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Bandits.avg_mean_reward_tendsto_of_sublinear_regret🔗

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Minimal Lean file

avg_mean_reward_tendsto_of_sublinear_regret🔗

LemmaBandits.avg_mean_reward_tendsto_of_sublinear_regret
Details

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]))
Type uses (1)

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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

Dependency graph

Type dependencies (1)

regret🔗

DefinitionBandits.regret
Details

Regret of a sequence of pulls k : ℕ → 𝓐 at time t for the reward kernel ν ; Kernel 𝓐 ℝ.

🔗def
Bandits.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) (t : ℕ) (ω : Ω) : ℝ
Bandits.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) (t : ℕ) (ω : Ω) : ℝ

Code

noncomputable
def regret (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  t * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]
Used by (11)

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