LeanMachineLearning exposition

Bandits.UCB.nextArm🔗

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Minimal Lean file

nextArm🔗

DefinitionBandits.UCB.nextArm
Details

Arm pulled by the UCB algorithm at time n + 1.

🔗def
Bandits.UCB.nextArm {K : ℕ} (hK : 0 < K) (c : ℝ) (n : ℕ)
  (h : ↥(Finset.Iic n) → Fin K × ℝ) : Fin K
Bandits.UCB.nextArm {K : ℕ} (hK : 0 < K)
  (c : ℝ) (n : ℕ)
  (h : ↥(Finset.Iic n) → Fin K × ℝ) :
  Fin K

Code

noncomputable
def UCB.nextArm (hK : 0 < K) (c : ℝ) (n : ℕ) (h : Iic n → Fin K × ℝ) : Fin K :=
  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
  if n < K - 1 then RoundRobin.nextAction hK n else
  argmax (fun a ↦ empMean' n h a + ucbWidth' c n h a)
Body uses (4)
Used by (7)

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Dependency graph

All dependencies, transitively (8)

nextAction🔗

DefinitionLearning.RoundRobin.nextAction
Details

Action chosen by the Round-Robin algorithm at time n + 1. This is action (n + 1) % K.

🔗def
Learning.RoundRobin.nextAction {K : ℕ} (hK : 0 < K) (n : ℕ) : Fin K
Learning.RoundRobin.nextAction {K : ℕ}
  (hK : 0 < K) (n : ℕ) : Fin K

Code

noncomputable
def RoundRobin.nextAction (hK : 0 < K) (n : ℕ) : Fin K := ⟨(n + 1) % K, Nat.mod_lt _ hK⟩
Used by (14)

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max🔗

DefinitionFunction.max
Details

The maximum value of a tuple.

🔗def
Function.max.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α
Function.max.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : α

Code

abbrev max : α := univ.sup' univ_nonempty f
Used by (8)

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exists_argmax🔗

Lemmaexists_argmax
Details

No docstring.

🔗theorem
exists_argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ∃ i, f i = Function.max f
exists_argmax.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) :
  ∃ i, f i = Function.max f

Code

lemma exists_argmax : ∃ i, f i = f.max
Type uses (1)
Used by (3)

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Proof
by
  obtain ⟨i, -, hi⟩ := Finset.exists_mem_eq_sup' (by simp : Finset.univ.Nonempty) f
  exact ⟨i, hi.symm⟩

argmax🔗

Definitionargmax
Details

The index of the maximum value of a tuple.

🔗def
argmax.{u_1, u_2} {ι : Type u_1} {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ι
argmax.{u_1, u_2} {ι : Type u_1}
  {α : Type u_2} [LinearOrder α]
  [Fintype ι] [Nonempty ι] (f : ι → α) : ι

Code

noncomputable def argmax := (exists_argmax f).choose
Body uses (2)
Used by (17)

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sumRewards'🔗

DefinitionLearning.sumRewards'
Details

Sum of rewards of arm a up to (and including) time n.

🔗def
Learning.sumRewards'.{u_1} {𝓐 : Type u_1} [DecidableEq 𝓐] (n : ℕ)
  (h : ↥(Finset.Iic n) → 𝓐 × ℝ) (a : 𝓐) : ℝ
Learning.sumRewards'.{u_1} {𝓐 : Type u_1}
  [DecidableEq 𝓐] (n : ℕ)
  (h : ↥(Finset.Iic n) → 𝓐 × ℝ) (a : 𝓐) :
  ℝ

Code

noncomputable
def sumRewards' (n : ℕ) (h : Iic n → 𝓐 × ℝ) (a : 𝓐) :=
  ∑ s, if (h s).1 = a then (h s).2 else 0
Used by (9)

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pullCount'🔗

DefinitionLearning.pullCount'
Details

Number of pulls of arm a up to (and including) time n. This is the number of entries in h in which the arm is a.

🔗def
Learning.pullCount'.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  [DecidableEq 𝓐] (n : ℕ) (h : ↥(Finset.Iic n) → 𝓐 × R) (a : 𝓐) : ℕ
Learning.pullCount'.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  [DecidableEq 𝓐] (n : ℕ)
  (h : ↥(Finset.Iic n) → 𝓐 × R) (a : 𝓐) :
  ℕ

Code

noncomputable
def pullCount' (n : ℕ) (h : Iic n → 𝓐 × R) (a : 𝓐) := #{s | (h s).1 = a}
Used by (29)

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empMean'🔗

DefinitionLearning.empMean'
Details

Empirical mean of arm a at time n.

🔗def
Learning.empMean'.{u_1} {𝓐 : Type u_1} [DecidableEq 𝓐] (n : ℕ)
  (h : ↥(Finset.Iic n) → 𝓐 × ℝ) (a : 𝓐) : ℝ
Learning.empMean'.{u_1} {𝓐 : Type u_1}
  [DecidableEq 𝓐] (n : ℕ)
  (h : ↥(Finset.Iic n) → 𝓐 × ℝ) (a : 𝓐) :
  ℝ

Code

noncomputable
def empMean' (n : ℕ) (h : Iic n → 𝓐 × ℝ) (a : 𝓐) :=
  (sumRewards' n h a) / (pullCount' n h a)
Body uses (2)
Used by (18)

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ucbWidth'🔗

DefinitionBandits.ucbWidth'
Details

The exploration bonus of the UCB algorithm, which corresponds to the width of a confidence interval.

🔗def
Bandits.ucbWidth' {K : ℕ} (c : ℝ) (n : ℕ)
  (h : ↥(Finset.Iic n) → Fin K × ℝ) (a : Fin K) : ℝ
Bandits.ucbWidth' {K : ℕ} (c : ℝ) (n : ℕ)
  (h : ↥(Finset.Iic n) → Fin K × ℝ)
  (a : Fin K) : ℝ

Code

noncomputable def ucbWidth' (c : ℝ) (n : ℕ) (h : Iic n → Fin K × ℝ) (a : Fin K) : ℝ :=
  √(2 * c * log (n + 2) / pullCount' n h a)
Body uses (1)
Used by (6)

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