Bandits.ArrayModel.hasCondDistrib

`hasCondDistrib_action'`🔗

LemmaBandits.ArrayModel.hasCondDistrib_action'

Details

No docstring.

theorem

Bandits.ArrayModel.hasCondDistrib_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] [Countable 𝓐]
  [StandardBorelSpace R] [Nonempty R] (alg : Learning.Algorithm 𝓐 R)
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] (n : ℕ) :
  ProbabilityTheory.HasCondDistrib (action alg (n + 1))
    (fun x => hist alg x n) (Learning.Algorithm.policy alg n)
    (arrayMeasure ν)
Bandits.ArrayModel.hasCondDistrib_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]
  [Countable 𝓐] [StandardBorelSpace R]
  [Nonempty R]
  (alg : Learning.Algorithm 𝓐 R)
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν]
  (n : ℕ) :
  ProbabilityTheory.HasCondDistrib
    (action alg (n + 1))
    (fun x => hist alg x n)
    (Learning.Algorithm.policy alg n)
    (arrayMeasure ν)

Code

lemma hasCondDistrib_action' (alg : Algorithm 𝓐 R) (ν : Kernel 𝓐 R) [IsMarkovKernel ν] (n : ℕ) :
    HasCondDistrib (action alg (n + 1)) (hist alg · n) (alg.policy n) (arrayMeasure ν)

Type uses (6)

Body uses (11)

Used by (1)

hasCondDistrib_action

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Proof

by
  rw [action_add_one_eq]
  have h_fun ω := algFunction_map alg n (hist alg ω n)
  refine ⟨by fun_prop, ?_⟩
  have h_indep : (arrayMeasure ν).map (fun ω ↦ (ω.1 (n + 1), hist alg ω n)) =
      (ℙ).prod ((arrayMeasure ν).map (hist alg · n)) := by
    have h_indep' := indepFun_fst_add_one_hist alg ν n
    rw [indepFun_iff_map_prod_eq_prod_map_map (by fun_prop) (by fun_prop)] at h_indep'
    rw [h_indep']
    congr
    simp only [arrayMeasure]
    calc ((Measure.infinitePi fun x ↦ ℙ).prod (streamMeasure ν)).map (fun ω ↦ ω.1 (n + 1))
    _ = (Measure.infinitePi fun x ↦ ℙ).map (Function.eval (n + 1)) := by
      nth_rw 2 [← Measure.fst_prod (μ := Measure.infinitePi fun x ↦ ℙ)
        (ν := streamMeasure ν)]
      rw [Measure.fst, Measure.map_map (by fun_prop) (by fun_prop)]
      rfl
    _ = ℙ := by rw [Measure.infinitePi_map_eval]
  have : (fun x ↦ (hist alg x n, algFunction alg n (hist alg x n) (x.1 (n + 1)))) =
      (fun p ↦ (p.2, algFunction alg n (p.2) (p.1))) ∘ (fun x ↦ (x.1 (n + 1), hist alg x n)) := rfl
  rw [this, ← Measure.map_map (by fun_prop) (by fun_prop), h_indep]
  have : (ℙ : Measure I).prod ((arrayMeasure ν).map (hist alg · n)) =
      ((Kernel.const _ ℙ) ×ₖ Kernel.id) ∘ₘ ((arrayMeasure ν).map (hist alg · n)) := by
    have h := Measure.compProd_const (μ := (arrayMeasure ν).map (hist alg · n))
      (ν := (ℙ : Measure I))
    rw [Measure.compProd_eq_comp_prod] at h
    rw [← Measure.prod_swap, ← h, ← Measure.deterministic_comp_eq_map (by fun_prop),
      Measure.comp_assoc, ← Kernel.swap, Kernel.swap_prod]
  rw [this, ← Measure.deterministic_comp_eq_map (by fun_prop),
    ← Measure.deterministic_comp_eq_map (by fun_prop), Measure.compProd_eq_comp_prod,
    Measure.comp_assoc, Measure.comp_assoc, Measure.comp_assoc]
  congr 2
  ext ω : 1
  simp only [Kernel.deterministic_comp_eq_map, Kernel.comp_deterministic_eq_comap, Kernel.coe_comap,
    Function.comp_apply]
  rw [Kernel.map_apply _ (by fun_prop), Kernel.prod_apply, Kernel.const_apply, Kernel.id_apply,
    Kernel.prod_apply, Kernel.id_apply, ← h_fun]
  calc (((ℙ).prod (Measure.dirac (hist alg ω n)))).map (fun p ↦ (p.2, algFunction alg n p.2 p.1))
  _ = (((ℙ).prod (Measure.dirac (hist alg ω n))).map Prod.swap).map
      (fun p ↦ (p.1, algFunction alg n p.1 p.2)) := by
    rw [Measure.map_map (by fun_prop) (by fun_prop)]
    rfl
  _ = ((Measure.dirac (hist alg ω n)).prod ℙ).map (fun p ↦ (p.1, algFunction alg n p.1 p.2)) := by
    rw [Measure.prod_swap]
  _ = (Measure.dirac (hist alg ω n)).prod ((ℙ).map (algFunction alg n (hist alg ω n))) := by
    ext s hs
    rw [Measure.map_apply (by fun_prop) hs, Measure.prod_apply, lintegral_dirac, Measure.prod_apply,
      lintegral_dirac, Measure.map_apply (by fun_prop)]
    · congr
    · exact hs.preimage (by fun_prop)
    · exact hs
    · exact hs.preimage (by fun_prop)

