`Bandits.ETC.pullCount_of_ge`🔗

Lemma

For n ≥ K * m, the number of pulls of each arm a at time n is equal to m plus n - K * m if arm a is the best arm after the exploration phase.

theorem

Bandits.ETC.pullCount_of_ge.{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}
  (h :
    Learning.IsAlgEnvSeq A R (etcAlgorithm hK m)
      (Learning.stationaryEnv ν) P)
  (a : Fin K) (hm : m ≠ 0) {n : ℕ} (hn : K * m ≤ n) :
  Learning.pullCount A a n =ᵐ[P] fun ω =>
    m +
      (n - K * m) * Set.indicator {ω' | A (K * m) ω' = a} (fun x => 1) ω
Bandits.ETC.pullCount_of_ge.{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}
  (h :
    Learning.IsAlgEnvSeq A R
      (etcAlgorithm hK m)
      (Learning.stationaryEnv ν) P)
  (a : Fin K) (hm : m ≠ 0) {n : ℕ}
  (hn : K * m ≤ n) :
  Learning.pullCount A a n =ᵐ[P] fun ω =>
    m +
      (n - K * m) *
        Set.indicator
          {ω' | A (K * m) ω' = a}
          (fun x => 1) ω

Code

lemma pullCount_of_ge (h : IsAlgEnvSeq A R (etcAlgorithm hK m) (stationaryEnv ν) P)
    (a : Fin K) (hm : m ≠ 0) {n : ℕ} (hn : K * m ≤ n) :
    pullCount A a n
      =ᵐ[P] fun ω ↦ m + (n - K * m) * {ω' | A (K * m) ω' = a}.indicator (fun _ ↦ 1) ω

Proof

by
  have h_ae n : K * m ≤ n → pullCount A a (n + 1)
      =ᵐ[P] fun ω ↦ pullCount A a n ω + {ω' | A (K * m) ω' = a}.indicator (fun _ ↦ 1) ω :=
    pullCount_add_one_of_ge h a hm
  simp_rw [Filter.EventuallyEq, ← ae_all_iff] at h_ae
  have h_ae_Km : pullCount A a (K * m) =ᵐ[P] fun _ ↦ m := pullCount_mul h a
  filter_upwards [h_ae_Km, h_ae] with ω h_Km h_ae
  induction n, hn using Nat.le_induction with
  | base => simp [h_Km]
  | succ n hmn h_ind =>
    rw [h_ae n hmn, h_ind, add_assoc, ← add_one_mul]
    congr
    grind

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, 117 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.pullCount_of_ge🔗

`Bandits.ETC.pullCount_of_ge`🔗