BanditRLProof.Exp3MixedSquareBernstein

Exact uncentered second moment of one mixed importance-weighted square.

theorem sum_prob_mul_sq_mixedSquaredImportanceWeightedLoss_eq {Action : Type u} [DecidableEq Action] (arms : Finset Action) (prob loss : Action -> Real) (hprob : forall action, action ∈ arms -> prob action ≠ 0) : arms.sum (fun chosen => prob chosen * (mixedSquaredImportanceWeightedLoss arms prob loss chosen) ^ 2) = arms.sum (fun action => (loss action) ^ 4 / prob action)

The centered mixed-square score has second moment at most `K / epsilon` under a uniform probability floor.

theorem sum_prob_mul_sq_mixedSquaredEstimatorDeviation_le_card_div_floor {Action : Type u} [DecidableEq Action] (arms : Finset Action) (prob loss : Action -> Real) (hdist : FiniteActionDistribution arms prob) (epsilon : Real) (hepsilon : 0 < epsilon) (hfloor : forall action, action ∈ arms -> epsilon <= prob action) (hloss : forall action, action ∈ arms -> loss action ∈ Set.Icc (0 : Real) 1) : arms.sum (fun chosen => prob chosen * (mixedSquaredImportanceWeightedLoss arms prob loss chosen - arms.sum (fun action => (loss action) ^ 2)) ^ 2) <= (arms.card : Real) / epsilon

Fixed-tilt MGF budget for a centered mixed estimator square under a finite sampling law. The quadratic coefficient is `K / epsilon`, while the admissible tilt is controlled by the sharper range cap `epsilon`.

theorem finiteActionMixedSquaredEstimator_hasMGFUpperBoundAt {Action : Type u} [MeasurableSpace Action] [MeasurableSingletonClass Action] [DecidableEq Action] (arms : Finset Action) (prob loss : Action -> Real) (hdist : FiniteActionDistribution arms prob) (epsilon : Real) (hepsilon : 0 < epsilon) (hfloor : forall action, action ∈ arms -> epsilon <= prob action) (hloss : forall action, action ∈ arms -> loss action ∈ Set.Icc (0 : Real) 1) (tilt : Real) (htilt_nonneg : 0 <= tilt) (htilt_le : tilt <= epsilon) : Concentration.HasMGFUpperBoundAt (fun chosen => mixedSquaredImportanceWeightedLoss arms prob loss chosen - arms.sum (fun action => (loss action) ^ 2)) tilt (tilt ^ 2 * ((arms.card : Real) / epsilon)) (finiteActionMeasure arms prob)

A finite conditional action law supplies the fixed-tilt mixed-square MGF budget with second-moment coefficient `K / epsilon`.

theorem mixedSquaredEstimator_hasCondMGFUpperBoundAt_of_condDistrib_ae_eq_finiteActionKernel {Omega : Type u} {History : Type v} {Action : Type*} [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) (tilt : Real) (htilt_nonneg : 0 <= tilt) (htilt_le : tilt <= epsilon) (hcond : condDistrib action history mu =ᵐ[mu.map history] finiteActionKernel arms prob source) : Concentration.HasCondMGFUpperBoundAt (mHistory.comap history) hhistory.comap_le (fun omega => mixedSquaredImportanceWeightedLoss arms (prob (history omega)) (loss (history omega)) (action omega) - arms.sum (fun candidate => (loss (history omega) candidate) ^ 2)) tilt (tilt ^ 2 * ((arms.card : Real) / epsilon)) mu

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

theorem sampledPredictableMixedSquaredDeviation_zero_hasCondMGFUpperBoundAt {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) (tilt : Real) (htilt_nonneg : 0 <= tilt) (htilt_le : tilt <= gamma / (arms.card : Real)) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment Concentration.HasCondMGFUpperBoundAt ((inferInstance : MeasurableSpace Env).comap (fun sample : Env × ((k : Nat) -> Action × Real) => sample.1)) measurable_fst.comap_le (sampledTrajectoryPredictableMixedSquaredDeviationAt arms eta gamma loss 0) tilt (tilt ^ 2 * ((arms.card : Real) / (gamma / (arms.card : Real)))) mu

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

theorem sampledPredictableMixedSquaredDeviation_succ_hasCondMGFUpperBoundAt {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) (n : Nat) (tilt : Real) (htilt_nonneg : 0 <= tilt) (htilt_le : tilt <= gamma / (arms.card : Real)) : 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) Concentration.HasCondMGFUpperBoundAt ((inferInstance : MeasurableSpace (Env × History.FinitePairHistory Action Real n)).comap history) (measurable_fst.prodMk ((Preorder.measurable_frestrictLe n).comp measurable_snd)).comap_le (sampledTrajectoryPredictableMixedSquaredDeviationAt arms eta gamma loss (n + 1)) tilt (tilt ^ 2 * ((arms.card : Real) / (gamma / (arms.card : Real)))) mu

Fixed-tilt Bernstein tail for the centered predictable mixed-square process on the generated EXP3 trajectory.

theorem sampledPredictableMixedSquaredDeviation_sum_tail_fixedTilt {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) (horizon : Nat) (tilt : Real) (htilt_nonneg : 0 <= tilt) (htilt_le : tilt <= gamma / (arms.card : Real)) (threshold : Real) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment mu {sample | threshold <= (Finset.range horizon).sum (fun i => sampledTrajectoryPredictableMixedSquaredDeviationAt arms eta gamma loss i sample)} <= ENNReal.ofReal (Real.exp (-tilt * threshold + (horizon : Real) * (tilt ^ 2 * ((arms.card : Real) / (gamma / (arms.card : Real))))))

Fixed-tilt Bernstein tail for the observed, uncentered mixed estimator-square sum on the generated trajectory.

theorem sampledPredictableObservedMixedSquared_sum_tail_bernstein_fixedTilt {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) (horizon : Nat) (tilt : Real) (htilt_nonneg : 0 <= tilt) (htilt_le : tilt <= gamma / (arms.card : Real)) (threshold : Real) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment mu {sample | (arms.card : Real) * (horizon : Real) + threshold <= sampledObservedMixedSquaredSum arms eta gamma horizon sample} <= ENNReal.ofReal (Real.exp (-tilt * threshold + (horizon : Real) * (tilt ^ 2 * ((arms.card : Real) / (gamma / (arms.card : Real))))))

Variance coefficient in the mixed-square fixed-tilt MGF budget.

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

Optimized Bernstein radius for the observed mixed estimator-square sum.

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

Delta-shaped variance-sensitive confidence bound for the generated observed mixed estimator-square sum. The square-root term uses the second-moment coefficient `K / epsilon`, while the linear correction uses the reciprocal range cap `1 / epsilon`.

theorem sampledPredictableObservedMixedSquared_sum_tail_bernstein_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) (horizon : Nat) (delta : Real) (hdelta : 0 < delta) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_le_one loss.environment mu {sample | (arms.card : Real) * (horizon : Real) + sampledMixedSquaredBernsteinConfidenceRadius arms gamma horizon delta <= sampledObservedMixedSquaredSum arms eta gamma horizon sample} <= ENNReal.ofReal delta