BanditRLProof.Exp3ComparatorConfidence

No declaration docstring is present; use the chapter context and exact statement below.

theorem comparatorEstimator_hasCondSubgaussianMGF_of_condDistrib_ae_eq_finiteActionKernel {Omega : Type u} {History : Type v} {Action : Type w} [mOmega : MeasurableSpace Omega] [StandardBorelSpace Omega] [Nonempty Omega] [mHistory : MeasurableSpace History] [StandardBorelSpace History] [mAction : MeasurableSpace Action] [MeasurableSingletonClass Action] [StandardBorelSpace Action] [Nonempty Action] [DecidableEq Action] (mu : Measure Omega) [IsFiniteMeasure mu] (history : Omega -> History) (hhistory : Measurable history) (action : Omega -> Action) (haction : Measurable action) (arms : Finset Action) (prob loss : History -> Action -> Real) (source : MeasurableFiniteActionDistribution arms prob) (epsilon : Real) (regularity : BoundedMeasurableLossWithProbabilityFloor arms prob loss epsilon) (comparator : Action) (hcomparator : comparator ∈ arms) (hcond : condDistrib action history mu =ᵐ[mu.map history] finiteActionKernel arms prob source) : ProbabilityTheory.HasCondSubgaussianMGF (mHistory.comap history) hhistory.comap_le (fun omega => importanceWeightedLoss (prob (history omega)) (loss (history omega)) (action omega) comparator - loss (history omega) comparator) (Concentration.intervalVarianceProxy 0 (1 / epsilon)) mu

No declaration docstring is present; use the chapter context and exact statement below.

noncomputable def sampledTrajectoryPredictableComparatorEstimatorDeviationAt {Env : Type u} {Action : Type v} [MeasurableSpace Env] [MeasurableSpace Action] [DecidableEq Action] (arms : Finset Action) (eta gamma : Real) (loss : PredictableLossVector Env Action) (comparator : Action) (t : Nat) (sample : Env × ((k : Nat) -> Action × Real)) : Real

No declaration docstring is present; use the chapter context and exact statement below.

noncomputable def sampledTrajectoryObservedComparatorEstimatorDeviationAt {Env : Type u} {Action : Type v} [MeasurableSpace Env] [MeasurableSpace Action] [DecidableEq Action] (arms : Finset Action) (eta gamma : Real) (loss : PredictableLossVector Env Action) (comparator : Action) (t : Nat) (sample : Env × ((k : Nat) -> Action × Real)) : Real

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledPredictableComparatorEstimatorDeviation_zero_hasCondSubgaussianMGF {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) [IsFiniteMeasure prior] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment ProbabilityTheory.HasCondSubgaussianMGF ((inferInstance : MeasurableSpace Env).comap (fun sample : Env × ((k : Nat) -> Action × Real) => sample.1)) measurable_fst.comap_le (sampledTrajectoryPredictableComparatorEstimatorDeviationAt arms eta gamma loss comparator 0) (Concentration.intervalVarianceProxy 0 (1 / (gamma / (arms.card : Real)))) mu

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledPredictableComparatorEstimatorDeviation_succ_hasCondSubgaussianMGF {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) [IsFiniteMeasure prior] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (n : Nat) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment let history := fun sample : Env × ((k : Nat) -> Action × Real) => (sample.1, Preorder.frestrictLe n sample.2) ProbabilityTheory.HasCondSubgaussianMGF ((inferInstance : MeasurableSpace (Env × History.FinitePairHistory Action Real n)).comap history) (measurable_fst.prodMk ((Preorder.measurable_frestrictLe n).comp measurable_snd)).comap_le (sampledTrajectoryPredictableComparatorEstimatorDeviationAt arms eta gamma loss comparator (n + 1)) (Concentration.intervalVarianceProxy 0 (1 / (gamma / (arms.card : Real)))) mu

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledObservedComparatorEstimatorDeviation_zero_hasCondSubgaussianMGF {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) [IsFiniteMeasure prior] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment ProbabilityTheory.HasCondSubgaussianMGF ((inferInstance : MeasurableSpace Env).comap (fun sample : Env × ((k : Nat) -> Action × Real) => sample.1)) measurable_fst.comap_le (sampledTrajectoryObservedComparatorEstimatorDeviationAt arms eta gamma loss comparator 0) (Concentration.intervalVarianceProxy 0 (1 / (gamma / (arms.card : Real)))) mu

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledObservedComparatorEstimatorDeviation_succ_hasCondSubgaussianMGF {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) [IsFiniteMeasure prior] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (n : Nat) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment let history := fun sample : Env × ((k : Nat) -> Action × Real) => (sample.1, Preorder.frestrictLe n sample.2) ProbabilityTheory.HasCondSubgaussianMGF ((inferInstance : MeasurableSpace (Env × History.FinitePairHistory Action Real n)).comap history) (measurable_fst.prodMk ((Preorder.measurable_frestrictLe n).comp measurable_snd)).comap_le (sampledTrajectoryObservedComparatorEstimatorDeviationAt arms eta gamma loss comparator (n + 1)) (Concentration.intervalVarianceProxy 0 (1 / (gamma / (arms.card : Real)))) mu

No declaration docstring is present; use the chapter context and exact statement below.

