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Learning.IsAlgEnvSeq.condIndepFun_feedback_history_action_actionπŸ”—

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condIndepFun_feedback_history_action_actionπŸ”—

LemmaLearning.IsAlgEnvSeq.condIndepFun_feedback_history_action_action
Details

No docstring.

πŸ”—theorem
Learning.IsAlgEnvSeq.condIndepFun_feedback_history_action_action.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ξ© : Type u_3} {mΞ© : MeasurableSpace Ξ©}
  {alg : Algorithm 𝓐 𝓨} {Ξ½ : ProbabilityTheory.Kernel 𝓐 𝓨}
  [ProbabilityTheory.IsMarkovKernel Ξ½] {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P] {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  [StandardBorelSpace Ξ©] [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  (h : IsAlgEnvSeq A Y alg (stationaryEnv Ξ½) P) (n : β„•) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap (A (n + 1)) inferInstance) β‹― (Y (n + 1))
    (fun Ο‰ => (history A Y n Ο‰, A (n + 1) Ο‰)) P
Learning.IsAlgEnvSeq.condIndepFun_feedback_history_action_action.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ξ© : Type u_3}
  {mΞ© : MeasurableSpace Ξ©}
  {alg : Algorithm 𝓐 𝓨}
  {Ξ½ : ProbabilityTheory.Kernel 𝓐 𝓨}
  [ProbabilityTheory.IsMarkovKernel Ξ½]
  {P : MeasureTheory.Measure Ξ©}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : β„• β†’ Ξ© β†’ 𝓐} {Y : β„• β†’ Ξ© β†’ 𝓨}
  [StandardBorelSpace Ξ©]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  (h :
    IsAlgEnvSeq A Y alg (stationaryEnv Ξ½)
      P)
  (n : β„•) :
  ProbabilityTheory.CondIndepFun
    (MeasurableSpace.comap (A (n + 1))
      inferInstance)
    β‹― (Y (n + 1))
    (fun Ο‰ =>
      (history A Y n Ο‰, A (n + 1) Ο‰))
    P

Code

lemma condIndepFun_feedback_history_action_action [StandardBorelSpace Ξ©]
    [StandardBorelSpace 𝓐] [Nonempty 𝓐] [StandardBorelSpace 𝓨] [Nonempty 𝓨]
    (h : IsAlgEnvSeq A Y alg (stationaryEnv Ξ½) P) (n : β„•) :
    Y (n + 1) βŸ‚α΅’[A (n + 1), h.measurable_action (n + 1); P]
      (fun Ο‰ ↦ (history A Y n Ο‰, A (n + 1) Ο‰))
Type uses (5)
Body uses (2)

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Proof
IsObliviousEnv.condIndepFun_feedback_history_action_action h n

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)
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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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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]
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All dependencies, transitively (1)

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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