LeanMachineLearning exposition

Learning.isAlgEnvSeq_unique๐Ÿ”—

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isAlgEnvSeq_unique๐Ÿ”—

LemmaLearning.isAlgEnvSeq_unique
Details

The law of the sequence of actions and observations generated by an algorithm-environment pair is unique: it does not depend on the probability space used.

๐Ÿ”—theorem
Learning.isAlgEnvSeq_unique.{u_1, u_2, u_4, u_5} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  {ฮฉ : Type u_4} {ฮฉ' : Type u_5} {mฮฉ : MeasurableSpace ฮฉ}
  {mฮฉ' : MeasurableSpace ฮฉ'} {alg : Algorithm ๐“ ๐“จ}
  {env : Environment ๐“ ๐“จ} {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P] {P' : MeasureTheory.Measure ฮฉ'}
  [MeasureTheory.IsProbabilityMeasure P'] {Aโ‚ : โ„• โ†’ ฮฉ โ†’ ๐“}
  {Rโ‚ : โ„• โ†’ ฮฉ โ†’ ๐“จ} {Aโ‚‚ : โ„• โ†’ ฮฉ' โ†’ ๐“} {Rโ‚‚ : โ„• โ†’ ฮฉ' โ†’ ๐“จ}
  (h1 : IsAlgEnvSeq Aโ‚ Rโ‚ alg env P)
  (h2 : IsAlgEnvSeq Aโ‚‚ Rโ‚‚ alg env P') :
  MeasureTheory.Measure.map (trajectory Aโ‚ Rโ‚) P =
    MeasureTheory.Measure.map (trajectory Aโ‚‚ Rโ‚‚) P'
Learning.isAlgEnvSeq_unique.{u_1, u_2,
    u_4, u_5}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {ฮฉ : Type u_4}
  {ฮฉ' : Type u_5} {mฮฉ : MeasurableSpace ฮฉ}
  {mฮฉ' : MeasurableSpace ฮฉ'}
  {alg : Algorithm ๐“ ๐“จ}
  {env : Environment ๐“ ๐“จ}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {P' : MeasureTheory.Measure ฮฉ'}
  [MeasureTheory.IsProbabilityMeasure P']
  {Aโ‚ : โ„• โ†’ ฮฉ โ†’ ๐“} {Rโ‚ : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  {Aโ‚‚ : โ„• โ†’ ฮฉ' โ†’ ๐“} {Rโ‚‚ : โ„• โ†’ ฮฉ' โ†’ ๐“จ}
  (h1 : IsAlgEnvSeq Aโ‚ Rโ‚ alg env P)
  (h2 : IsAlgEnvSeq Aโ‚‚ Rโ‚‚ alg env P') :
  MeasureTheory.Measure.map
      (trajectory Aโ‚ Rโ‚) P =
    MeasureTheory.Measure.map
      (trajectory Aโ‚‚ Rโ‚‚) P'

Code

lemma isAlgEnvSeq_unique (h1 : IsAlgEnvSeq Aโ‚ Rโ‚ alg env P)
    (h2 : IsAlgEnvSeq Aโ‚‚ Rโ‚‚ alg env P') :
    P.map (trajectory Aโ‚ Rโ‚) = P'.map (trajectory Aโ‚‚ Rโ‚‚)
Type uses (4)
Body uses (2)
Used by (3)

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Proof
by
  rw [h1.map_trajectory, h2.map_trajectory]

Dependency graph

Type dependencies (4)

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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Environment๐Ÿ”—

StructureLearning.Environment
Details

A stochastic environment.

๐Ÿ”—structure
Learning.Environment.{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.Environment.{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 Environment (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : โ„•) โ†’ Kernel ((Iic n โ†’ ๐“ ร— ๐“จ) ร— ๐“) ๐“จ
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : โˆ€ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ฮฝ0 : Kernel ๐“ ๐“จ
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ฮฝ0]
Used by (128)

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IsAlgEnvSeq๐Ÿ”—

StructureLearning.IsAlgEnvSeq
Details

An algorithm-environment sequence: a sequence of actions and feedbacks generated by an algorithm interacting with an environment.

๐Ÿ”—structure
Learning.IsAlgEnvSeq.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ)
  (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ)
  (P : MeasureTheory.Measure ฮฉ) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsAlgEnvSeq.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (alg : Algorithm ๐“ ๐“จ)
  (env : Environment ๐“ ๐“จ)
  (P : MeasureTheory.Measure ฮฉ)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsAlgEnvSeq
    (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ)
    (P : Measure ฮฉ) [IsFiniteMeasure P] : Prop where
  /-- The action sequence is measurable. -/
  measurable_action n : Measurable (A n) := by fun_prop
  /-- The feedback sequence is measurable. -/
  measurable_feedback n : Measurable (Y n) := by fun_prop
  /-- The first action has the correct law. -/
  hasLaw_action_zero : HasLaw (fun ฯ‰ โ†ฆ (A 0 ฯ‰)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.ฮฝ0 P
  /-- The next action has the correct conditional distribution given the history. -/
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (history A Y n) (alg.policy n) P
  /-- The next feedback has the correct conditional distribution given the history and
  next action. -/
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun ฯ‰ โ†ฆ (history A Y n ฯ‰, A (n + 1) ฯ‰))
      (env.feedback n) P
Type uses (3)
Used by (111)

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trajectory๐Ÿ”—

DefinitionLearning.trajectory
Details

A random variable that gives the sequence of action-feedback pairs.

๐Ÿ”—def
Learning.trajectory.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (ฯ‰ : ฮฉ) : โ„• โ†’ ๐“ ร— ๐“จ
Learning.trajectory.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (ฯ‰ : ฮฉ) : โ„• โ†’ ๐“ ร— ๐“จ

Code

def trajectory (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (ฯ‰ : ฮฉ) : โ„• โ†’ ๐“ ร— ๐“จ := fun n โ†ฆ (A n ฯ‰, Y n ฯ‰)
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All dependencies, transitively (1)

history๐Ÿ”—

DefinitionLearning.history
Details

History of the algorithm-environment sequence up to time n.

๐Ÿ”—def
Learning.history.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ
Learning.history.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ

Code

def history (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) : Iic n โ†’ ๐“ ร— ๐“จ :=
  fun i โ†ฆ (A i ฯ‰, Y i ฯ‰)
Used by (72)

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