noncomputable def sampledObservedComparatorEstimatorDeviationProcess {Env : Type u} {Action : Type v} [MeasurableSpace Env] [MeasurableSpace Action] [DecidableEq Action] (arms : Finset Action) (eta gamma : Real) (loss : PredictableLossVector Env Action) (comparator : Action) : Nat -> Env × ((k : Nat) -> Action × Real) -> Real | 0, _sample => 0 | i + 1, sample => sampledTrajectoryObservedComparatorEstimatorDeviationAt arms eta gamma loss comparator i sample noncomputable def sampledComparatorEstimatorVarianceProxy {Action : Type v} (arms : Finset Action) (gamma : Real) : NNReal

No declaration docstring is present; use the chapter context and exact statement below.

noncomputable def sampledComparatorEstimatorVarianceProxy {Action : Type v} (arms : Finset Action) (gamma : Real) : NNReal

No declaration docstring is present; use the chapter context and exact statement below.

noncomputable def sampledComparatorEstimatorDeviationProxy {Action : Type v} (arms : Finset Action) (gamma : Real) : Nat -> NNReal | 0 => 0 | _i + 1 => sampledComparatorEstimatorVarianceProxy arms gamma theorem sampledObservedComparatorEstimatorDeviationProcess_stronglyAdapted {Env : Type u} {Action : Type v} [MeasurableSpace Env] [MeasurableSpace Action] [MeasurableSingletonClass Action] [DecidableEq Action] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (hgamma_nonneg : 0 <= gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) : StronglyAdapted (sampledPredictableDeviationFiltration Env Action) (sampledObservedComparatorEstimatorDeviationProcess arms eta gamma loss comparator)

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledObservedComparatorEstimatorDeviationProcess_stronglyAdapted {Env : Type u} {Action : Type v} [MeasurableSpace Env] [MeasurableSpace Action] [MeasurableSingletonClass Action] [DecidableEq Action] (arms : Finset Action) (harms : arms.Nonempty) (eta gamma : Real) (hgamma_nonneg : 0 <= gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) : StronglyAdapted (sampledPredictableDeviationFiltration Env Action) (sampledObservedComparatorEstimatorDeviationProcess arms eta gamma loss comparator)

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledObservedComparatorEstimatorDeviationProcess_sum_range_succ {Env : Type u} {Action : Type v} [MeasurableSpace Env] [MeasurableSpace Action] [DecidableEq Action] (arms : Finset Action) (eta gamma : Real) (loss : PredictableLossVector Env Action) (comparator : Action) (horizon : Nat) (sample : Env × ((k : Nat) -> Action × Real)) : (Finset.range (horizon + 1)).sum (fun i => sampledObservedComparatorEstimatorDeviationProcess arms eta gamma loss comparator i sample) = (Finset.range horizon).sum (fun i => sampledTrajectoryObservedComparatorEstimatorDeviationAt arms eta gamma loss comparator i sample)

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledComparatorEstimatorDeviationProxy_sum_range_succ {Action : Type v} (arms : Finset Action) (gamma : Real) (horizon : Nat) : (Finset.range (horizon + 1)).sum (sampledComparatorEstimatorDeviationProxy arms gamma) = (horizon : NNReal) * sampledComparatorEstimatorVarianceProxy arms gamma

Fixed-comparator concentration for the observed importance-weighted EXP3 estimator minus its true predictable comparator loss.

theorem sampledObservedComparatorEstimatorDeviation_sum_tail_ennreal {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) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon : Nat) {eps : Real} (heps : 0 <= eps) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment mu {sample | eps <= (Finset.range horizon).sum (fun i => sampledTrajectoryObservedComparatorEstimatorDeviationAt arms eta gamma loss comparator i sample)} <= ENNReal.ofReal (Real.exp (-eps ^ 2 / (2 * ((((horizon : NNReal) * sampledComparatorEstimatorVarianceProxy arms gamma : NNReal)) : Real))))

No declaration docstring is present; use the chapter context and exact statement below.

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

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledComparatorEstimatorVarianceProxy_pos {Action : Type v} (arms : Finset Action) (harms : arms.Nonempty) (gamma : Real) (hgamma_pos : 0 < gamma) : 0 < ((sampledComparatorEstimatorVarianceProxy arms gamma : NNReal) : Real)

No declaration docstring is present; use the chapter context and exact statement below.

theorem sampledObservedComparatorEstimatorDeviation_sum_tail_exp_neg_budget {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) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon : Nat) (hhorizon : 0 < horizon) (budget : Real) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment mu {sample | Real.sqrt (2 * ((((horizon : NNReal) * sampledComparatorEstimatorVarianceProxy arms gamma : NNReal)) : Real) * budget) <= (Finset.range horizon).sum (fun i => sampledTrajectoryObservedComparatorEstimatorDeviationAt arms eta gamma loss comparator i sample)} <= ENNReal.ofReal (Real.exp (-budget))

Delta-shaped one-sided confidence bound for a fixed comparator's observed importance-weighted estimator against its true predictable loss.

theorem sampledObservedComparatorEstimatorDeviation_sum_tail_delta {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) (hgamma_pos : 0 < gamma) (hgamma_le_one : gamma <= 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon : Nat) (hhorizon : 0 < horizon) (delta : Real) (hdelta : 0 < delta) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment mu {sample | sampledComparatorEstimatorConfidenceRadius arms gamma horizon delta <= (Finset.range horizon).sum (fun i => observedImportanceWeightedLossAt arms eta gamma i sample comparator - predictableLossAt loss i sample comparator)} <= ENNReal.ofReal delta