Lean module · EXP3
BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedMarkovAllHorizon
# All-horizon sparse-loss predictable-variance EXP3 route The explicit sparse-loss schedule has its refined `14 * gamma * T` threshold when four horizon inequalities make clipping inactive. Outside that regime this module reuses the compiled almost-sure horizon bound and the strict `T + 1` zero-probability threshold. This closes active clipping for the current pathwise sparse Markov route. It does not improve Markov's polynomial confidence dependence, convert armwise aggregate sparse loss to best-arm first-order regret, or prove Freedman/EXP3.P.
Module map
Imports
BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedMarkovExplicitTuning, BanditRLProof.Exp3BernsteinAllHorizon
Imported by
BanditRLProof, BanditRLProof.Exp3MixedSquarePredictableVarianceSparseLossRealizedMarkovAESparsityAllHorizon
Declarations
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def
BanditRLProof.Exp3.sparseLossPredictableVarianceLargeHorizonCondition
Compiled
The regime in which all four components of the sparse-loss predictable-variance exploration schedule are at most one half.
def sparseLossPredictableVarianceLargeHorizonCondition (K S T delta : Real) : Prop
def
BanditRLProof.Exp3.sparseLossPredictableVarianceAllHorizonRegretThreshold
Compiled
All-horizon threshold for the sparse-loss predictable-variance route: use the explicit large-horizon rate in its valid regime and `T + 1` otherwise.
noncomputable def sparseLossPredictableVarianceAllHorizonRegretThreshold {Action : Type v} (arms : Finset Action) (horizon sparsity : Nat) (delta : Real) : Real
theorem
BanditRLProof.Exp3.sampledPredictable_allHorizonSparseLossPredictableVarianceRealizedMarkovRegret_tail
Compiled
Generated sparse-loss realized-regret tail for every positive horizon under the exact eta and clipped gamma schedules. The refined threshold is used precisely in the four-contract regime; the complementary branch is the genuine zero-probability `T + 1` fallback.
theorem sampledPredictable_allHorizonSparseLossPredictableVarianceRealizedMarkovRegret_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) (hsparse : ∀ sample t, t < horizon → (sampledPredictableLossSupport arms loss t sample).card <= sparsity) : let gamma := sparseLossPredictableVarianceClippedExplorationRate (arms.card : Real) (sparsity : Real) (horizon : Real) delta let eta := sparseLossPredictableVarianceHighProbabilityLearningRate arms gamma horizon sparsity delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (sparseLossPredictableVarianceClippedExplorationRate_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 (sparseLossPredictableVarianceClippedExplorationRate_le_half (arms.card : Real) (sparsity : Real) (horizon : Real) delta).trans (by norm_num)) loss.environment mu {sample | sparseLossPredictableVarianceAllHorizonRegretThreshold 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