LeanMachineLearning exposition

Learning.stepsUntil_pullCount_le🔗

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Minimal Lean file

stepsUntil_pullCount_le🔗

LemmaLearning.stepsUntil_pullCount_le
Details

No docstring.

🔗theorem
Learning.stepsUntil_pullCount_le.{u_1, u_3} {𝓐 : Type u_1}
  {Ω : Type u_3} [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} (ω : Ω) (a : 𝓐)
  (t : ℕ) : stepsUntil A a (pullCount A a (t + 1) ω) ω ≤ ↑t
Learning.stepsUntil_pullCount_le.{u_1,
    u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] {A : ℕ → Ω → 𝓐} (ω : Ω)
  (a : 𝓐) (t : ℕ) :
  stepsUntil A a (pullCount A a (t + 1) ω)
      ω ≤
    ↑t

Code

lemma stepsUntil_pullCount_le (ω : Ω) (a : 𝓐) (t : ℕ) :
    stepsUntil A a (pullCount A a (t + 1) ω) ω ≤ t
Type uses (2)
Used by (2)

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Proof
by
  rw [stepsUntil]
  exact csInf_le (OrderBot.bddBelow _) ⟨t, rfl, rfl⟩

Dependency graph

Type dependencies (2)

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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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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