`Bandits.ETC.prob_arm_mul_eq_le`🔗

Lemma

The probability that at time K * m the ETC algorithm chooses arm a is at most exp(- m * Δ_a^2 / 4).

theorem

Bandits.ETC.prob_arm_mul_eq_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) (hm : m ≠ 0) :
  MeasureTheory.Measure.real P {ω | A (K * m) ω = a} ≤
    Real.exp (-↑m * gap ν a ^ 2 / (4 * ↑σ2))
Bandits.ETC.prob_arm_mul_eq_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) (hm : m ≠ 0) :
  MeasureTheory.Measure.real P
      {ω | A (K * m) ω = a} ≤
    Real.exp
      (-↑m * gap ν a ^ 2 / (4 * ↑σ2))

Code

lemma prob_arm_mul_eq_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)
    (hm : m ≠ 0) :
    P.real {ω | A (K * m) ω = a} ≤ Real.exp (- (m : ℝ) * gap ν a ^ 2 / (4 * σ2))

Proof

by
  have h_pos : 0 < K * m := Nat.mul_pos hK hm.bot_lt
  have h_le : P.real {ω | A (K * m) ω = a}
      ≤ P.real {ω | sumRewards A R (bestArm ν) (K * m) ω ≤ sumRewards A R a (K * m) ω} := by
    simp_rw [measureReal_def]
    gcongr 1
    · simp
    refine measure_mono_ae ?_
    exact sumRewards_bestArm_le_of_arm_mul_eq h a hm
  exact h_le.trans (probReal_sumRewards_le_sumRewards_le h hν a)

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: 17 project declarations, 137 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.prob_arm_mul_eq_le🔗

`Bandits.ETC.prob_arm_mul_eq_le`🔗