`Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

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Minimal Lean file

`integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

LemmaBandits.integral_regret_eq_sum_gap_mul_integral_pullCount

Details

No docstring.

theorem

Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {n : ℕ} [StandardBorelSpace 𝓐] [Fintype 𝓐]
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  (hA : ∀ (n : ℕ), Measurable (A n)) :
  ∫ (x : Ω), regret ν A n x ∂P =
    ∑ a,
      gap ν a * ∫ (x : Ω), (fun ω => ↑(Learning.pullCount A a n ω)) x ∂P
Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {n : ℕ}
  [StandardBorelSpace 𝓐] [Fintype 𝓐]
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  (hA : ∀ (n : ℕ), Measurable (A n)) :
  ∫ (x : Ω), regret ν A n x ∂P =
    ∑ a,
      gap ν a *
        ∫ (x : Ω),
          (fun ω =>
              ↑(Learning.pullCount A a n
                  ω))
            x ∂P

Code

lemma integral_regret_eq_sum_gap_mul_integral_pullCount
    [StandardBorelSpace 𝓐] [Fintype 𝓐] {P : Measure Ω} [IsProbabilityMeasure P]
    (hA : ∀ n, Measurable (A n)) :
    P[regret ν A n] = ∑ a, gap ν a * P[fun ω ↦ (pullCount A a n ω : ℝ)]

Type uses (3)

Body uses (2)

Used by (1)

integral_regret_le_of_forall_integral_pullCount_le

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Proof

by
  simp_rw [regret_eq_sum_pullCount_mul_gap]
  rw [integral_finsetSum]
  swap; · exact fun i _ ↦ (integrable_pullCount hA i n).mul_const _
  congr with a
  rw [integral_mul_const, mul_comm]

Dependency graph

Type dependencies (3)

`regret`🔗

DefinitionBandits.regret

Details

Regret of a sequence of pulls k : ℕ → 𝓐 at time t for the reward kernel ν ; Kernel 𝓐 ℝ.

def

Bandits.regret.{u_1, u_2} {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ
Bandits.regret.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ

Code

noncomputable
def regret (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  t * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]

Used by (11)

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`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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`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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Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount🔗

integral_regret_eq_sum_gap_mul_integral_pullCount🔗

regret🔗

gap🔗

pullCount🔗

`Bandits.integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

`integral_regret_eq_sum_gap_mul_integral_pullCount`🔗

`regret`🔗

`gap`🔗

`pullCount`🔗