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Learning.feedbackFunZero_detEnvironmentπŸ”—

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feedbackFunZero_detEnvironmentπŸ”—

LemmaLearning.feedbackFunZero_detEnvironment
Details

No docstring.

πŸ”—theorem
Learning.feedbackFunZero_detEnvironment.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {f0 : 𝓐 β†’ 𝓨} {hf0 : Measurable f0}
  {f : (n : β„•) β†’ (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨}
  {hf : βˆ€ (n : β„•), Measurable (f n)}
  [MeasurableSpace.SeparatesPoints 𝓨] :
  feedbackFunZero (detEnvironment f0 hf0 f hf) = f0
Learning.feedbackFunZero_detEnvironment.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {f0 : 𝓐 β†’ 𝓨}
  {hf0 : Measurable f0}
  {f :
    (n : β„•) β†’
      (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨}
  {hf : βˆ€ (n : β„•), Measurable (f n)}
  [MeasurableSpace.SeparatesPoints 𝓨] :
  feedbackFunZero
      (detEnvironment f0 hf0 f hf) =
    f0

Code

lemma feedbackFunZero_detEnvironment [MeasurableSpace.SeparatesPoints 𝓨] :
    feedbackFunZero (detEnvironment f0 hf0 f hf) = f0
Type uses (3)
Body uses (4)

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Proof
by
  simpa [detEnvironment] using (Ξ½0_eq_deterministic (detEnvironment f0 hf0 f hf)).symm

Dependency graph

Type dependencies (3)

feedbackFunZeroπŸ”—

DefinitionLearning.feedbackFunZero
Details

The initial feedback function of a deterministic environment.

πŸ”—def
Learning.feedbackFunZero.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) [h_det : IsDeterministicEnv env] : 𝓐 β†’ 𝓨
Learning.feedbackFunZero.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨)
  [h_det : IsDeterministicEnv env] : 𝓐 β†’ 𝓨

Code

noncomputable
def feedbackFunZero (env : Environment 𝓐 𝓨) [h_det : IsDeterministicEnv env] : 𝓐 β†’ 𝓨 :=
  h_det.exists_f0.choose
Type uses (2)
Used by (6)

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detEnvironmentπŸ”—

DefinitionLearning.detEnvironment
Details

A deterministic environment, where the feedback is given by evaluating fixed measurable functions.

πŸ”—def
Learning.detEnvironment.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} (f0 : 𝓐 β†’ 𝓨)
  (hf0 : Measurable f0)
  (f : (n : β„•) β†’ (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨)
  (hf : βˆ€ (n : β„•), Measurable (f n)) : Environment 𝓐 𝓨
Learning.detEnvironment.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (f0 : 𝓐 β†’ 𝓨)
  (hf0 : Measurable f0)
  (f :
    (n : β„•) β†’
      (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨)
  (hf : βˆ€ (n : β„•), Measurable (f n)) :
  Environment 𝓐 𝓨

Code

noncomputable def detEnvironment
    (f0 : 𝓐 β†’ 𝓨) (hf0 : Measurable f0)
    (f : (n : β„•) β†’ ((Iic n β†’ 𝓐 Γ— 𝓨) Γ— 𝓐) β†’ 𝓨) (hf : βˆ€ n, Measurable (f n)) :
    Environment 𝓐 𝓨 where
  feedback n := (Kernel.deterministic (f n) (hf n))
  Ξ½0 := Kernel.deterministic f0 hf0
Type uses (1)
Used by (3)

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instIsDeterministicEnvDetEnvironmentπŸ”—

InstanceLearning.instIsDeterministicEnvDetEnvironment
Details

No docstring.

πŸ”—theorem
Learning.instIsDeterministicEnvDetEnvironment.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {f0 : 𝓐 β†’ 𝓨} {hf0 : Measurable f0}
  {f : (n : β„•) β†’ (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨}
  {hf : βˆ€ (n : β„•), Measurable (f n)} :
  IsDeterministicEnv (detEnvironment f0 hf0 f hf)
Learning.instIsDeterministicEnvDetEnvironment.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {f0 : 𝓐 β†’ 𝓨}
  {hf0 : Measurable f0}
  {f :
    (n : β„•) β†’
      (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨}
  {hf : βˆ€ (n : β„•), Measurable (f n)} :
  IsDeterministicEnv
    (detEnvironment f0 hf0 f hf)

Code

instance : IsDeterministicEnv (detEnvironment f0 hf0 f hf) where
  exists_f0
Type uses (2)
Body uses (1)
Used by (2)

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Proof
⟨f0, hf0, rfl⟩
  exists_f n := ⟨f n, hf n, rfl⟩

All dependencies, transitively (2)

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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IsDeterministicEnvπŸ”—

Type ClassLearning.IsDeterministicEnv
Details

An environment is deterministic if its initial feedbacks are determined by measurable functions (and not possibly random kernels).

πŸ”—type class
Learning.IsDeterministicEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) : Prop
Learning.IsDeterministicEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) : Prop

Code

class IsDeterministicEnv (env : Environment 𝓐 𝓨) : Prop where
  exists_f0 : βˆƒ (f0 : 𝓐 β†’ 𝓨) (hf0 : Measurable f0), env.Ξ½0 = Kernel.deterministic f0 hf0
  exists_f : βˆ€ n, βˆƒ (f : ((Iic n β†’ 𝓐 Γ— 𝓨) Γ— 𝓐) β†’ 𝓨) (hf : Measurable f),
    env.feedback n = Kernel.deterministic f hf
Type uses (1)
Used by (11)

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