`Bandits.ETC.probReal_sumRewards_le_sumRewards_le`🔗

Lemma

No docstring.

theorem

Bandits.ETC.probReal_sumRewards_le_sumRewards_le.{u_1} {K : ℕ}
  {hK : 0 < K} {m : ℕ} {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {Ω : Type u_1}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ} {σ2 : NNReal} [Nonempty (Fin K)]
  (h :
    Learning.IsAlgEnvSeq A R (etcAlgorithm hK m)
      (Learning.stationaryEnv ν) P)
  (hν :
    ∀ (a : Fin K),
      ProbabilityTheory.HasSubgaussianMGF
        (fun x => x - ∫ (x : ℝ), id x ∂ν a) σ2 (ν a))
  (a : Fin K) :
  MeasureTheory.Measure.real P
      {ω |
        Learning.sumRewards A R (bestArm ν) (K * m) ω ≤
          Learning.sumRewards A R a (K * m) ω} ≤
    Real.exp (-↑m * gap ν a ^ 2 / (4 * ↑σ2))
Bandits.ETC.probReal_sumRewards_le_sumRewards_le.{u_1}
  {K : ℕ} {hK : 0 < K} {m : ℕ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ}
  {σ2 : NNReal} [Nonempty (Fin K)]
  (h :
    Learning.IsAlgEnvSeq A R
      (etcAlgorithm hK m)
      (Learning.stationaryEnv ν) P)
  (hν :
    ∀ (a : Fin K),
      ProbabilityTheory.HasSubgaussianMGF
        (fun x =>
          x - ∫ (x : ℝ), id x ∂ν a)
        σ2 (ν a))
  (a : Fin K) :
  MeasureTheory.Measure.real P
      {ω |
        Learning.sumRewards A R
            (bestArm ν) (K * m) ω ≤
          Learning.sumRewards A R a
            (K * m) ω} ≤
    Real.exp
      (-↑m * gap ν a ^ 2 / (4 * ↑σ2))

Code

lemma probReal_sumRewards_le_sumRewards_le [Nonempty (Fin K)]
    (h : IsAlgEnvSeq A R (etcAlgorithm hK m) (stationaryEnv ν) P)
    (hν : ∀ a, HasSubgaussianMGF (fun x ↦ x - (ν a)[id]) σ2 (ν a)) (a : Fin K) :
    P.real {ω | sumRewards A R (bestArm ν) (K * m) ω ≤ sumRewards A R a (K * m) ω} ≤
      Real.exp (-↑m * gap ν a ^ 2 / (4 * σ2))

Proof

by
  have h1 := Bandits.probReal_sumRewards_le_sumRewards_le h a (K * m) m m
  have h2 := probReal_sum_le_sum_streamMeasure hν a m
  refine le_trans (le_of_eq ?_) (h1.trans h2)
  simp_rw [measureReal_def]
  congr 1
  refine measure_congr ?_
  rw [Filter.eventuallyEq_set]
  filter_upwards [pullCount_mul h a, pullCount_mul h (bestArm ν)] with ω ha h_best
  simp [ha, h_best]

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 5th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 19 project declarations, 147 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Bandits.ETC.probReal_sumRewards_le_sumRewards_le🔗

`Bandits.ETC.probReal_sumRewards_le_sumRewards_le`🔗