Lean module · OFUL
BanditRLProof.OFULExpectedRegretAsymptotics
# Asymptotic OFUL expected pseudo-regret This module upgrades the explicit finite-window canonical OFUL expected pseudo-regret theorem to an `IsBigO` statement at `Filter.atTop`. The feature dimension and model parameters are fixed while the horizon varies.
Module map
Imports
BanditRLProof.OFULExpectedRegretRate
Imported by
BanditRLProof, BanditRLProof.OFULExpectedRegretConsistency, BanditRLProof.OFULScheduledBoundedStoppingTimeExpectedRegretAsymptotics
Declarations
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theorem
BanditRLProof.OFUL.standardScalarLogDetBudget_isBigO_log
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
theorem standardScalarLogDetBudget_isBigO_log {Feature : Type u} [Fintype Feature] [Nonempty Feature] (lambda : Real) (hlambda : 0 < lambda) (L2 : Real) (hL2 : 0 <= L2) : (fun horizon : Nat => standardScalarLogDetBudget (Feature
theorem
BanditRLProof.OFUL.standardExpectedRegretLogBudget_isBigO_log
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
theorem standardExpectedRegretLogBudget_isBigO_log {Feature : Type u} [Fintype Feature] [Nonempty Feature] (lambda : Real) (hlambda : 0 < lambda) (L2 : Real) (hL2 : 0 <= L2) : (fun horizon : Nat => standardExpectedRegretLogBudget (Feature
theorem
BanditRLProof.OFUL.sqrt_log_succ_isBigO_log_succ
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
theorem sqrt_log_succ_isBigO_log_succ : (fun horizon : Nat => Real.sqrt (Real.log (((horizon + 1 : Nat) : Real)))) =O[atTop] (fun horizon : Nat => Real.log (((horizon + 1 : Nat) : Real)))
theorem
BanditRLProof.OFUL.standardExpectedPseudoRegretBound_isBigO_sqrt_mul_log
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
theorem standardExpectedPseudoRegretBound_isBigO_sqrt_mul_log {Feature : Type u} [Fintype Feature] [Nonempty Feature] (R : Real) (lambda : Real) (hlambda : 0 < lambda) (S : Real) (L2 : Real) (hL2 : 0 <= L2) : (fun horizon : Nat => standardExpectedPseudoRegretBound (Feature
def
BanditRLProof.OFUL.canonicalStandardExpectedPseudoRegret
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No declaration docstring is present; use the chapter context and exact statement below.
noncomputable def canonicalStandardExpectedPseudoRegret {K : Nat} {Feature : Type u} [Fintype Feature] [DecidableEq Feature] (hK : 0 < K) (lambda : Real) (thetaStar : Feature -> Real) (actionFeature : Fin K -> Feature -> Real) (R S : Real) (environment : Thompson.HistoryEnvironment (Fin K) Real) (best : Fin K) (horizon : Nat) : Real
theorem
BanditRLProof.OFUL.canonicalStandardExpectedPseudoRegret_nonneg_and_le
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
theorem canonicalStandardExpectedPseudoRegret_nonneg_and_le {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) (S : Real) (hS : 0 <= S) (environment : Thompson.HistoryEnvironment (Fin K) Real) (horizon : Nat) (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) : 0 <= canonicalStandardExpectedPseudoRegret hK lambda thetaStar actionFeature R S environment best horizon ∧ canonicalStandardExpectedPseudoRegret hK lambda thetaStar actionFeature R S environment best horizon <= standardExpectedPseudoRegretBound (Feature
theorem
BanditRLProof.OFUL.canonicalStandardExpectedPseudoRegret_isBigO_sqrt_mul_log
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
theorem canonicalStandardExpectedPseudoRegret_isBigO_sqrt_mul_log {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) (S : Real) (hS : 0 <= S) (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) : (canonicalStandardExpectedPseudoRegret hK lambda thetaStar actionFeature R S environment best) =O[atTop] (fun horizon : Nat => Real.sqrt (((horizon + 1 : Nat) : Real)) * Real.log (((horizon + 1 : Nat) : Real)))