Lean module · EXP3
BanditRLProof.Exp3MixedSquareExponentialHighProbabilityRegret
# Predictable EXP3 regret with exponential mixed-square confidence This module replaces the Markov estimator-square event in the generated predictable EXP3 regret assembly by the compiled exponential mixed-square confidence theorem. The pure-cross and fixed-comparator confidence terms remain the existing variance-sensitive Bernstein radii. The square radius still uses the interval proxy of order `(|arms| / gamma)^2`; this route is logarithmic in the square-event failure probability but is not a Freedman or ideal EXP3.P theorem.
Module map
Imports
BanditRLProof.Exp3MixedSquareConfidence
Imported by
BanditRLProof, BanditRLProof.Exp3MixedSquareBernsteinHighProbabilityRegret, BanditRLProof.Exp3MixedSquareExponentialRealizedHighProbabilityRegret
Declarations
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def
BanditRLProof.Exp3.sampledPredictableExponentialSquareBernsteinHighProbabilityRegretBudget
Compiled
Predictable regret budget using the exponential mixed-square threshold instead of the Markov `|arms| * T / deltaSquare` threshold.
noncomputable def sampledPredictableExponentialSquareBernsteinHighProbabilityRegretBudget {Action : Type v} (arms : Finset Action) (eta gamma : Real) (horizon : Nat) (deltaSquare deltaConfidence : Real) : Real
theorem
BanditRLProof.Exp3.sampledPredictable_exponentialSquareBernsteinHighProbabilityRegret_tail
Compiled
Generated predictable EXP3 regret with an exponential estimator-square event and the two compiled Bernstein confidence events.
theorem sampledPredictable_exponentialSquareBernsteinHighProbabilityRegret_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) (eta gamma : Real) (heta : 0 < eta) (hgamma_pos : 0 < gamma) (hgamma_lt_one : gamma < 1) (loss : PredictableLossVector Env Action) (comparator : Action) (hcomparator : comparator ∈ arms) (horizon : Nat) (hhorizon : 0 < horizon) (deltaSquare deltaConfidence : Real) (hdeltaSquare : 0 < deltaSquare) (hdeltaConfidence : 0 < deltaConfidence) : let mu := prior ⊗ₘ sampledImportanceWeightedTrajectoryKernel arms harms eta gamma hgamma_pos.le hgamma_lt_one.le loss.environment mu {sample | sampledPredictableExponentialSquareBernsteinHighProbabilityRegretBudget arms eta gamma horizon deltaSquare deltaConfidence <= (Finset.range horizon).sum (fun t => sampledTrajectoryExploredPredictableLossAt arms eta gamma loss t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator)} <= (ENNReal.ofReal deltaSquare + ENNReal.ofReal deltaConfidence) + ENNReal.ofReal deltaConfidence
theorem
BanditRLProof.Exp3.sampledPredictable_exponentialSquareBernsteinHighProbabilityRegret_tail_total_delta
Compiled
Total-failure form: the exponential square event and both Bernstein confidence events each receive `delta / 3`.
theorem sampledPredictable_exponentialSquareBernsteinHighProbabilityRegret_tail_total_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) (heta : 0 < eta) (hgamma_pos : 0 < gamma) (hgamma_lt_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_lt_one.le loss.environment mu {sample | sampledPredictableExponentialSquareBernsteinHighProbabilityRegretBudget arms eta gamma horizon (delta / 3) (delta / 3) <= (Finset.range horizon).sum (fun t => sampledTrajectoryExploredPredictableLossAt arms eta gamma loss t sample) - (Finset.range horizon).sum (fun t => predictableLossAt loss t sample comparator)} <= ENNReal.ofReal delta