Lean module · EXP3
BanditRLProof.Exp3MixedSquarePredictableVarianceRealizedDoublePredictableVarianceHighProbabilityRegret
# Realized EXP3 regret with two pathwise predictable variances The predictable-regret component retains the mixed-square estimator variance, while the realized-minus-predictable selected-loss component retains its own exact predictable variance. This replaces the fixed Hoeffding proxy in the realized component without changing the existing predictable-regret route.
Module map
Imports
BanditRLProof.Exp3MixedSquarePredictableVarianceHighProbabilityRegret, BanditRLProof.Exp3RealizedPredictableVarianceTail
Imported by
Declarations
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def
BanditRLProof.Exp3.sampledPredictableDoubleVarianceRealizedHighProbabilityRegretBudget
Compiled
No declaration docstring is present; use the chapter context and exact statement below.
noncomputable def sampledPredictableDoubleVarianceRealizedHighProbabilityRegretBudget {Action : Type v} [DecidableEq Action] (arms : Finset Action) (eta gamma : Real) (horizon : Nat) (mixedVarianceBudget realizedVarianceBudget deltaSquare deltaConfidence deltaRealized : Real) : Real
theorem
BanditRLProof.Exp3.sampledPredictable_doublePredictableVarianceRealizedHighProbabilityRegret_tail_joint
Compiled
Joint realized-regret tail on simultaneous pathwise budgets for the mixed-square and selected-loss predictable variances.
theorem sampledPredictable_doublePredictableVarianceRealizedHighProbabilityRegret_tail_joint {Env : Type u} {Action : Type v} [MeasurableSpace Env] [StandardBorelSpace Env] [Nonempty Env] [MeasurableSpace Action] [MeasurableSingletonClass Action] [StandardBorelSpace Action] [Nonempty Action] [DecidableEq Action] (prior : Measure Env) [IsProbabilityMeasure prior] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (heta : 0 < eta) (hgamma_pos : 0 < gamma) (hgamma_lt_one : gamma < 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon : Nat) (mixedVarianceBudget realizedVarianceBudget deltaSquare deltaConfidence deltaRealized : Real) (hmixedVarianceBudget : 0 < mixedVarianceBudget) (hrealizedVarianceBudget : 0 < realizedVarianceBudget) (hdeltaSquare : 0 < deltaSquare) (hdeltaConfidence : 0 < deltaConfidence) (hdeltaRealized : 0 < deltaRealized) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_lt_one.le loss.environment mu {sample | sampledPredictableDoubleVarianceRealizedHighProbabilityRegretBudget arms eta gamma horizon mixedVarianceBudget realizedVarianceBudget deltaSquare deltaConfidence deltaRealized ≤ (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator) ∧ (Finset.range horizon).sum (fun i => sampledTrajectoryPredictableMixedSquaredVarianceAt arms eta gamma loss i sample) ≤ mixedVarianceBudget ∧ (Finset.range horizon).sum (fun i => sampledTrajectoryPredictableRealizedVarianceAt arms eta gamma loss i sample) ≤ realizedVarianceBudget} ≤ ((ENNReal.ofReal deltaSquare + ENNReal.ofReal deltaConfidence) + ENNReal.ofReal deltaConfidence) + ENNReal.ofReal deltaRealized