BanditRLProof.OFULScheduledPowerOfTwoForcedScalarChargeBound

A pointwise prescribed-arm gap ceiling bounds the forced charge by its cardinality.

theorem powerOfTwoForcedActionSuccessorPseudoRegret_le_card_mul {K : Nat} {Feature : Type u} [Fintype Feature] (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (best : Fin K) (forcedAction : Nat -> Fin K) (horizon : Nat) (forcedGapBound : Real) (hforcedGap : forall exponent, linearValue thetaStar (actionFeature best) - linearValue thetaStar (actionFeature (forcedAction exponent)) <= forcedGapBound) : powerOfTwoForcedActionSuccessorPseudoRegret thetaStar actionFeature best forcedAction horizon <= ((powerOfTwoForcedIndexSet horizon).card : Real) * forcedGapBound

The forced charge is at most `Nat.log2 horizon + 1` times a gap ceiling.

theorem powerOfTwoForcedActionSuccessorPseudoRegret_le_log2_add_one_mul {K : Nat} {Feature : Type u} [Fintype Feature] (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (best : Fin K) (forcedAction : Nat -> Fin K) (horizon : Nat) (forcedGapBound : Real) (hforcedGapBound : 0 <= forcedGapBound) (hforcedGap : forall exponent, linearValue thetaStar (actionFeature best) - linearValue thetaStar (actionFeature (forcedAction exponent)) <= forcedGapBound) : powerOfTwoForcedActionSuccessorPseudoRegret thetaStar actionFeature best forcedAction horizon <= ((Nat.log2 horizon + 1 : Nat) : Real) * forcedGapBound

The standard parameter and arm envelopes instantiate the generic gap ceiling for every prescribed arm.

theorem powerOfTwoForcedActionSuccessorPseudoRegret_le_log2_add_one_mul_two_mul_parameterFeatureBound {K : Nat} {Feature : Type u} [Fintype Feature] (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (S L2 : Real) (hS : 0 <= S) (htheta : euclideanLength thetaStar <= S) (hactionFeatureBound : forall action, dotProduct (actionFeature action) (actionFeature action) <= L2) (best : Fin K) (forcedAction : Nat -> Fin K) (horizon : Nat) : powerOfTwoForcedActionSuccessorPseudoRegret thetaStar actionFeature best forcedAction horizon <= ((Nat.log2 horizon + 1 : Nat) : Real) * (2 * S * Real.sqrt L2)

Fully scalar all-horizon violation event for the power-of-two forced policy. At horizon zero the forced set is empty while `Nat.log2 0 + 1 = 1`, so this uses a harmless conservative envelope rather than an exact zero charge.

noncomputable def powerOfTwoForcedCanonicalStandardHighProbabilityPseudoRegretScalarAllHorizonViolationSet {K : Nat} {Feature : Type u} [Fintype Feature] (lambda : Real) (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (R delta S L2 : Real) (best : Fin K) : Set (Nat -> Fin K × Real)

The logarithmic scalar-budget violation event is contained in the explicit-charge event.

theorem powerOfTwoForcedCanonicalStandardHighProbabilityPseudoRegretScalarAllHorizonViolationSet_subset {K : Nat} {Feature : Type u} [Fintype Feature] (lambda : Real) (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (R delta S L2 : Real) (hS : 0 <= S) (htheta : euclideanLength thetaStar <= S) (hactionFeatureBound : forall action, dotProduct (actionFeature action) (actionFeature action) <= L2) (forcedAction : Nat -> Fin K) (best : Fin K) : powerOfTwoForcedCanonicalStandardHighProbabilityPseudoRegretScalarAllHorizonViolationSet lambda thetaStar actionFeature R delta S L2 best ⊆ powerOfTwoForcedCanonicalStandardHighProbabilityPseudoRegretAllHorizonViolationSet lambda thetaStar actionFeature R delta S L2 forcedAction best

Complete all-horizon theorem with the prescribed-action charge replaced by the logarithmic power-of-two count and the common linear arm-gap envelope.

theorem powerOfTwoForcedCanonicalStandardHighProbabilityPseudoRegret_nonneg_and_scalarAllHorizon_tail_le_explicitBound_of_linearSubgaussianEnvironment_of_featureBound_le_regularization {K : Nat} {Feature : Type u} [Fintype Feature] [DecidableEq Feature] [Nonempty Feature] (hK : 0 < K) (lambda : Real) (hlambda : 0 < lambda) (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (R : Real) (hR : 0 < R) (delta : Real) (hdelta : 0 < delta) (hdelta_one : delta <= 1) (S : Real) (hS : 0 <= S) (forcedAction : Nat -> Fin K) (environment : Thompson.HistoryEnvironment (Fin K) Real) (L2 : Real) (hL2 : 0 <= L2) (hactionFeatureBound : forall action, dotProduct (actionFeature action) (actionFeature action) <= L2) (hL2lambda : L2 <= lambda) (best : Fin K) (hbest : IsOptimalLinearArm thetaStar actionFeature best) (source : CanonicalLinearSubgaussianEnvironmentLaw hK thetaStar actionFeature R S environment) : (forall horizon trajectory, 0 <= canonicalStandardHighProbabilityPseudoRegret thetaStar actionFeature best horizon trajectory) ∧ Thompson.canonicalHistoryTrajectoryMeasure (finiteHistoryPowerOfTwoForcedTelescopingScalarRidgeAlgorithm hK lambda actionFeature R delta S forcedAction) environment (powerOfTwoForcedCanonicalStandardHighProbabilityPseudoRegretScalarAllHorizonViolationSet lambda thetaStar actionFeature R delta S L2 best) <= ENNReal.ofReal delta