`Learning.IsAlgEnvSeqUntil.hasLaw_action_zero_detAlgorithm`🔗

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Minimal Lean file

`hasLaw_action_zero_detAlgorithm`🔗

LemmaLearning.IsAlgEnvSeqUntil.hasLaw_action_zero_detAlgorithm

Details

No docstring.

theorem

Learning.IsAlgEnvSeqUntil.hasLaw_action_zero_detAlgorithm.{u_1, u_2,
    u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {nextA : (n : ℕ) → (↥(Finset.Iic n) → 𝓐 × 𝓨) → 𝓐}
  {h_next : ∀ (n : ℕ), Measurable (nextA n)} {action0 : 𝓐}
  {env : Environment 𝓐 𝓨} {Ω : Type u_3} {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨} {N : ℕ} [MeasurableEq 𝓐]
  (h :
    IsAlgEnvSeqUntil A Y (detAlgorithm nextA h_next action0) env P N) :
  ProbabilityTheory.HasLaw (A 0) (MeasureTheory.Measure.dirac action0) P
Learning.IsAlgEnvSeqUntil.hasLaw_action_zero_detAlgorithm.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {nextA :
    (n : ℕ) →
      (↥(Finset.Iic n) → 𝓐 × 𝓨) → 𝓐}
  {h_next :
    ∀ (n : ℕ), Measurable (nextA n)}
  {action0 : 𝓐} {env : Environment 𝓐 𝓨}
  {Ω : Type u_3} {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨} {N : ℕ}
  [MeasurableEq 𝓐]
  (h :
    IsAlgEnvSeqUntil A Y
      (detAlgorithm nextA h_next action0)
      env P N) :
  ProbabilityTheory.HasLaw (A 0)
    (MeasureTheory.Measure.dirac action0)
    P

Code

lemma hasLaw_action_zero_detAlgorithm [MeasurableEq 𝓐]
    (h : IsAlgEnvSeqUntil A Y (detAlgorithm nextA h_next action0) env P N) :
    HasLaw (A 0) (Measure.dirac action0) P

Type uses (3)

Body uses (4)

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Proof

by
  simpa using IsDeterministicAlg.hasLaw_action_zero_of_IsAlgEnvSeqUntil h

Dependency graph

Type dependencies (3)

`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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`IsAlgEnvSeqUntil`🔗

StructureLearning.IsAlgEnvSeqUntil

Details

An algorithm-environment sequence: a sequence of actions and feedbacks generated by an algorithm interacting with an environment.

structure

Learning.IsAlgEnvSeqUntil.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨)
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P]
  (N : ℕ) : Prop
Learning.IsAlgEnvSeqUntil.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ω)
  [MeasureTheory.IsFiniteMeasure P]
  (N : ℕ) : Prop

Code

structure IsAlgEnvSeqUntil
    (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
    (P : Measure Ω) [IsFiniteMeasure P] (N : ℕ) : 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 ω ↦ (A 0 ω)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.ν0 P
  /-- The next action has the correct conditional distribution given the history. -/
  hasCondDistrib_action n (hn : n < 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 (hn : n < N) :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, A (n + 1) ω))
      (env.feedback n) P

Type uses (3)

Used by (22)

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`detAlgorithm`🔗

DefinitionLearning.detAlgorithm

Details

A deterministic algorithm, which chooses the action given by the function nextAction.

def

Learning.detAlgorithm.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (nextA : (n : ℕ) → (↥(Finset.Iic n) → 𝓐 × 𝓨) → 𝓐)
  (h_next : ∀ (n : ℕ), Measurable (nextA n)) (action0 : 𝓐) :
  Algorithm 𝓐 𝓨
Learning.detAlgorithm.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (nextA :
    (n : ℕ) →
      (↥(Finset.Iic n) → 𝓐 × 𝓨) → 𝓐)
  (h_next :
    ∀ (n : ℕ), Measurable (nextA n))
  (action0 : 𝓐) : Algorithm 𝓐 𝓨

Code

noncomputable
def detAlgorithm (nextA : (n : ℕ) → (Iic n → 𝓐 × 𝓨) → 𝓐)
    (h_next : ∀ n, Measurable (nextA n)) (action0 : 𝓐) :
    Algorithm 𝓐 𝓨 where
  policy n := Kernel.deterministic (nextA n) (h_next n)
  p0 := Measure.dirac action0

Type uses (1)

Algorithm

Used by (15)

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All dependencies, transitively (2)

`Algorithm`🔗

StructureLearning.Algorithm

Details

A stochastic, sequential algorithm.

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 : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ 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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`history`🔗

DefinitionLearning.history

Details

History of the algorithm-environment sequence up to time n.

def

Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def history (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) : Iic n → 𝓐 × 𝓨 :=
  fun i ↦ (A i ω, Y i ω)

Used by (72)

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Learning.IsAlgEnvSeqUntil.hasLaw_action_zero_detAlgorithm🔗

hasLaw_action_zero_detAlgorithm🔗

Environment🔗

IsAlgEnvSeqUntil🔗

detAlgorithm🔗

Algorithm🔗

history🔗

`Learning.IsAlgEnvSeqUntil.hasLaw_action_zero_detAlgorithm`🔗

`hasLaw_action_zero_detAlgorithm`🔗

`Environment`🔗

`IsAlgEnvSeqUntil`🔗

`detAlgorithm`🔗

`Algorithm`🔗

`history`🔗