`Bandits.ArrayModel.truePast`🔗

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Minimal Lean file

`truePast`🔗

DefinitionBandits.ArrayModel.truePast

Details

All random variables in the space, except for the unseen rewards for action a after time n.

def

Bandits.ArrayModel.truePast.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐] [Nonempty R]
  (alg : Learning.Algorithm 𝓐 R) (a : 𝓐) (n : ℕ) (ω : probSpace 𝓐 R) :
  probSpace 𝓐 R
Bandits.ArrayModel.truePast.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  [Nonempty R]
  (alg : Learning.Algorithm 𝓐 R) (a : 𝓐)
  (n : ℕ) (ω : probSpace 𝓐 R) :
  probSpace 𝓐 R

Code

noncomputable
def truePast (alg : Algorithm 𝓐 R) (a : 𝓐) (n : ℕ) (ω : probSpace 𝓐 R) :
    probSpace 𝓐 R :=
  (ω.1, fun i b ↦ if b = a then if pullCount (action alg) a (n + 1) ω ≠ 0 then
      ω.2 (min i ((pullCount (action alg) a (n + 1) ω) - 1)) a else Nonempty.some inferInstance
    else ω.2 i b)

Type uses (2)

Body uses (2)

Used by (6)

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Dependency graph

Type dependencies (2)

`Algorithm`🔗

StructureLearning.Algorithm

Details

A stochastic, sequential algorithm.

structure

Learning.Algorithm.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]

Used by (216)

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`probSpace`🔗

DefinitionBandits.ArrayModel.probSpace

Details

Probability space for the array model of stochastic bandits.

def

Bandits.ArrayModel.probSpace.{u_1, u_2} (𝓐 : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)
Bandits.ArrayModel.probSpace.{u_1, u_2}
  (𝓐 : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)

Code

def probSpace : Type _ := (ℕ → I) × (ℕ → 𝓐 → R)

Used by (64)

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All dependencies, transitively (8)

`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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`instIsProbabilityMeasureP0`🔗

InstanceLearning.instIsProbabilityMeasureP0

Details

No docstring.

theorem

Learning.instIsProbabilityMeasureP0.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm 𝓐 𝓨) : IsProbabilityMeasure alg.p0

Type uses (1)

Algorithm

Used by (13)

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Proof

alg.hp0

`initAlgFunction`🔗

DefinitionBandits.ArrayModel.initAlgFunction

Details

The initial action is the image of a uniform random variable by this function.

def

Bandits.ArrayModel.initAlgFunction.{u_1, u_2} {𝓐 : Type u_1}
  {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  [Nonempty 𝓐] [StandardBorelSpace 𝓐] (alg : Learning.Algorithm 𝓐 R) :
  ↑unitInterval → 𝓐
Bandits.ArrayModel.initAlgFunction.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐]
  (alg : Learning.Algorithm 𝓐 R) :
  ↑unitInterval → 𝓐

Code

noncomputable
def initAlgFunction (alg : Algorithm 𝓐 R) : I → 𝓐 :=
  (Measure.exists_measurable_map_eq alg.p0).choose

Type uses (1)

Algorithm

Body uses (1)

instIsProbabilityMeasureP0

Used by (12)

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`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy

Details

No docstring.

theorem

Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (alg.policy n)

Type uses (1)

Algorithm

Used by (14)

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Proof

alg.h_policy n

`algFunction`🔗

DefinitionBandits.ArrayModel.algFunction

Details

The next action is the image of the history and a uniform random variable by this function.

def

Bandits.ArrayModel.algFunction.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] (alg : Learning.Algorithm 𝓐 R) (n : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × R) → ↑unitInterval → 𝓐
Bandits.ArrayModel.algFunction.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × R) →
    ↑unitInterval → 𝓐

Code

noncomputable
def algFunction (alg : Algorithm 𝓐 R) (n : ℕ) :
    (Iic n → 𝓐 × R) → I → 𝓐 :=
  (Kernel.exists_measurable_map_eq_unitInterval (alg.policy n)).choose

Type uses (1)

Algorithm

Body uses (1)

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy

Used by (17)

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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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`hist`🔗

DefinitionBandits.ArrayModel.hist

Details

History of actions and rewards up to time n in the array model.

def

Bandits.ArrayModel.hist.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐] (alg : Learning.Algorithm 𝓐 R)
  (ω : probSpace 𝓐 R) (n : ℕ) : ↥(Finset.Iic n) → 𝓐 × R
Bandits.ArrayModel.hist.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  (alg : Learning.Algorithm 𝓐 R)
  (ω : probSpace 𝓐 R) (n : ℕ) :
  ↥(Finset.Iic n) → 𝓐 × R

Code

noncomputable
def hist [DecidableEq 𝓐] (alg : Algorithm 𝓐 R) (ω : probSpace 𝓐 R) : (n : ℕ) → Iic n → 𝓐 × R
| 0 => fun _ ↦ (initAlgFunction alg (ω.1 0), ω.2 0 (initAlgFunction alg (ω.1 0)))
| n + 1 =>
  let hn : Iic n → 𝓐 × R := hist alg ω n
  let a : 𝓐 := algFunction alg n hn (ω.1 (n + 1))
  fun i ↦ if hin : i ≤ n then hn ⟨i, by simp [hin]⟩ else (a, ω.2 (pullCount' n hn a) a)

Type uses (2)

Body uses (3)

Used by (30)

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`action`🔗

DefinitionBandits.ArrayModel.action

Details

Action taken at time n in the array model.

def

Bandits.ArrayModel.action.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐] (alg : Learning.Algorithm 𝓐 R)
  (n : ℕ) (ω : probSpace 𝓐 R) : 𝓐
Bandits.ArrayModel.action.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : ℕ)
  (ω : probSpace 𝓐 R) : 𝓐

Code

noncomputable
def action [DecidableEq 𝓐] (alg : Algorithm 𝓐 R) (n : ℕ) (ω : probSpace 𝓐 R) : 𝓐 :=
  (hist alg ω n ⟨n, by simp⟩).1

Type uses (2)

Body uses (1)

hist

Used by (43)

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Bandits.ArrayModel.truePast🔗

truePast🔗

Algorithm🔗

probSpace🔗

pullCount🔗

instIsProbabilityMeasureP0🔗

initAlgFunction🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

algFunction🔗

pullCount'🔗

hist🔗

action🔗

`Bandits.ArrayModel.truePast`🔗

`truePast`🔗

`Algorithm`🔗

`probSpace`🔗

`pullCount`🔗

`instIsProbabilityMeasureP0`🔗

`initAlgFunction`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`algFunction`🔗

`pullCount'`🔗

`hist`🔗

`action`🔗