LeanMachineLearning exposition

Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards๐Ÿ”—

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identDistrib_pullCount_sumRewards๐Ÿ”—

LemmaLearning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards
Details

No docstring.

๐Ÿ”—theorem
Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards.{u_1, u_2, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_2} {ฮฉ' : Type u_3} [DecidableEq ๐“]
  {m๐“ : MeasurableSpace ๐“} {mฮฉ : MeasurableSpace ฮฉ}
  {mฮฉ' : MeasurableSpace ฮฉ'} {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P] {P' : MeasureTheory.Measure ฮฉ'}
  [MeasureTheory.IsProbabilityMeasure P'] {alg : Algorithm ๐“ โ„}
  {ฮฝ : ProbabilityTheory.Kernel ๐“ โ„}
  [ProbabilityTheory.IsMarkovKernel ฮฝ] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {R : โ„• โ†’ ฮฉ โ†’ โ„}
  {Aโ‚‚ : โ„• โ†’ ฮฉ' โ†’ ๐“} {Rโ‚‚ : โ„• โ†’ ฮฉ' โ†’ โ„} [MeasurableSingletonClass ๐“]
  (h1 : IsAlgEnvSeq A R alg (stationaryEnv ฮฝ) P)
  (h2 : IsAlgEnvSeq Aโ‚‚ Rโ‚‚ alg (stationaryEnv ฮฝ) P') :
  ProbabilityTheory.IdentDistrib
    (fun ฯ‰ n a => (pullCount A a n ฯ‰, sumRewards A R a n ฯ‰))
    (fun ฯ‰' n a => (pullCount Aโ‚‚ a n ฯ‰', sumRewards Aโ‚‚ Rโ‚‚ a n ฯ‰')) P P'
Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_2}
  {ฮฉ' : Type u_3} [DecidableEq ๐“]
  {m๐“ : MeasurableSpace ๐“}
  {mฮฉ : MeasurableSpace ฮฉ}
  {mฮฉ' : MeasurableSpace ฮฉ'}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsProbabilityMeasure P]
  {P' : MeasureTheory.Measure ฮฉ'}
  [MeasureTheory.IsProbabilityMeasure P']
  {alg : Algorithm ๐“ โ„}
  {ฮฝ : ProbabilityTheory.Kernel ๐“ โ„}
  [ProbabilityTheory.IsMarkovKernel ฮฝ]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {R : โ„• โ†’ ฮฉ โ†’ โ„}
  {Aโ‚‚ : โ„• โ†’ ฮฉ' โ†’ ๐“} {Rโ‚‚ : โ„• โ†’ ฮฉ' โ†’ โ„}
  [MeasurableSingletonClass ๐“]
  (h1 :
    IsAlgEnvSeq A R alg (stationaryEnv ฮฝ)
      P)
  (h2 :
    IsAlgEnvSeq Aโ‚‚ Rโ‚‚ alg
      (stationaryEnv ฮฝ) P') :
  ProbabilityTheory.IdentDistrib
    (fun ฯ‰ n a =>
      (pullCount A a n ฯ‰,
        sumRewards A R a n ฯ‰))
    (fun ฯ‰' n a =>
      (pullCount Aโ‚‚ a n ฯ‰',
        sumRewards Aโ‚‚ Rโ‚‚ a n ฯ‰'))
    P P'

Code

lemma _root_.Learning.IsAlgEnvSeq.identDistrib_pullCount_sumRewards [MeasurableSingletonClass ๐“]
    (h1 : IsAlgEnvSeq A R alg (stationaryEnv ฮฝ) P)
    (h2 : IsAlgEnvSeq Aโ‚‚ Rโ‚‚ alg (stationaryEnv ฮฝ) P') :
    IdentDistrib (fun ฯ‰ n a โ†ฆ (pullCount A a n ฯ‰, sumRewards A R a n ฯ‰))
      (fun ฯ‰' n a โ†ฆ (pullCount Aโ‚‚ a n ฯ‰', sumRewards Aโ‚‚ Rโ‚‚ a n ฯ‰')) P P'
Type uses (5)
Body uses (2)
Used by (1)

