LeanMachineLearning exposition

Learning.feedbackFunπŸ”—

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Minimal Lean file

feedbackFunπŸ”—

DefinitionLearning.feedbackFun
Details

The feedback function of a deterministic environment at step n.

πŸ”—def
Learning.feedbackFun.{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] (n : β„•) :
  (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨
Learning.feedbackFun.{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]
  (n : β„•) :
  (β†₯(Finset.Iic n) β†’ 𝓐 Γ— 𝓨) Γ— 𝓐 β†’ 𝓨

Code

noncomputable
def feedbackFun (env : Environment 𝓐 𝓨) [h_det : IsDeterministicEnv env] (n : β„•) :
    ((Iic n β†’ 𝓐 Γ— 𝓨) Γ— 𝓐) β†’ 𝓨 :=
  (h_det.exists_f n).choose
Type uses (2)
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Dependency graph

Type dependencies (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]
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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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