`Learning.measurable_nextAction`🔗

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Minimal Lean file

`measurable_nextAction`🔗

LemmaLearning.measurable_nextAction

Details

No docstring.

theorem

Learning.measurable_nextAction.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) [IsDeterministicAlg alg] (n : ℕ) :
  Measurable (nextAction alg n)
Learning.measurable_nextAction.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  [IsDeterministicAlg alg] (n : ℕ) :
  Measurable (nextAction alg n)

Code

lemma measurable_nextAction (alg : Algorithm 𝓐 𝓨) [IsDeterministicAlg alg] (n : ℕ) :
    Measurable (nextAction alg n)

Type uses (3)

Used by (3)

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Proof

(IsDeterministicAlg.exists_nextAction n).choose_spec.choose

Dependency graph

Type dependencies (3)

`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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`IsDeterministicAlg`🔗

Type ClassLearning.IsDeterministicAlg

Details

An algorithm is deterministic if its initial action and subsequent actions are determined by measurable functions (and not possibly random kernels).

type class

Learning.IsDeterministicAlg.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) : Prop
Learning.IsDeterministicAlg.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) : Prop

Code

class IsDeterministicAlg (alg : Algorithm 𝓐 𝓨) : Prop where
  exists_action0 : ∃ action0, alg.p0 = Measure.dirac action0
  exists_nextAction n : ∃ (nextAction : (Iic n → 𝓐 × 𝓨) → 𝓐) (h_meas : Measurable nextAction),
    alg.policy n = Kernel.deterministic nextAction h_meas

Type uses (1)

Algorithm

Used by (14)

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`nextAction`🔗

DefinitionLearning.nextAction

Details

The next action of a deterministic algorithm after step n.

def

Learning.nextAction.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) [h_det : IsDeterministicAlg alg] (n : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × 𝓨) → 𝓐
Learning.nextAction.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨)
  [h_det : IsDeterministicAlg alg]
  (n : ℕ) : (↥(Finset.Iic n) → 𝓐 × 𝓨) → 𝓐

Code

noncomputable
def nextAction (alg : Algorithm 𝓐 𝓨) [h_det : IsDeterministicAlg alg] (n : ℕ) :
    (Iic n → 𝓐 × 𝓨) → 𝓐 :=
  (h_det.exists_nextAction n).choose

Type uses (2)

Used by (9)

Actions: Source · Open Issue

Learning.measurable_nextAction🔗

measurable_nextAction🔗

Algorithm🔗

IsDeterministicAlg🔗

nextAction🔗

`Learning.measurable_nextAction`🔗

`measurable_nextAction`🔗

`Algorithm`🔗

`IsDeterministicAlg`🔗

`nextAction`🔗