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Proof
by
  let f (ฯ„ : โ„• โ†’ ๐“ ร— โ„) (n : โ„•) (a : ๐“) : โ„• ร— โ„ :=
    (โˆ‘ i โˆˆ range n, if (ฯ„ i).1 = a then 1 else 0,
     โˆ‘ i โˆˆ range n, if (ฯ„ i).1 = a then (ฯ„ i).2 else 0)
  have hc1 : (fun ฯ‰ n a โ†ฆ (pullCount A a n ฯ‰, sumRewards A R a n ฯ‰)) =
      f โˆ˜ (trajectory A R) := by
    ext ฯ‰ n a : 3
    simp_rw [Function.comp, f, pullCount, card_filter, sumRewards, trajectory]
    rfl
  have hc2 : (fun ฯ‰' n a โ†ฆ (pullCount Aโ‚‚ a n ฯ‰', sumRewards Aโ‚‚ Rโ‚‚ a n ฯ‰')) =
      f โˆ˜ (trajectory Aโ‚‚ Rโ‚‚) := by
    ext ฯ‰' n a : 3
    simp_rw [Function.comp, f, pullCount, card_filter, sumRewards, trajectory]
    rfl
  have hf : Measurable f := by
    simp_rw [f, measurable_pi_iff]
    intro n a
    apply Measurable.prod
    ยท dsimp only
      exact measurable_sum _
        (fun _ _ โ†ฆ Measurable.ite (by measurability) (by fun_prop) (by fun_prop))
    ยท dsimp only
      exact measurable_sum _
        (fun _ _ โ†ฆ Measurable.ite (by measurability) (by fun_prop) (by fun_prop))
  rw [hc1, hc2]
  exact (h1.identDistrib_trajectory h2).comp hf

Dependency graph

Type dependencies (5)

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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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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stationaryEnv๐Ÿ”—

DefinitionLearning.stationaryEnv
Details

A stationary environment, in which the distribution of the next feedback depends only on the last action.

๐Ÿ”—def
Learning.stationaryEnv.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ ๐“จ)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] : Environment ๐“ ๐“จ
Learning.stationaryEnv.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ ๐“จ)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] :
  Environment ๐“ ๐“จ

Code

def stationaryEnv (ฮฝ : Kernel ๐“ ๐“จ) [IsMarkovKernel ฮฝ] : Environment ๐“ ๐“จ := obliviousEnv fun _ โ†ฆ ฮฝ
Type uses (1)
Body uses (1)
Used by (81)

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pullCount๐Ÿ”—

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

๐Ÿ”—def
Learning.pullCount.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„•
Learning.pullCount.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (t : โ„•) (ฯ‰ : ฮฉ) : โ„•

Code

noncomputable
def pullCount (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„• :=
  #(filter (fun s โ†ฆ A s ฯ‰ = a) (range t))
Used by (146)

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sumRewards๐Ÿ”—

DefinitionLearning.sumRewards
Details

Sum of rewards obtained when pulling action a up to time t (exclusive).

๐Ÿ”—def
Learning.sumRewards.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•)
  (ฯ‰ : ฮฉ) : โ„
Learning.sumRewards.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•)
  (ฯ‰ : ฮฉ) : โ„

Code

def sumRewards (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„ :=
  โˆ‘ s โˆˆ range t, if A s ฯ‰ = a then R' s ฯ‰ else 0
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All dependencies, transitively (3)

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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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‰)
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obliviousEnv๐Ÿ”—

DefinitionLearning.obliviousEnv
Details

An oblivious environment, in which the distribution of the next feedback depends only on the last action, but in a possibly time-dependent manner.

๐Ÿ”—def
Learning.obliviousEnv.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (ฮฝ : โ„• โ†’ ProbabilityTheory.Kernel ๐“ ๐“จ)
  [โˆ€ (n : โ„•), ProbabilityTheory.IsMarkovKernel (ฮฝ n)] : Environment ๐“ ๐“จ
Learning.obliviousEnv.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (ฮฝ : โ„• โ†’ ProbabilityTheory.Kernel ๐“ ๐“จ)
  [โˆ€ (n : โ„•),
      ProbabilityTheory.IsMarkovKernel
        (ฮฝ n)] :
  Environment ๐“ ๐“จ

Code

def obliviousEnv (ฮฝ : โ„• โ†’ Kernel ๐“ ๐“จ) [โˆ€ n, IsMarkovKernel (ฮฝ n)] : Environment ๐“ ๐“จ where
  feedback n := (ฮฝ (n + 1)).prodMkLeft _
  ฮฝ0 := ฮฝ 0
Type uses (1)
Used by (10)

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