`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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`instMeasurableSpaceProbSpace`🔗

InstanceBandits.ArrayModel.instMeasurableSpaceProbSpace

Details

No docstring.

def

Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3, u_4}
  {𝓐 : Type u_3} {R : Type u_4} [MeasurableSpace R] :
  MeasurableSpace (probSpace 𝓐 R)
Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3,
    u_4}
  {𝓐 : Type u_3} {R : Type u_4}
  [MeasurableSpace R] :
  MeasurableSpace (probSpace 𝓐 R)

Code

instance {𝓐 R : Type*} [MeasurableSpace R] : MeasurableSpace (probSpace 𝓐 R)

Type uses (1)

probSpace

Used by (41)

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Proof

inferInstanceAs (MeasurableSpace ((ℕ → I) × (ℕ → 𝓐 → R)))

`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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`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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`arrayMeasure`🔗

DefinitionBandits.ArrayModel.arrayMeasure

Details

Probability measure for the array model of stochastic bandits.

def

Bandits.ArrayModel.arrayMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (probSpace 𝓐 R)
Bandits.ArrayModel.arrayMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (probSpace 𝓐 R)

Code

noncomputable
def arrayMeasure (ν : Kernel 𝓐 R) : Measure (probSpace 𝓐 R) :=
  (Measure.infinitePi fun _ ↦ volume).prod (streamMeasure ν)

Type uses (2)

Body uses (1)

streamMeasure

Used by (29)

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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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`streamMeasure`🔗

DefinitionBandits.streamMeasure

Details

Measure of an infinite stream of rewards from each action.

def

Bandits.streamMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) : MeasureTheory.Measure (ℕ → 𝓐 → R)
Bandits.streamMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (ℕ → 𝓐 → R)

Code

noncomputable
def streamMeasure (ν : Kernel 𝓐 R) : Measure (ℕ → 𝓐 → R) :=
  Measure.infinitePi fun _ ↦ Measure.infinitePi ν

Used by (56)

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LeanMachineLearning exposition

`Bandits.ArrayModel.hasCondDistrib_action'`🔗

`hasCondDistrib_action'`🔗

`Algorithm`🔗

`probSpace`🔗

`instMeasurableSpaceProbSpace`🔗

`action`🔗

`hist`🔗

`arrayMeasure`🔗

`instIsProbabilityMeasureP0`🔗

`initAlgFunction`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`algFunction`🔗

`pullCount'`🔗

`streamMeasure`🔗

Bandits.ArrayModel.hasCondDistrib_action'🔗

hasCondDistrib_action'🔗

Algorithm🔗

probSpace🔗

instMeasurableSpaceProbSpace🔗

action🔗

hist🔗

arrayMeasure🔗

instIsProbabilityMeasureP0🔗

initAlgFunction🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

algFunction🔗

pullCount'🔗

streamMeasure🔗

`Bandits.ArrayModel.hasCondDistrib_action'`🔗

`hasCondDistrib_action'`🔗

`Algorithm`🔗

`probSpace`🔗

`instMeasurableSpaceProbSpace`🔗

`action`🔗

`hist`🔗

`arrayMeasure`🔗

`instIsProbabilityMeasureP0`🔗

`initAlgFunction`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`algFunction`🔗

`pullCount'`🔗

`streamMeasure`🔗