LeanMachineLearning exposition

Learning.actionZero_detAlgorithm๐Ÿ”—

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Minimal Lean file

actionZero_detAlgorithm๐Ÿ”—

LemmaLearning.actionZero_detAlgorithm
Details

No docstring.

๐Ÿ”—theorem
Learning.actionZero_detAlgorithm.{u_1, u_2} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  {nextA : (n : โ„•) โ†’ (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“}
  {h_next : โˆ€ (n : โ„•), Measurable (nextA n)} {action0 : ๐“}
  [MeasurableSpace.SeparatesPoints ๐“] :
  actionZero (detAlgorithm nextA h_next action0) = action0
Learning.actionZero_detAlgorithm.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {nextA :
    (n : โ„•) โ†’
      (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“}
  {h_next :
    โˆ€ (n : โ„•), Measurable (nextA n)}
  {action0 : ๐“}
  [MeasurableSpace.SeparatesPoints ๐“] :
  actionZero
      (detAlgorithm nextA h_next
        action0) =
    action0

Code

lemma actionZero_detAlgorithm [MeasurableSpace.SeparatesPoints ๐“] :
    actionZero (detAlgorithm nextA h_next action0) = action0
Type uses (3)
Body uses (3)
Used by (5)

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Proof
by
  have h_eq := p0_eq_dirac (detAlgorithm nextA h_next action0)
  simp only [detAlgorithm] at h_eq
  rw [dirac_eq_dirac_iff] at h_eq
  exact h_eq.symm

Dependency graph

Type dependencies (3)

actionZero๐Ÿ”—

DefinitionLearning.actionZero
Details

The initial action of a deterministic algorithm.

๐Ÿ”—def
Learning.actionZero.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) [h_det : IsDeterministicAlg alg] : ๐“
Learning.actionZero.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  [h_det : IsDeterministicAlg alg] : ๐“

Code

noncomputable
def actionZero (alg : Algorithm ๐“ ๐“จ) [h_det : IsDeterministicAlg alg] : ๐“ :=
  h_det.exists_action0.choose
Type uses (2)
Used by (12)

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detAlgorithm๐Ÿ”—

DefinitionLearning.detAlgorithm
Details

A deterministic algorithm, which chooses the action given by the function nextAction.

๐Ÿ”—def
Learning.detAlgorithm.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (nextA : (n : โ„•) โ†’ (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“)
  (h_next : โˆ€ (n : โ„•), Measurable (nextA n)) (action0 : ๐“) :
  Algorithm ๐“ ๐“จ
Learning.detAlgorithm.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (nextA :
    (n : โ„•) โ†’
      (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“)
  (h_next :
    โˆ€ (n : โ„•), Measurable (nextA n))
  (action0 : ๐“) : Algorithm ๐“ ๐“จ

Code

noncomputable
def detAlgorithm (nextA : (n : โ„•) โ†’ (Iic n โ†’ ๐“ ร— ๐“จ) โ†’ ๐“)
    (h_next : โˆ€ n, Measurable (nextA n)) (action0 : ๐“) :
    Algorithm ๐“ ๐“จ where
  policy n := Kernel.deterministic (nextA n) (h_next n)
  p0 := Measure.dirac action0
Type uses (1)
Used by (15)

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instIsDeterministicAlgDetAlgorithm๐Ÿ”—

InstanceLearning.instIsDeterministicAlgDetAlgorithm
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsDeterministicAlgDetAlgorithm.{u_1, u_2} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  {nextA : (n : โ„•) โ†’ (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“}
  {h_next : โˆ€ (n : โ„•), Measurable (nextA n)} {action0 : ๐“} :
  IsDeterministicAlg (detAlgorithm nextA h_next action0)
Learning.instIsDeterministicAlgDetAlgorithm.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {nextA :
    (n : โ„•) โ†’
      (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“}
  {h_next :
    โˆ€ (n : โ„•), Measurable (nextA n)}
  {action0 : ๐“} :
  IsDeterministicAlg
    (detAlgorithm nextA h_next action0)

Code

instance : IsDeterministicAlg (detAlgorithm nextA h_next action0) where
  exists_action0
Type uses (2)
Body uses (1)
Used by (9)

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Proof
โŸจaction0, rflโŸฉ
  exists_nextAction n := โŸจnextA n, h_next n, rflโŸฉ

All dependencies, transitively (2)

Algorithm๐Ÿ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

๐Ÿ”—structure
Learning.Algorithm.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Algorithm (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : โ„•) โ†’ Kernel (Iic n โ†’ ๐“ ร— ๐“จ) ๐“
  /-- The policy is a Markov kernel. -/
  [h_policy : โˆ€ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure ๐“
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
Used by (216)

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IsDeterministicAlg๐Ÿ”—

Type ClassLearning.IsDeterministicAlg
Details

An algorithm is deterministic if its initial action and subsequent actions are determined by measurable functions (and not possibly random kernels).

๐Ÿ”—type class
Learning.IsDeterministicAlg.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) : Prop
Learning.IsDeterministicAlg.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) : Prop

Code

class IsDeterministicAlg (alg : Algorithm ๐“ ๐“จ) : Prop where
  exists_action0 : โˆƒ action0, alg.p0 = Measure.dirac action0
  exists_nextAction n : โˆƒ (nextAction : (Iic n โ†’ ๐“ ร— ๐“จ) โ†’ ๐“) (h_meas : Measurable nextAction),
    alg.policy n = Kernel.deterministic nextAction h_meas
Type uses (1)
Used by (14)

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