BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedDoublePathwiseVarianceProbabilisticSparsityBestArmAllHorizon

Best-arm exact double-variance threshold. The fixed-comparator schedule receives the armwise confidence share `delta / K`.

noncomputable def doubleVarianceProbabilisticSparseLossBestArmAllHorizonRegretThreshold {Action : Type v} (arms : Finset Action) (horizon sparsity : Nat) (delta : Real) : Real

Best-arm all-horizon tail away from the common support-sparsity failure event. The comparator union spends only the armwise confidence shares.

theorem sampledPredictable_allHorizonDoubleVarianceProbabilisticSparseLossBestArmRealizedRegret_tail_off_sparsityFailure {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) (hcard_two : 2 <= arms.card) (loss : PredictableLossVector Env Action) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta : Real) (hdelta : 0 < delta) (hdelta_le_one : delta <= 1) : let deltaArm := delta / (arms.card : Real) let gamma := doubleVarianceProbabilisticSparseLossClippedExplorationRate (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm let eta := pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity deltaArm let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (doubleVarianceProbabilisticSparseLossClippedExplorationRate_pos (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm (by exact_mod_cast hcard_two) (by exact_mod_cast hsparsity) (by exact_mod_cast hhorizon)).le (by exact (doubleVarianceProbabilisticSparseLossClippedExplorationRate_le_half (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm).trans (by norm_num)) loss.environment mu ({sample | doubleVarianceProbabilisticSparseLossBestArmAllHorizonRegretThreshold arms horizon sparsity delta <= (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - sampledPredictableBestArmCumulativeLoss arms harms loss horizon sample} \ sampledPredictableSparsityFailure arms loss horizon sparsity) <= ENNReal.ofReal delta

Best-arm all-horizon residual theorem. The common support-sparsity failure event is charged exactly once after the comparator union.

theorem sampledPredictable_allHorizonDoubleVarianceProbabilisticSparseLossBestArmRealizedRegret_tail {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) (hcard_two : 2 <= arms.card) (loss : PredictableLossVector Env Action) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta : Real) (hdelta : 0 < delta) (hdelta_le_one : delta <= 1) : let deltaArm := delta / (arms.card : Real) let gamma := doubleVarianceProbabilisticSparseLossClippedExplorationRate (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm let eta := pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity deltaArm let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (doubleVarianceProbabilisticSparseLossClippedExplorationRate_pos (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm (by exact_mod_cast hcard_two) (by exact_mod_cast hsparsity) (by exact_mod_cast hhorizon)).le (by exact (doubleVarianceProbabilisticSparseLossClippedExplorationRate_le_half (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm).trans (by norm_num)) loss.environment mu {sample | doubleVarianceProbabilisticSparseLossBestArmAllHorizonRegretThreshold arms horizon sparsity delta <= (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - sampledPredictableBestArmCumulativeLoss arms harms loss horizon sample} <= ENNReal.ofReal delta + mu (sampledPredictableSparsityFailure arms loss horizon sparsity)

Practical exact double-variance best-arm theorem with a single charge for the common support-sparsity failure event.

theorem sampledPredictable_allHorizonDoubleVarianceProbabilisticSparseLossBestArmRealizedRegret_tail_of_sparsityFailure_le {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) (hcard_two : 2 <= arms.card) (loss : PredictableLossVector Env Action) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta epsilon : Real) (hdelta : 0 < delta) (hdelta_le_one : delta <= 1) : let deltaArm := delta / (arms.card : Real) let gamma := doubleVarianceProbabilisticSparseLossClippedExplorationRate (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm let eta := pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity deltaArm let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (doubleVarianceProbabilisticSparseLossClippedExplorationRate_pos (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm (by exact_mod_cast hcard_two) (by exact_mod_cast hsparsity) (by exact_mod_cast hhorizon)).le (by exact (doubleVarianceProbabilisticSparseLossClippedExplorationRate_le_half (arms.card : Real) (sparsity : Real) (horizon : Real) deltaArm).trans (by norm_num)) loss.environment mu (sampledPredictableSparsityFailure arms loss horizon sparsity) <= ENNReal.ofReal epsilon → mu {sample | doubleVarianceProbabilisticSparseLossBestArmAllHorizonRegretThreshold arms horizon sparsity delta <= (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - sampledPredictableBestArmCumulativeLoss arms harms loss horizon sample} <= ENNReal.ofReal delta + ENNReal.ofReal epsilon