LeanMachineLearning exposition

Bandits.gap_bestArm🔗

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

gap_bestArm🔗

LemmaBandits.gap_bestArm
Details

No docstring.

🔗theorem
Bandits.gap_bestArm.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν (bestArm ν) = 0
Bandits.gap_bestArm.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [Fintype 𝓐] [Nonempty 𝓐] :
  gap ν (bestArm ν) = 0

Code

lemma gap_bestArm : gap ν (bestArm ν) = 0
Type uses (2)
Body uses (1)
Used by (1)

Actions: Source · Open Issue

Proof
by
  rw [gap_eq_bestArm_sub, sub_self]

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)

Actions: Source · Open Issue

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)

Actions: Source · Open Issue