LeanMachineLearning exposition

Learning.instIsObliviousEnvStationaryEnvπŸ”—

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instIsObliviousEnvStationaryEnvπŸ”—

InstanceLearning.instIsObliviousEnvStationaryEnv
Details

No docstring.

πŸ”—theorem
Learning.instIsObliviousEnvStationaryEnv.{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] :
  IsObliviousEnv (stationaryEnv Ξ½)
Learning.instIsObliviousEnvStationaryEnv.{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] :
  IsObliviousEnv (stationaryEnv Ξ½)

Code

instance (Ξ½ : Kernel 𝓐 𝓨) [IsMarkovKernel Ξ½] : IsObliviousEnv (stationaryEnv Ξ½) where
  exists_eq_prodMkLeft
Type uses (2)
Body uses (1)
Used by (5)

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Proof
⟨fun _ ↦ Ξ½, inferInstance, rfl, fun _ ↦ rfl⟩

Dependency graph

Type dependencies (2)

IsObliviousEnvπŸ”—

Type ClassLearning.IsObliviousEnv
Details

An environment is oblivious if the distribution of the next feedback depends only on the last action and not on the past history.

πŸ”—type class
Learning.IsObliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) : Prop
Learning.IsObliviousEnv.{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 IsObliviousEnv (env : Environment 𝓐 𝓨) : Prop where
  exists_eq_prodMkLeft : βˆƒ Ξ½ : β„• β†’ Kernel 𝓐 𝓨, (βˆ€ n, IsMarkovKernel (Ξ½ n)) ∧
    (env.Ξ½0 = Ξ½ 0) ∧ (βˆ€ n, env.feedback n = (Ξ½ (n + 1)).prodMkLeft _)
Type uses (1)
Used by (13)

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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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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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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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