`Bandits.pullCount_rate_tendsto_of_sublinear_regret`🔗

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Minimal Lean file

`pullCount_rate_tendsto_of_sublinear_regret`🔗

LemmaBandits.pullCount_rate_tendsto_of_sublinear_regret

Details

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)

Type uses (3)

Body uses (3)

Actions: Source · Open Issue

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)

Dependency graph

Type dependencies (3)

`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)

Actions: Source · Open Issue

`gap`🔗

DefinitionBandits.gap

Details

Gap of an action a: difference between the highest mean of the actions and the mean of a.

def

Bandits.gap.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ) (a : 𝓐) : ℝ
Bandits.gap.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (a : 𝓐) : ℝ

Code

noncomputable
def gap (ν : Kernel 𝓐 ℝ) (a : 𝓐) : ℝ := (⨆ i, (ν i)[id]) - (ν a)[id]

Used by (27)

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`pullCount`🔗

DefinitionLearning.pullCount

Details

Number of times action a was chosen up to time t (excluding t).

def

Learning.pullCount.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ
Learning.pullCount.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (t : ℕ) (ω : Ω) : ℕ

Code

noncomputable
def pullCount (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ :=
  #(filter (fun s ↦ A s ω = a) (range t))

Used by (146)

Actions: Source · Open Issue

Bandits.pullCount_rate_tendsto_of_sublinear_regret🔗

pullCount_rate_tendsto_of_sublinear_regret🔗

regret🔗

gap🔗

pullCount🔗

`Bandits.pullCount_rate_tendsto_of_sublinear_regret`🔗

`pullCount_rate_tendsto_of_sublinear_regret`🔗

`regret`🔗

`gap`🔗

`pullCount`🔗