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Bandits.condDistrib_reward''πŸ”—

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condDistrib_reward''πŸ”—

LemmaBandits.condDistrib_reward''
Details

No docstring.

πŸ”—theorem
Bandits.condDistrib_reward''.{u_1, u_2} {𝓐 : Type u_1} {Ξ© : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mΞ© : MeasurableSpace Ξ©} {A : β„• β†’ Ξ© β†’ 𝓐}
  {R : β„• β†’ Ξ© β†’ ℝ} {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {Ξ½ : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel Ξ½] [Countable 𝓐]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv Ξ½) P)
  (n : β„•) :
  ⇑𝓛[fun Ο‰ => R n (Prod.fst Ο‰) | fun Ο‰ => A n (Prod.fst Ο‰);
        MeasureTheory.Measure.prod P
          (streamMeasure
            Ξ½)] =ᡐ[MeasureTheory.Measure.map (fun Ο‰ => A n (Prod.fst Ο‰))
      (MeasureTheory.Measure.prod P (streamMeasure Ξ½))]
    ⇑ν
Bandits.condDistrib_reward''.{u_1, u_2}
  {𝓐 : Type u_1} {Ξ© : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mΞ© : MeasurableSpace Ξ©} {A : β„• β†’ Ξ© β†’ 𝓐}
  {R : β„• β†’ Ξ© β†’ ℝ}
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {Ξ½ : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel Ξ½]
  [Countable 𝓐]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv Ξ½) P)
  (n : β„•) :
  ⇑𝓛[fun Ο‰ => R n (Prod.fst Ο‰) | fun Ο‰ =>
        A n (Prod.fst Ο‰);
        MeasureTheory.Measure.prod P
          (streamMeasure
            ν)] =ᡐ[MeasureTheory.Measure.map
      (fun Ο‰ => A n (Prod.fst Ο‰))
      (MeasureTheory.Measure.prod P
        (streamMeasure Ξ½))]
    ⇑ν

Code

lemma condDistrib_reward'' [Countable 𝓐]
    (h : IsAlgEnvSeq A R alg (stationaryEnv Ξ½) P) (n : β„•) :
    𝓛[fun Ο‰ ↦ R n Ο‰.1 | fun Ο‰ ↦ A n Ο‰.1; 𝔓] =ᡐ[(𝔓).map (fun Ο‰ ↦ A n Ο‰.1)] Ξ½
Type uses (5)
Body uses (2)
Used by (1)

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Proof
by
  have hA := h.measurable_action
  have hR := h.measurable_feedback
  have h_ra' : 𝓛[R n | A n; P] =ᡐ[P.map (A n)] Ξ½ := h.condDistrib_feedback_stationaryEnv n
  have h_law : (𝔓).map (fun Ο‰ ↦ A n Ο‰.1) = P.map (A n) := by
    change ((𝔓).map (A n ∘ Prod.fst)) = _
    rw [← Measure.map_map (by fun_prop) (by fun_prop), ← Measure.fst, Measure.fst_prod]
  rw [h_law]
  have h_prod : 𝓛[fun Ο‰ ↦ R n Ο‰.1 | fun Ο‰ ↦ A n Ο‰.1; 𝔓]
      =ᡐ[P.map (A n)] 𝓛[R n | A n; P] :=
    condDistrib_fst_prod _ (by fun_prop) _
  filter_upwards [h_ra', h_prod] with Ο‰ h_eq h_prod
  rw [h_prod, h_eq]

Dependency graph

Type dependencies (5)

AlgorithmπŸ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

πŸ”—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 : β„•) β†’ Kernel (Iic n β†’ 𝓐 Γ— 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : βˆ€ 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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IsAlgEnvSeqπŸ”—

StructureLearning.IsAlgEnvSeq
Details

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

πŸ”—structure
Learning.IsAlgEnvSeq.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {mΞ© : MeasurableSpace Ξ©} (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨)
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ξ©) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsAlgEnvSeq.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΞ© : MeasurableSpace Ξ©} (A : β„• β†’ Ξ© β†’ 𝓐)
  (Y : β„• β†’ Ξ© β†’ 𝓨) (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ξ©)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsAlgEnvSeq
    (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
    (P : Measure Ξ©) [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 Ο‰ ↦ (A 0 Ο‰)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.Ξ½0 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 Ο‰ ↦ (history A Y n Ο‰, A (n + 1) Ο‰))
      (env.feedback n) P
Type uses (3)
Used by (111)

