`Learning.feedbackCondAction_evalEnv`🔗

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

`feedbackCondAction_evalEnv`🔗

LemmaLearning.feedbackCondAction_evalEnv

Details

No docstring.

theorem

Learning.feedbackCondAction_evalEnv.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {f : 𝓐 → 𝓨} {hf : Measurable f} (n : ℕ) :
  feedbackCondAction (evalEnv f hf) n =
    ProbabilityTheory.Kernel.deterministic f hf
Learning.feedbackCondAction_evalEnv.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {f : 𝓐 → 𝓨}
  {hf : Measurable f} (n : ℕ) :
  feedbackCondAction (evalEnv f hf) n =
    ProbabilityTheory.Kernel.deterministic
      f hf

Code

lemma feedbackCondAction_evalEnv (n : ℕ) :
    feedbackCondAction (evalEnv f hf) n = Kernel.deterministic f hf

Type uses (3)

Body uses (1)

feedbackCondAction_onlineEvalEnv

Used by (1)

hascondDistrib_feedback_evalEnv

Actions: Source · Open Issue

Proof

by simp [evalEnv]

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)

Actions: Source · Open Issue

`evalEnv`🔗

DefinitionLearning.evalEnv

Details

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

def

Learning.evalEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨} (f : 𝓐 → 𝓨)
  (hf : Measurable f) : Environment 𝓐 𝓨
Learning.evalEnv.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (f : 𝓐 → 𝓨)
  (hf : Measurable f) : Environment 𝓐 𝓨

Code

noncomputable def evalEnv (f : 𝓐 → 𝓨) (hf : Measurable f) := onlineEvalEnv (fun _ ↦ f) (fun _ ↦ hf)

Type uses (1)

Environment

Body uses (1)

onlineEvalEnv

Used by (9)

Actions: Source · Open Issue

`instIsObliviousEnvEvalEnv`🔗

InstanceLearning.instIsObliviousEnvEvalEnv

Details

No docstring.

theorem

Learning.instIsObliviousEnvEvalEnv.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {f : 𝓐 → 𝓨} {hf : Measurable f} : IsObliviousEnv (evalEnv f hf)
Learning.instIsObliviousEnvEvalEnv.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {f : 𝓐 → 𝓨}
  {hf : Measurable f} :
  IsObliviousEnv (evalEnv f hf)

Code

instance : IsObliviousEnv (evalEnv f hf)

Type uses (2)

Body uses (2)

Used by (2)

Actions: Source · Open Issue

Proof

by unfold evalEnv; infer_instance

All dependencies, transitively (5)

`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)

Actions: Source · Open Issue

`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)

Environment

Used by (13)

Actions: Source · Open Issue

`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)

Environment

Used by (10)

Actions: Source · Open Issue

`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)

Environment

Body uses (1)

obliviousEnv

Used by (11)

Actions: Source · Open Issue

`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)

Environment

Used by (3)

Actions: Source · Open Issue

Proof

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

Learning.feedbackCondAction_evalEnv🔗

feedbackCondAction_evalEnv🔗

feedbackCondAction🔗

evalEnv🔗

instIsObliviousEnvEvalEnv🔗

Environment🔗

IsObliviousEnv🔗

obliviousEnv🔗

onlineEvalEnv🔗

instIsObliviousEnvOnlineEvalEnv🔗

`Learning.feedbackCondAction_evalEnv`🔗

`feedbackCondAction_evalEnv`🔗

`feedbackCondAction`🔗

`evalEnv`🔗

`instIsObliviousEnvEvalEnv`🔗

`Environment`🔗

`IsObliviousEnv`🔗

`obliviousEnv`🔗

`onlineEvalEnv`🔗

`instIsObliviousEnvOnlineEvalEnv`🔗