Lean module · EXP3
BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedMarkovProbabilisticSparsityAllHorizon
# All-horizon predictable-variance EXP3 under probabilistic sparse losses The explicit probabilistic-sparsity schedule has its refined `14 * gamma * T` threshold when four horizon inequalities make clipping inactive. Outside that regime this module uses the strict `T + 1` zero-probability threshold under exactly the same internal eta, gamma, and generated trajectory measure. The resulting theorem covers every positive horizon and preserves the exact support-sparsity failure residual. Its practical endpoint consumes `mu(sparsityFailure) <= ofReal epsilon` and returns `ofReal delta + ofReal epsilon`. The refined branch still uses the global `K * T` Markov envelope. This is not a pathwise-sparse variance, best-arm first-order, Freedman, anytime, or ideal EXP3.P theorem.
Module map
Imports
BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedMarkovProbabilisticSparsityExplicitTuning, BanditRLProof.Exp3BernsteinAllHorizon
Imported by
Declarations
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def
BanditRLProof.Exp3.probabilisticSparseLossPredictableVarianceLargeHorizonCondition
Compiled
Regime in which all four components of the probabilistic-sparsity predictable-variance exploration schedule are at most one half.
def probabilisticSparseLossPredictableVarianceLargeHorizonCondition (K S T delta : Real) : Prop
def
BanditRLProof.Exp3.probabilisticSparseLossPredictableVarianceAllHorizonRegretThreshold
Compiled
All-horizon threshold for the probabilistic-sparsity route: use the explicit large-horizon threshold in its valid regime and `T + 1` otherwise.
noncomputable def probabilisticSparseLossPredictableVarianceAllHorizonRegretThreshold {Action : Type v} (arms : Finset Action) (horizon sparsity : Nat) (delta : Real) : Real
theorem
BanditRLProof.Exp3.sampledPredictable_allHorizonProbabilisticSparseLossPredictableVarianceRealizedMarkovRegret_tail
Compiled
Generated all-horizon realized-regret theorem with the exact support-sparsity failure residual.
theorem sampledPredictable_allHorizonProbabilisticSparseLossPredictableVarianceRealizedMarkovRegret_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) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta : Real) (hdelta : 0 < delta) (hdelta_le_one : delta <= 1) : let gamma := probabilisticSparseLossPredictableVarianceClippedExplorationRate (arms.card : Real) (sparsity : Real) (horizon : Real) delta let eta := probabilisticSparseLossPredictableVarianceHighProbabilityLearningRate arms gamma horizon sparsity delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (probabilisticSparseLossPredictableVarianceClippedExplorationRate_pos (arms.card : Real) (sparsity : Real) (horizon : Real) delta (by exact_mod_cast hcard_two) (by exact_mod_cast hsparsity) (by exact_mod_cast hhorizon)).le (by exact (probabilisticSparseLossPredictableVarianceClippedExplorationRate_le_half (arms.card : Real) (sparsity : Real) (horizon : Real) delta).trans (by norm_num)) loss.environment mu {sample | probabilisticSparseLossPredictableVarianceAllHorizonRegretThreshold arms 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)
theorem
BanditRLProof.Exp3.sampledPredictable_allHorizonProbabilisticSparseLossPredictableVarianceRealizedMarkovRegret_tail_of_sparsityFailure_le
Compiled
Practical all-horizon `delta + epsilon` theorem under an exact bound on the support-sparsity failure event for the same internally tuned measure.
theorem sampledPredictable_allHorizonProbabilisticSparseLossPredictableVarianceRealizedMarkovRegret_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) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon sparsity : Nat) (hhorizon : 0 < horizon) (hsparsity : 0 < sparsity) (delta epsilon : Real) (hdelta : 0 < delta) (hdelta_le_one : delta <= 1) : let gamma := probabilisticSparseLossPredictableVarianceClippedExplorationRate (arms.card : Real) (sparsity : Real) (horizon : Real) delta let eta := probabilisticSparseLossPredictableVarianceHighProbabilityLearningRate arms gamma horizon sparsity delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (probabilisticSparseLossPredictableVarianceClippedExplorationRate_pos (arms.card : Real) (sparsity : Real) (horizon : Real) delta (by exact_mod_cast hcard_two) (by exact_mod_cast hsparsity) (by exact_mod_cast hhorizon)).le (by exact (probabilisticSparseLossPredictableVarianceClippedExplorationRate_le_half (arms.card : Real) (sparsity : Real) (horizon : Real) delta).trans (by norm_num)) loss.environment mu (sampledPredictableSparsityFailure arms loss horizon sparsity) <= ENNReal.ofReal epsilon -> mu {sample | probabilisticSparseLossPredictableVarianceAllHorizonRegretThreshold arms 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