`Bandits.pullCount_rate_tendsto_of_sublinear_regret`🔗

Lemma

If the regret is sublinear, the rate of suboptimal arm pulls tends to zero.

theorem

Bandits.pullCount_rate_tendsto_of_sublinear_regret.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {ω : Ω} {a : 𝓐}
  [Finite 𝓐]
  (hr : (fun x => regret ν A x ω) =o[Filter.atTop] fun t => ↑t)
  (hg : 0 < gap ν a) :
  Filter.Tendsto (fun t => ↑(Learning.pullCount A a t ω) / ↑t)
    Filter.atTop (nhds 0)
Bandits.pullCount_rate_tendsto_of_sublinear_regret.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω} {a : 𝓐}
  [Finite 𝓐]
  (hr :
    (fun x =>
        regret ν A x ω) =o[Filter.atTop]
      fun t => ↑t)
  (hg : 0 < gap ν a) :
  Filter.Tendsto
    (fun t =>
      ↑(Learning.pullCount A a t ω) / ↑t)
    Filter.atTop (nhds 0)

Code

lemma pullCount_rate_tendsto_of_sublinear_regret [Finite 𝓐]
    (hr : (regret ν A · ω) =o[atTop] fun t ↦ (t : ℝ)) (hg : 0 < gap ν a) :
    Tendsto (fun t ↦ (pullCount A a t ω : ℝ) / t) atTop (nhds 0)

Proof

by
  have := Fintype.ofFinite 𝓐
  have hb (t : ℕ) : (pullCount A a t ω : ℝ) * gap ν a ≤ regret ν A t ω := by
    rw [regret_eq_sum_pullCount_mul_gap]
    exact single_le_sum (f := fun a ↦ pullCount A a t ω * gap ν a)
      (fun _ _ ↦ mul_nonneg (Nat.cast_nonneg _) gap_nonneg) (mem_univ a)
  have hb' (t : ℕ) : (pullCount A a t ω : ℝ) / t ≤ regret ν A t ω / t / gap ν a := by
    obtain ht | ht := eq_or_ne t 0
    · simp [ht]
    · calc (pullCount A a t ω : ℝ) / t
          = pullCount A a t ω * gap ν a / gap ν a / t := by field_simp
        _ ≤ regret ν A t ω / gap ν a / t := by gcongr; exact hb t
        _ = regret ν A t ω / t / gap ν a := by ring
  apply squeeze_zero' (Eventually.of_forall fun _ ↦ by positivity) (Eventually.of_forall hb')
  simpa using hr.tendsto_div_nhds_zero.div_const (gap ν a)

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Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 2th 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: 3 project declarations, 51 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.pullCount_rate_tendsto_of_sublinear_regret🔗

`Bandits.pullCount_rate_tendsto_of_sublinear_regret`🔗