order ties recommend observation, and the actual expected regret is exactly 1/5. Conditional reward given W=1 is 3/4, distinct from intervention reward 7/10. These are local model diagnostics, not numerical claims attributed to the paper.","Construct the integer rarity r, prove at most r strict rare interventions, allocate 1/(2r) to each and 1-D to observation. Actual intervention product laws give P_a<=2r Q, including deterministic roots; derive coverage and design cost<=2r. Injective actual parent-law transport connects both repaired and attained optimal designs to the real learner. Actual threshold uses its true cost; RHS relaxes to 2r. The N=2 fair and deterministic witnesses have exact costs 2 and 4."]}},{"id":"multi-agent","title":"Multi-agent 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unknown-N learner: actual local feedback and expected regret","semantic_status":"accepted-with-explicit-delta","topic_complete":false,"source":{"title":"Multi-Player Bandits -- a Musical Chairs Approach","authors":"Jonathan Rosenski, Ohad Shamir and Liran Szlak","year":2016,"url":"https://proceedings.mlr.press/v48/rosenski16.pdf","sha256":"c56474f0c3e689c1819b11e2bca4d3e5dcaef6eddfd0173bb55ff6a84f11d99d"},"source_commit":"3d0265b41717401463ca35b832a87906c17b80e2","scope":"Actual independent exploration/reward law, local inverse-collision population estimate and empirical ranking, learned continuation and full chronological visible feedback yield all-horizon expected pseudo and unconditional realized regret <=min(nH,nS+8n^2+delta*nH).","qualifications":["Primitive model: synchronized0<n<k, independent stationary arm laws supported on[0,1], separately sensed collision bit, known k/epsilon/delta and positive nth-next gap. True n/means are not learner inputs.","Exploration uses exactly S=ceil max(16k/epsilon^2 log(4k^2/delta),50k^2 log(4k/delta)); second logarithm is an explicit all-player budget repair. Inclusive source-loop indexing is reconciled to S observations.","All-collision fallback, zero-count mean, rounding/cap and fixed arm tie order make local candidates total/nonempty on bad histories. Correct candidates are derived on an actual probability>=1-delta event.","Actual learned kernel and feedback prefixes establish the selected static finite learner. No general filtration/condDistrib or cross-horizon projective-consistency theorem is claimed.","Pseudo-regret preserves n*min(H,S),8n^2,delta*nH and the cap nH; conditional normalized pseudo-regret and source residual/inverse-horizon corollaries retain their hypotheses. Realized/visible expectation equality is unconditional; reward-dependent-event conditioning is excluded.","Public-root noisy canary passed shared-root integration: actual Bernoulli means(3/4,1/2,1/4), exact S, allH visible bound, full reward-law transient event mass>=3/64 and pseudo charge5/4. Zero-bit/ranking fixtures are separate diagnostics, not measured full-trace events.","Combined root/Tests passed9151 jobs;410-name public-environment audit uses only standard axioms. Full harness passed437 tests(7 skipped); site/books and371 canonical links/anchors passed. Bounded recent-source selection and descriptive case are independently reviewed. Direct outside-case ABRL proof/value references:0;902 cross-module pairs are within this new case. No efficiency effect estimated. All-topic evaluation and manuscript integration remain open; topic incomplete; no merge/deployment."],"review_receipts":[{"path":"runs/extended-topics-20260920/multi-agent-occupation-review.json","sha256":"bf0ce4bc440a3efce02d8dc013f8d02081f8603de48ec28f4dc91c3e627920a0"},{"path":"runs/extended-topics-20260920/multi-agent-regret-review.json","sha256":"618ab5fa065cc5696f8eb2d1c7e9a8a7a9027faf4f80ca3f983c3c26f6120eb3"},{"path":"runs/extended-topics-20260920/multi-agent-learner-promotion-review.json","sha256":"c528ee256556cdbe9fbd3a8ad4d93337893bacb8c256dd871e45a2ca7b7db224"}],"hash_normalization_records":[{"path":"runs/extended-topics-20260921/topic-receipt-hash-normalization.json","sha256":"e43bfd99b7ce3f878498839a403decf0d493acb1e16dfa6bf0d1832da6fbba4f"}],"declarations":[{"name":"BanditRLProof.MusicalChairs.State","statement_sha256":"05b7cec51921f41234b762f7422b144d7263d58a74b1fe8ece225a59feefca5d","role":"model","source_locator":"Static