LeanMachineLearning exposition

Learning.feedbackCondAction_onlineEvalEnv🔗

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Minimal Lean file

feedbackCondAction_onlineEvalEnv🔗

LemmaLearning.feedbackCondAction_onlineEvalEnv
Details

No docstring.

🔗theorem
Learning.feedbackCondAction_onlineEvalEnv.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {g : ℕ → 𝓐 → 𝓨} {hg : ∀ (n : ℕ), Measurable (g n)} (n : ℕ) :
  feedbackCondAction (onlineEvalEnv g hg) n =
    ProbabilityTheory.Kernel.deterministic (g n) ⋯
Learning.feedbackCondAction_onlineEvalEnv.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {g : ℕ → 𝓐 → 𝓨}
  {hg : ∀ (n : ℕ), Measurable (g n)}
  (n : ℕ) :
  feedbackCondAction (onlineEvalEnv g hg)
      n =
    ProbabilityTheory.Kernel.deterministic
      (g n) ⋯

Code

lemma feedbackCondAction_onlineEvalEnv (n : ℕ) :
    feedbackCondAction (onlineEvalEnv g hg) n = Kernel.deterministic (g n) (hg n)
Type uses (3)
Body uses (1)
Used by (2)

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Proof
by
  simp [onlineEvalEnv]

Dependency graph

Type dependencies (3)

feedbackCondAction🔗

DefinitionLearning.feedbackCondAction
Details

The kernel representing the conditional distribution of the feedback given the action at time n in an oblivious environment.

🔗def
Learning.feedbackCondAction.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨) [h_obl : IsObliviousEnv env] (n : ℕ) :
  ProbabilityTheory.Kernel 𝓐 𝓨
Learning.feedbackCondAction.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (env : Environment 𝓐 𝓨)
  [h_obl : IsObliviousEnv env] (n : ℕ) :
  ProbabilityTheory.Kernel 𝓐 𝓨

Code

noncomputable
def feedbackCondAction (env : Environment 𝓐 𝓨) [h_obl : IsObliviousEnv env] (n : ℕ) : Kernel 𝓐 𝓨 :=
  h_obl.exists_eq_prodMkLeft.choose n
Type uses (2)
Used by (12)

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onlineEvalEnv🔗

DefinitionLearning.onlineEvalEnv
Details

The evaluation environment where the feedback is given by evaluating a fixed measurable function f at the chosen action.

🔗def
Learning.onlineEvalEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} (g : ℕ → 𝓐 → 𝓨)
  (hg : ∀ (n : ℕ), Measurable (g n)) : Environment 𝓐 𝓨
Learning.onlineEvalEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (g : ℕ → 𝓐 → 𝓨)
  (hg : ∀ (n : ℕ), Measurable (g n)) :
  Environment 𝓐 𝓨

Code

noncomputable def onlineEvalEnv (g : ℕ → 𝓐 → 𝓨) (hg : ∀ n, Measurable (g n)) :=
  obliviousEnv (fun n ↦ Kernel.deterministic (g n) (hg n))
Type uses (1)
Body uses (1)
Used by (11)

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instIsObliviousEnvOnlineEvalEnv🔗

InstanceLearning.instIsObliviousEnvOnlineEvalEnv
Details

No docstring.

🔗theorem
Learning.instIsObliviousEnvOnlineEvalEnv.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {g : ℕ → 𝓐 → 𝓨} {hg : ∀ (n : ℕ), Measurable (g n)} :
  IsObliviousEnv (onlineEvalEnv g hg)
Learning.instIsObliviousEnvOnlineEvalEnv.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {g : ℕ → 𝓐 → 𝓨}
  {hg : ∀ (n : ℕ), Measurable (g n)} :
  IsObliviousEnv (onlineEvalEnv g hg)

Code

instance : IsObliviousEnv (onlineEvalEnv g hg)
Type uses (2)
Body uses (1)
Used by (3)

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Proof
⟨⟨fun n ↦ Kernel.deterministic (g n) (hg n), fun _ ↦ inferInstance, rfl, fun _ ↦ rfl⟩⟩

All dependencies, transitively (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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IsObliviousEnv🔗

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.

🔗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 : ∃ ν : ℕ → Kernel 𝓐 𝓨, (∀ n, IsMarkovKernel (ν n)) ∧
    (env.ν0 = ν 0) ∧ (∀ n, env.feedback n = (ν (n + 1)).prodMkLeft _)
Type uses (1)
Used by (13)

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obliviousEnv🔗

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.

🔗def
Learning.obliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (ν n)] : Environment 𝓐 𝓨
Learning.obliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [∀ (n : ℕ),
      ProbabilityTheory.IsMarkovKernel
        (ν n)] :
  Environment 𝓐 𝓨

Code

def obliviousEnv (ν : ℕ → Kernel 𝓐 𝓨) [∀ n, IsMarkovKernel (ν n)] : Environment 𝓐 𝓨 where
  feedback n := (ν (n + 1)).prodMkLeft _
  ν0 := ν 0
Type uses (1)
Used by (10)

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