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Bandits.gap_eq_bestArm_sub🔗

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Minimal Lean file

gap_eq_bestArm_sub🔗

LemmaBandits.gap_eq_bestArm_sub
Details

No docstring.

🔗theorem
Bandits.gap_eq_bestArm_sub.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐} [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν a = ∫ (x : ℝ), id x ∂ν (bestArm ν) - ∫ (x : ℝ), id x ∂ν a
Bandits.gap_eq_bestArm_sub.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν a =
    ∫ (x : ℝ), id x ∂ν (bestArm ν) -
      ∫ (x : ℝ), id x ∂ν a

Code

lemma gap_eq_bestArm_sub : gap ν a = (ν (bestArm ν))[id] - (ν a)[id]
Type uses (2)
Body uses (1)
Used by (4)

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Proof
by
  rw [gap]
  congr
  refine le_antisymm ?_ (le_ciSup (f := fun a ↦ (ν a)[id]) (by simp) (bestArm ν))
  exact ciSup_le le_bestArm

Dependency graph

Type dependencies (2)

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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bestArm🔗

DefinitionBandits.bestArm
Details

action with the highest mean.

🔗def
Bandits.bestArm.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  [Fintype 𝓐] [Nonempty 𝓐] (ν : ProbabilityTheory.Kernel 𝓐 ℝ) : 𝓐
Bandits.bestArm.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐} [Fintype 𝓐]
  [Nonempty 𝓐]
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ) : 𝓐

Code

noncomputable def bestArm (ν : Kernel 𝓐 ℝ) : 𝓐 :=
  (exists_max_image univ (fun a ↦ (ν a)[id]) (univ_nonempty_iff.mpr inferInstance)).choose
Used by (18)

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