Algorithm2 / supplement A.1 Lemma4 coordination component; 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Restriction-event counting gives Pr(all draw_i in B_i)=product_i |C_i intersect B_i|/|C_i|. The transition keeps old fixed arms and fixes a new player exactly on collision-free feedback.","If the common candidate set has n arms and player i is unfixed, at most n-1 fixed labels are occupied. Cardinality produces an unused arm. A specific isolated-draw event yields actual fixation probability at least(1/n)((n-1)/n)^(n-1).","For n>=2, m=n-1 and x=m/(m+1) satisfy m log x>=-1 by1-1/x<=log x. Thus x^m>=exp(-1)>=1/4; n=1 is handled directly. This proves the actual statewise hazard1/(4n).","Fixed players never return to unfixed. The actual PMF bind recursion gives Pr(i unfixed at t)<=(1-1/(4n))^t. Finite geometric summation yields sum_(t<T)Pr(i unfixed)<=4n.","For U(s)=number of unfixed players, finite sum interchange proves sum_s p_t(s)U(s)=sum_i Pr(i unfixed). Summing gives total finite expected occupation<=4n². No independence of players waiting times, expected completion time, or full regret statement is inferred.","The actual real deficit is sum of means over S minus the sum of means of collision-free played actions. A fixed player hit by a collision must be hit by an unfixed player. Choosing a partner gives an injection, because two fixed players cannot share an arm. Thus at most U fixed players are hit, and at most2U reference arms are missing from successful fixed actions. Means in[0,1] give the actual per-round deficit<=2U.","All successful actions are distinct and, on the actual state/draw support, belong to S. Their sum is at most the S comparator, proving nonnegative supported deficit. Consequently the finite ENNReal expectation equals the ordinary real PMF-weighted deficit sum after toReal; no negative supported values are discarded. Summing the2U charge against the same constructed state/draw law and consuming occupation<=4n^2 gives actual finite expected coordination mean deficit<=8n^2. The integrated exploration producer and learned-kernel factorization now supply the actual true top-n interface.","Actual iid uniform exploration gives q=(1/k)(1-1/k)^(n-1)>=1/(4k). Conditioning on the reward-independent action schedule fixes collision-free sample times; bounded independent reward MGFs and the count PGF yield P(|mean-mu|>=epsilon/2)<=2exp(-S epsilon^2/(16k)). The all-player/arm union pays delta/2.","Collision-frequency concentration gives |pHat-p|<=1/(10k). With b=1-1/k and q=b^(n-1)>=1/4, real-exponent bounds put the inverse estimate in[n-2/5,n+2/5], strictly within its rounding cell. The repaired all-player log(4k/delta) pays the remaining delta/2.","Finite sorting with a fixed tie order constructs local candidates. The numeric nth-next boundary gap plus strict epsilon/2 errors preserves selected/outside separation, so all actual local population estimates and top sets are correct on an event G with probability>=1-delta; no correct-set oracle is assumed.","The measurable learned path kernel is total on every exploration history. On G it factors to independent common-top-set draws. Actual path/state marginals transfer the8n^2 coordination bound. The all-history comparator and exact phase split then give E[R_H]<=min(nH,n*min(H,S)+8n^2+delta*nH), including H<S and H=0.","Fresh continuation reward arrays extend the same learned law. Finite schedule first moments, signed integrability and Fubini give unconditional expected realized=pseudo regret. Measurable chronological feedback pulls back the exact observed reward sum; its pushforward law supplies the visible regret endpoint. Conditioning this identity on reward-dependent G is not asserted.","The genuine noisy n2/k3 canary proves actual means/gap, source-good-event probability>=3/4 and all-horizon visible bound. A two-round collide-then-separate path has conditional mass1/16 and joint/full reward-law mass>=3/64, with mean coordination charge5/4. 