`Bandits.avg_mean_reward_tendsto_of_sublinear_regret`🔗

Lemma

If the regret is sublinear, the average mean reward tends to the highest mean of the arms.

theorem

Bandits.avg_mean_reward_tendsto_of_sublinear_regret.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {A : ℕ → Ω → 𝓐} {ω : Ω}
  (hr : (fun x => regret ν A x ω) =o[Filter.atTop] fun t => ↑t) :
  Filter.Tendsto
    (fun t => (∑ s ∈ Finset.range t, ∫ (x : ℝ), id x ∂ν (A s ω)) / ↑t)
    Filter.atTop (nhds (⨆ a, ∫ (x : ℝ), id x ∂ν a))
Bandits.avg_mean_reward_tendsto_of_sublinear_regret.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {ω : Ω}
  (hr :
    (fun x =>
        regret ν A x ω) =o[Filter.atTop]
      fun t => ↑t) :
  Filter.Tendsto
    (fun t =>
      (∑ s ∈ Finset.range t,
          ∫ (x : ℝ), id x ∂ν (A s ω)) /
        ↑t)
    Filter.atTop
    (nhds (⨆ a, ∫ (x : ℝ), id x ∂ν a))

Code

lemma avg_mean_reward_tendsto_of_sublinear_regret
    (hr : (regret ν A · ω) =o[atTop] fun t ↦ (t : ℝ)) :
    Tendsto (fun t ↦ (∑ s ∈ range t, (ν (A s ω))[id]) / (t : ℝ))
      atTop (nhds (⨆ a, (ν a)[id]))

Proof

by
  have ht : Tendsto (fun t ↦ (⨆ a, (ν a)[id]) - regret ν A t ω / t)
      atTop (nhds (⨆ a, (ν a)[id])) := by
    simpa using tendsto_const_nhds.sub hr.tendsto_div_nhds_zero
  apply ht.congr'
  filter_upwards [eventually_ne_atTop 0] with t ht
  rw [regret]
  field_simp
  ring

Actions: Source · Open Issue

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: 1 project declarations, 41 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.avg_mean_reward_tendsto_of_sublinear_regret🔗

`Bandits.avg_mean_reward_tendsto_of_sublinear_regret`🔗