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stationaryEnvπŸ”—

DefinitionLearning.stationaryEnv
Details

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

πŸ”—def
Learning.stationaryEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : ProbabilityTheory.Kernel 𝓐 𝓨)
  [ProbabilityTheory.IsMarkovKernel Ξ½] : Environment 𝓐 𝓨
Learning.stationaryEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : ProbabilityTheory.Kernel 𝓐 𝓨)
  [ProbabilityTheory.IsMarkovKernel Ξ½] :
  Environment 𝓐 𝓨

Code

def stationaryEnv (Ξ½ : Kernel 𝓐 𝓨) [IsMarkovKernel Ξ½] : Environment 𝓐 𝓨 := obliviousEnv fun _ ↦ Ξ½
Type uses (1)
Body uses (1)
Used by (81)

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streamMeasureπŸ”—

DefinitionBandits.streamMeasure
Details

Measure of an infinite stream of rewards from each action.

πŸ”—def
Bandits.streamMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (Ξ½ : ProbabilityTheory.Kernel 𝓐 R) : MeasureTheory.Measure (β„• β†’ 𝓐 β†’ R)
Bandits.streamMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (Ξ½ : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (β„• β†’ 𝓐 β†’ R)

Code

noncomputable
def streamMeasure (Ξ½ : Kernel 𝓐 R) : Measure (β„• β†’ 𝓐 β†’ R) :=
  Measure.infinitePi fun _ ↦ Measure.infinitePi Ξ½
Used by (56)

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instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernelπŸ”—

InstanceBandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel
Details

No docstring.

πŸ”—theorem
Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} (Ξ½ : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel Ξ½] :
  MeasureTheory.IsProbabilityMeasure (streamMeasure Ξ½)
Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (Ξ½ : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel Ξ½] :
  MeasureTheory.IsProbabilityMeasure
    (streamMeasure Ξ½)

Code

instance (Ξ½ : Kernel 𝓐 R) [IsMarkovKernel Ξ½] : IsProbabilityMeasure (streamMeasure Ξ½)
Type uses (1)
Used by (22)

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Proof
by
  unfold streamMeasure
  infer_instance

All dependencies, transitively (3)

EnvironmentπŸ”—

StructureLearning.Environment
Details

A stochastic environment.

πŸ”—structure
Learning.Environment.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Environment.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : β„•) β†’ Kernel ((Iic n β†’ 𝓐 Γ— 𝓨) Γ— 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : βˆ€ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  Ξ½0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel Ξ½0]
Used by (128)

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historyπŸ”—

DefinitionLearning.history
Details

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

πŸ”—def
Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) :
  β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ξ© : Type u_3} (A : β„• β†’ Ξ© β†’ 𝓐)
  (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) :
  β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨

Code

def history (A : β„• β†’ Ξ© β†’ 𝓐) (Y : β„• β†’ Ξ© β†’ 𝓨) (n : β„•) (Ο‰ : Ξ©) : Iic n β†’ 𝓐 Γ— 𝓨 :=
  fun i ↦ (A i Ο‰, Y i Ο‰)
Used by (72)

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obliviousEnvπŸ”—

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.

πŸ”—def
Learning.obliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : β„• β†’ ProbabilityTheory.Kernel 𝓐 𝓨)
  [βˆ€ (n : β„•), ProbabilityTheory.IsMarkovKernel (Ξ½ n)] : Environment 𝓐 𝓨
Learning.obliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (Ξ½ : β„• β†’ ProbabilityTheory.Kernel 𝓐 𝓨)
  [βˆ€ (n : β„•),
      ProbabilityTheory.IsMarkovKernel
        (Ξ½ n)] :
  Environment 𝓐 𝓨

Code

def obliviousEnv (Ξ½ : β„• β†’ Kernel 𝓐 𝓨) [βˆ€ n, IsMarkovKernel (Ξ½ n)] : Environment 𝓐 𝓨 where
  feedback n := (Ξ½ (n + 1)).prodMkLeft _
  Ξ½0 := Ξ½ 0
Type uses (1)
Used by (10)

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