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endentdriftingmeanrewardlaw/index.html#decl-4d2bb200f38d","status":"compiled","statement_sha256":"c477d59a008ed50c2f2b5eb8d4267c6e8feb5a0fa5534b08932ef689346309d2","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.independentLossStateTimeVaryingMeanGap_finiteArmIndependentRewardVectorLoss_eq_mean_sub","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-2d57941d6bfd","status":"compiled","statement_sha256":"f8e5e75ffe8175602a3b3425d8cc3804f0ebd4e48aaaa459f82cbf964f452f2d","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.abs_independentLossStateTimeVaryingMeanGap_sub_modelGap_le","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-c5f3d4860500","status":"compiled","statement_sha256":"8bbd796d3dadb88706dadea9204a637f0d86a0cec00aca0c723a97957c4dbaf5","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.finiteArmIndependentMeanDeviationBudget","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-817344452f5e","status":"compiled","statement_sha256":"3fc047fffc719562f023836425fedd63fe6bc6c932e050b252ca1d78df6fbc5c","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.finiteArmIndependentMeanDeviationBudget_eq","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-a62ce5f94e76","status":"compiled","statement_sha256":"9b2b3b583d650a6cfa187690d632450e9686b0e403a727176820503d0d8d28fc","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.finiteArmIndependentMeanDeviationBudget_zero","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-31f50339e81a","status":"compiled","statement_sha256":"3246e502c08a0eb71518338b52fabc54bf429bba7ff87f35f72fc7542390ae93","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.integral_sampledScheduledHalfTsallisFiniteArmIndependentDriftingMeanRewardLaw_hasSelfBounding","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-e5e74d4373fc","status":"compiled","statement_sha256":"db929447c793c4ba8a0ae7920b8d44ee175ee9793680e3b7c2f5853a5471096d","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.integral_sampledScheduledHalfTsallisFiniteArmIndependentDriftingMeanRewardLawRegret_le_log","module":"BanditRLProof.TsallisFiniteArmIndependentDriftingMeanRewardLaw","url":"modules/banditrlproof-tsallisfinitearmindependentdriftingmeanrewardlaw/index.html#decl-342348bd8d36","status":"compiled","statement_sha256":"76aea65da86775c6106b48204cad85b8922671489f1d513b36a4ae3c16b06a8e","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.finiteArmIndependentCumulativeGlobalMeanSwitchCount_nonneg","module":"BanditRLProof.TsallisFiniteArmIndependentGlobalMeanSwitchCountCompressedDynamicRegret","url":"modules/banditrlproof-tsallisfinitearmindependentglobalmeanswitchcountcompresseddynamicregret/index.html#decl-6ab97a8ce376","status":"compiled","statement_sha256":"b1aba47df16201515ce09bac51815581be87f34a3eda8f24fd901d439713c208","identity_basis":"source-qualified-name","books":["bandit","online-learning"],"chapters":["teaching:tsallis"],"settings":[]},{"id":"declaration:BanditRLProof.Tsallis.finiteArmIndependentCumulativeGlobalMeanSwitchCount_mono","module":"BanditRLProof.TsallisFiniteArmIndependentGlobalMeanSwitchCountCompressedDynamicRegret","url":"modules/banditrlproof-tsallisfinitearmindependentglobalmeanswitchcountcompresseddynamicregret/index.html#decl-3aca4dba19c2","status":"compiled","statement_sha256":"11a9e1a7c9d6c085f0d7bd9b5b84114a05f8b8a2ec18942040adbeac73dc65d1","identity_basis":"source-qualified-name","books":["bandit","online-learni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