Lean module · UCB
BanditRLProof.Algorithms.UCBBoundedFiniteArmRewardLaw
# Canonical UCB expected regret for bounded finite-arm reward laws This module instantiates the centered-kernel canonical Real theorem with stationary action-indexed reward laws. It supports both one common interval and arm-dependent nondegenerate intervals.
Module map
Imports
BanditRLProof.Algorithms.UCBConditionalRewardLawCenteredKernelReal, BanditRLProof.FiniteArmRewardKernelLaw
Imported by
Declarations
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theorem
BanditRLProof.UCB.integral_real_pseudoRegret_selectedPolicySuccessorGeneratedUCBRegretAction_le_textbookGapSum_boundedFiniteArmLaws
Compiled
Canonical Real expected pseudo-regret bound for finite-arm stationary reward laws supported almost surely on one nondegenerate interval. The initial reward is sampled from the default arm law. Successor rewards use the context-independent action law kernel, while the context is `Unit`.
theorem integral_real_pseudoRegret_selectedPolicySuccessorGeneratedUCBRegretAction_le_textbookGapSum_boundedFiniteArmLaws {K : Nat} (model : FiniteBanditModel K) (armLaw : Fin K -> Measure Rat) (hprob : forall arm, IsProbabilityMeasure (armLaw arm)) (lo hi : Real) (hlohi : lo < hi) (hmeas : forall arm, AEMeasurable (fun reward : Rat => ((reward : Rat) : Real)) (armLaw arm)) (hbound : forall arm, Filter.Eventually (fun reward : Rat => Set.Icc lo hi ((reward : Rat) : Real)) (ae (armLaw arm))) (hmean : forall arm, integral (armLaw arm) (fun reward : Rat => ((reward : Rat) : Real)) = ((model.mean arm : Rat) : Real)) (defaultAction : Fin K) (T : Nat) (hT : 0 < T) (delta : Real) (hdelta : 0 < delta) : let sigma2 := Concentration.intervalVarianceProxy lo hi let rewardKernel := RewardKernel.contextIndependentOfActionLaws (Context
theorem
BanditRLProof.UCB.integral_real_pseudoRegret_selectedPolicySuccessorGeneratedUCBRegretAction_le_textbookGapSum_armwiseBoundedFiniteArmLaws
Compiled
Canonical Real expected pseudo-regret bound for finite-arm stationary reward laws with arm-dependent nondegenerate support intervals. The UCB variance parameter is the maximum of the armwise Hoeffding proxies, computed internally with `Finset.sup` over all arms.
theorem integral_real_pseudoRegret_selectedPolicySuccessorGeneratedUCBRegretAction_le_textbookGapSum_armwiseBoundedFiniteArmLaws {K : Nat} (model : FiniteBanditModel K) (armLaw : Fin K -> Measure Rat) (hprob : forall arm, IsProbabilityMeasure (armLaw arm)) (lo hi : Fin K -> Real) (hlohi : forall arm, lo arm < hi arm) (hmeas : forall arm, AEMeasurable (fun reward : Rat => ((reward : Rat) : Real)) (armLaw arm)) (hbound : forall arm, Filter.Eventually (fun reward : Rat => Set.Icc (lo arm) (hi arm) ((reward : Rat) : Real)) (ae (armLaw arm))) (hmean : forall arm, integral (armLaw arm) (fun reward : Rat => ((reward : Rat) : Real)) = ((model.mean arm : Rat) : Real)) (defaultAction : Fin K) (T : Nat) (hT : 0 < T) (delta : Real) (hdelta : 0 < delta) : let sigma2 := Concentration.finiteArmIntervalVarianceProxy lo hi let rewardKernel := RewardKernel.contextIndependentOfActionLaws (Context