Lean module · EXP3
BanditRLProof.Exp3MixedSquareExponentialRealizedAllHorizon
# All-horizon exponential mixed-square realized EXP3 route The explicit exponential-square 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. The fallback closes the active-clipping presentation without claiming a sharp short-horizon, Freedman, or EXP3.P rate.
Module map
Imports
BanditRLProof.Exp3MixedSquareExponentialRealizedExplicitTuning, BanditRLProof.Exp3BernsteinAllHorizon
Imported by
Declarations
Open an item to read its exact compact statement and source link. Detailed teaching notes are linked when registered.
def
BanditRLProof.Exp3.exponentialSquareBernsteinLargeHorizonCondition
Compiled
The regime in which all four components of the exponential-square exploration schedule are at most one half.
def exponentialSquareBernsteinLargeHorizonCondition (K T delta : Real) : Prop
def
BanditRLProof.Exp3.exponentialSquareBernsteinAllHorizonRegretThreshold
Compiled
All-horizon threshold for the exponential-square route: use the explicit large-horizon rate in its valid regime and `T + 1` otherwise.
noncomputable def exponentialSquareBernsteinAllHorizonRegretThreshold {Action : Type v} (arms : Finset Action) (horizon : Nat) (delta : Real) : Real
theorem
BanditRLProof.Exp3.sampledPredictable_allHorizonExponentialSquareBernsteinRealizedRegret_tail
Compiled
Generated realized-regret tail for every positive horizon under the exact exponential-square learning rate and clipped exploration schedule. The refined threshold is used precisely in the four-contract regime; the complementary branch is the genuine zero-probability `T + 1` fallback.
theorem sampledPredictable_allHorizonExponentialSquareBernsteinRealizedRegret_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 : Nat) (hhorizon : 0 < horizon) (delta : Real) (hdelta : 0 < delta) (hdelta_le_one : delta <= 1) : let gamma := exponentialSquareBernsteinClippedExplorationRate (arms.card : Real) (horizon : Real) delta let eta := exponentialSquareHighProbabilityLearningRate arms gamma horizon delta let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma (exponentialSquareBernsteinClippedExplorationRate_pos (arms.card : Real) (horizon : Real) delta (by exact_mod_cast hcard_two) (by exact_mod_cast hhorizon)).le (by exact (exponentialSquareBernsteinClippedExplorationRate_le_half (arms.card : Real) (horizon : Real) delta).trans (by norm_num)) loss.environment mu {sample | exponentialSquareBernsteinAllHorizonRegretThreshold arms horizon delta <= (Finset.range horizon).sum (fun t => sampledTrajectoryRealizedLossAt t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator)} <= ENNReal.ofReal delta