LeanMachineLearning exposition

Learning.IT.condDistrib_feedback🔗

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Minimal Lean file

condDistrib_feedback🔗

LemmaLearning.IT.condDistrib_feedback
Details

No docstring.

🔗theorem
Learning.IT.condDistrib_feedback.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  [StandardBorelSpace 𝓨] [Nonempty 𝓨] (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ⇑𝓛[feedback (n + 1) | fun ω => (hist n ω, action (n + 1) ω);
        trajMeasure alg
          env] =ᵐ[MeasureTheory.Measure.map
      (fun ω => (hist n ω, action (n + 1) ω)) (trajMeasure alg env)]
    ⇑(Environment.feedback env n)
Learning.IT.condDistrib_feedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ⇑𝓛[feedback (n + 1) | fun ω =>
        (hist n ω, action (n + 1) ω);
        trajMeasure alg
          env] =ᵐ[MeasureTheory.Measure.map
      (fun ω =>
        (hist n ω, action (n + 1) ω))
      (trajMeasure alg env)]
    ⇑(Environment.feedback env n)

Code

lemma condDistrib_feedback [StandardBorelSpace 𝓨] [Nonempty 𝓨]
    (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
    condDistrib (feedback (n + 1)) (fun ω ↦ (hist n ω, action (n + 1) ω)) (trajMeasure alg env)
      =ᵐ[(trajMeasure alg env).map (fun ω ↦ (hist n ω, action (n + 1) ω))] env.feedback n
Type uses (7)
Body uses (3)

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Proof
(hasCondDistrib_feedback alg env n).condDistrib_eq

Dependency graph

Type dependencies (7)

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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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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hist🔗

DefinitionLearning.IT.hist
Details

hist n is the history up to time n. This is a random variable on the measurable space ℕ → 𝓐 × 𝓨.

🔗def
Learning.IT.hist.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.IT.hist.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} (n : ℕ) (h : ℕ → 𝓐 × 𝓨) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def hist (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : Iic n → 𝓐 × 𝓨 := fun i ↦ h i
Used by (23)

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action🔗

DefinitionLearning.IT.action
Details

action n is the action pulled at time n. This is a random variable on the measurable space ℕ → 𝓐 × 𝓨.

🔗def
Learning.IT.action.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓐
Learning.IT.action.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓐

Code

def action (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓐 := (h n).1
Used by (31)

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trajMeasure🔗

DefinitionLearning.trajMeasure
Details

Measure on the sequence of actions and observations generated by the algorithm/environment.

🔗def
Learning.trajMeasure.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓨)
Learning.trajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.Measure (ℕ → 𝓐 × 𝓨)

Code

noncomputable
def trajMeasure (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :
    Measure (ℕ → 𝓐 × 𝓨) :=
  Kernel.trajMeasure (alg.p0 ⊗ₘ env.ν0) (stepKernel alg env)
deriving IsProbabilityMeasure
Type uses (2)
Body uses (2)
Used by (19)

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feedback🔗

DefinitionLearning.IT.feedback
Details

feedback n is the feedback at time n. This is a random variable on the measurable space ℕ → 𝓐 × 𝓨.

🔗def
Learning.IT.feedback.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓨
Learning.IT.feedback.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} (n : ℕ)
  (h : ℕ → 𝓐 × 𝓨) : 𝓨

Code

def feedback (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓨 := (h n).2
Used by (16)

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instIsProbabilityMeasureForallNatProdTrajMeasure🔗

InstanceLearning.instIsProbabilityMeasureForallNatProdTrajMeasure
Details

No docstring.

🔗theorem
Learning.instIsProbabilityMeasureForallNatProdTrajMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (trajMeasure alg env)
Learning.instIsProbabilityMeasureForallNatProdTrajMeasure.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (trajMeasure alg env)

Code

deriving IsProbabilityMeasure
Type uses (3)
Body uses (4)
Used by (8)

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Proof
deriving IsProbabilityMeasure

All dependencies, transitively (6)

stepKernel🔗

DefinitionLearning.stepKernel
Details

Kernel describing the distribution of the next action-feedback pair given the history up to n.

🔗def
Learning.stepKernel.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel (↥(Finset.Iic n) → 𝓐 × 𝓨) (𝓐 × 𝓨)
Learning.stepKernel.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.Kernel
    (↥(Finset.Iic n) → 𝓐 × 𝓨) (𝓐 × 𝓨)

Code

noncomputable
def stepKernel (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :
    Kernel (Iic n → 𝓐 × 𝓨) (𝓐 × 𝓨) :=
  alg.policy n ⊗ₖ env.feedback n
deriving IsMarkovKernel
Type uses (2)
Used by (17)

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instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (alg.policy n)
Type uses (1)
Used by (14)

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Proof
alg.h_policy n

instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback🔗

InstanceLearning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Environment.feedback env n)
Learning.instIsMarkovKernelProdForallSubtypeNatMemFinsetIicFeedback.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.feedback env n)

Code

instance (env : Environment 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (env.feedback n)
Type uses (1)
Used by (5)

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Proof
env.h_feedback n

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (n : ℕ) : ProbabilityTheory.IsMarkovKernel (stepKernel alg env n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdStepKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (stepKernel alg env n)

Code

deriving IsMarkovKernel
Type uses (3)
Body uses (2)
Used by (10)

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Proof
deriving IsMarkovKernel

instIsProbabilityMeasureP0🔗

InstanceLearning.instIsProbabilityMeasureP0
Details

No docstring.

🔗theorem
Learning.instIsProbabilityMeasureP0.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm 𝓐 𝓨) : IsProbabilityMeasure alg.p0
Type uses (1)
Used by (13)

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Proof
alg.hp0

instIsMarkovKernelν0🔗

InstanceLearning.instIsMarkovKernelν0
Details

No docstring.

🔗theorem
Learning.instIsMarkovKernelν0.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) :
  ProbabilityTheory.IsMarkovKernel (Environment.ν0 env)
Learning.instIsMarkovKernelν0.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) :
  ProbabilityTheory.IsMarkovKernel
    (Environment.ν0 env)

Code

instance (env : Environment 𝓐 𝓨) : IsMarkovKernel env.ν0
Type uses (1)
Used by (8)

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Proof
env.hp0