BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedDoublePathwiseVarianceProbabilisticSparsityTuning

Eta-tuned threshold with both sparse pathwise predictable variances.

noncomputable def pathwiseVarianceProbabilisticSparseLossDoubleVarianceRealizedTunedThreshold {Action : Type v} [DecidableEq Action] (arms : Finset Action) (gamma : Real) (horizon sparsity : Nat) (delta : Real) : Real

The sparse small-loss double-variance budget is bounded by the eta-tuned threshold.

theorem sampledPredictableDoubleVarianceProbabilisticSparseLossRealizedHighProbabilityRegretBudget_le_tunedThreshold {Action : Type v} [DecidableEq Action] (arms : Finset Action) (hcard_two : 2 ≤ arms.card) (gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_half : gamma ≤ 1 / 2) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta : Real) : sampledPredictableDoubleVarianceProbabilisticSparseLossRealizedHighProbabilityRegretBudget arms (pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity delta) gamma horizon sparsity delta ≤ pathwiseVarianceProbabilisticSparseLossDoubleVarianceRealizedTunedThreshold arms gamma horizon sparsity delta

Eta-tuned generated regret away from the common sparsity-failure event.

theorem sampledPredictable_tunedDoubleVarianceProbabilisticSparseLossRealizedRegret_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) (gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_half : gamma ≤ 1 / 2) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta : Real) (hdelta : 0 < delta) : let eta := pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le (by linarith : gamma ≤ 1) loss.environment mu ({sample | pathwiseVarianceProbabilisticSparseLossDoubleVarianceRealizedTunedThreshold arms gamma horizon sparsity delta ≤ (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator)} \ sampledPredictableSparsityFailure arms loss horizon sparsity) ≤ ENNReal.ofReal delta

Eta-tuned generated regret with the exact sparsity-failure residual.

theorem sampledPredictable_tunedDoubleVarianceProbabilisticSparseLossRealizedRegret_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) (gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_half : gamma ≤ 1 / 2) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta : Real) (hdelta : 0 < delta) : let eta := pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le (by linarith : gamma ≤ 1) loss.environment mu {sample | pathwiseVarianceProbabilisticSparseLossDoubleVarianceRealizedTunedThreshold arms gamma horizon sparsity delta ≤ (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator)} ≤ ENNReal.ofReal delta + mu (sampledPredictableSparsityFailure arms loss horizon sparsity)

Practical eta-tuned `delta + epsilon` theorem under the internally tuned generated measure.

theorem sampledPredictable_tunedDoubleVarianceProbabilisticSparseLossRealizedRegret_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) (gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_half : gamma ≤ 1 / 2) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta epsilon : Real) (hdelta : 0 < delta) : let eta := pathwiseVarianceProbabilisticSparseLossHighProbabilityLearningRate arms gamma horizon sparsity delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le (by linarith : gamma ≤ 1) loss.environment mu (sampledPredictableSparsityFailure arms loss horizon sparsity) ≤ ENNReal.ofReal epsilon → mu {sample | pathwiseVarianceProbabilisticSparseLossDoubleVarianceRealizedTunedThreshold arms gamma horizon sparsity delta ≤ (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator)} ≤ ENNReal.ofReal delta + ENNReal.ofReal epsilon