LeanMachineLearning exposition

Learning.measurable_stepsUntil'🔗

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Minimal Lean file

measurable_stepsUntil'🔗

LemmaLearning.measurable_stepsUntil'
Details

No docstring.

🔗theorem
Learning.measurable_stepsUntil'.{u_1, u_2, u_3} {𝓐 : Type u_1}
  {R : Type u_2} {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} {mΩ : MeasurableSpace Ω} [DecidableEq 𝓐]
  {A : ℕ → Ω → 𝓐} [MeasurableSingletonClass 𝓐]
  (hA : ∀ (n : ℕ), Measurable (A n)) (a : 𝓐) (m : ℕ) :
  Measurable fun ω => stepsUntil A a m (Prod.fst ω)
Learning.measurable_stepsUntil'.{u_1, u_2,
    u_3}
  {𝓐 : Type u_1} {R : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  {mΩ : MeasurableSpace Ω} [DecidableEq 𝓐]
  {A : ℕ → Ω → 𝓐}
  [MeasurableSingletonClass 𝓐]
  (hA : ∀ (n : ℕ), Measurable (A n))
  (a : 𝓐) (m : ℕ) :
  Measurable fun ω =>
    stepsUntil A a m (Prod.fst ω)

Code

lemma measurable_stepsUntil' [MeasurableSingletonClass 𝓐]
    (hA : ∀ n, Measurable (A n)) (a : 𝓐) (m : ℕ) :
    Measurable (fun ω : Ω × (ℕ → 𝓐 → R) ↦ stepsUntil A a m ω.1)
Type uses (1)
Body uses (1)
Used by (1)

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Proof
(measurable_stepsUntil hA a m).comp measurable_fst

Dependency graph

Type dependencies (1)

stepsUntil🔗

DefinitionLearning.stepsUntil
Details

Number of steps until action a was pulled exactly m times.

🔗def
Learning.stepsUntil.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (m : ℕ) (ω : Ω) : ℕ∞
Learning.stepsUntil.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (m : ℕ) (ω : Ω) : ℕ∞

Code

noncomputable
def stepsUntil (A : ℕ → Ω → 𝓐) (a : 𝓐) (m : ℕ) (ω : Ω) : ℕ∞ :=
  sInf ((↑) '' {s | pullCount A a (s + 1) ω = m})
Body uses (1)
Used by (46)

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All dependencies, transitively (1)

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)

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