LeanMachineLearning exposition

Learning.feedback_eq_deterministic🔗

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Minimal Lean file

feedback_eq_deterministic🔗

LemmaLearning.feedback_eq_deterministic
Details

No docstring.

🔗theorem
Learning.feedback_eq_deterministic.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) [IsDeterministicEnv env] (n : ℕ) :
  Environment.feedback env n =
    ProbabilityTheory.Kernel.deterministic (feedbackFun env n) ⋯
Learning.feedback_eq_deterministic.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨)
  [IsDeterministicEnv env] (n : ℕ) :
  Environment.feedback env n =
    ProbabilityTheory.Kernel.deterministic
      (feedbackFun env n) ⋯

Code

lemma feedback_eq_deterministic (env : Environment 𝓐 𝓨) [IsDeterministicEnv env] (n : ℕ) :
    env.feedback n = Kernel.deterministic (feedbackFun env n) (measurable_feedbackFun env n)
Type uses (4)
Used by (3)

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Proof
(IsDeterministicEnv.exists_f n).choose_spec.choose_spec

Dependency graph

Type dependencies (4)

Environment🔗

StructureLearning.Environment
Details

A stochastic environment.

🔗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 : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ν0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ν0]
Used by (128)

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IsDeterministicEnv🔗

Type ClassLearning.IsDeterministicEnv
Details

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

🔗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 : ∃ (f0 : 𝓐 → 𝓨) (hf0 : Measurable f0), env.ν0 = Kernel.deterministic f0 hf0
  exists_f : ∀ n, ∃ (f : ((Iic n → 𝓐 × 𝓨) × 𝓐) → 𝓨) (hf : Measurable f),
    env.feedback n = Kernel.deterministic f hf
Type uses (1)
Used by (11)

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feedbackFun🔗

DefinitionLearning.feedbackFun
Details

The feedback function of a deterministic environment at step n.

🔗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 : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × 𝓨) × 𝓐 → 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 : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × 𝓨) × 𝓐 → 𝓨

Code

noncomputable
def feedbackFun (env : Environment 𝓐 𝓨) [h_det : IsDeterministicEnv env] (n : ℕ) :
    ((Iic n → 𝓐 × 𝓨) × 𝓐) → 𝓨 :=
  (h_det.exists_f n).choose
Type uses (2)
Used by (6)

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measurable_feedbackFun🔗

LemmaLearning.measurable_feedbackFun
Details

No docstring.

🔗theorem
Learning.measurable_feedbackFun.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) [IsDeterministicEnv env] (n : ℕ) :
  Measurable (feedbackFun env n)
Learning.measurable_feedbackFun.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨)
  [IsDeterministicEnv env] (n : ℕ) :
  Measurable (feedbackFun env n)

Code

lemma measurable_feedbackFun (env : Environment 𝓐 𝓨) [IsDeterministicEnv env] (n : ℕ) :
    Measurable (feedbackFun env n)
Type uses (3)
Used by (4)

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Proof
(IsDeterministicEnv.exists_f n).choose_spec.choose