LeanMachineLearning exposition

Learning.detAlgorithm_policy๐Ÿ”—

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detAlgorithm_policy๐Ÿ”—

LemmaLearning.detAlgorithm_policy
Details

No docstring.

๐Ÿ”—theorem
Learning.detAlgorithm_policy.{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“) (n : โ„•) :
  Algorithm.policy (detAlgorithm nextA h_next action0) n =
    ProbabilityTheory.Kernel.deterministic (nextA n) โ‹ฏ
Learning.detAlgorithm_policy.{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“) (n : โ„•) :
  Algorithm.policy
      (detAlgorithm nextA h_next action0)
      n =
    ProbabilityTheory.Kernel.deterministic
      (nextA n) โ‹ฏ

Code

theorem detAlgorithm_policy : โˆ€ {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“) (n : โ„•),
  (Learning.detAlgorithm nextA h_next action0).policy n = ProbabilityTheory.Kernel.deterministic (nextA n) โ‹ฏ
Type uses (2)
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Proof
@[simps]

Dependency graph

Type dependencies (2)

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)
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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]
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