`Learning.forall_feedback_evalEnv_ae_eq_eval_action`🔗

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`forall_feedback_evalEnv_ae_eq_eval_action`🔗

LemmaLearning.forall_feedback_evalEnv_ae_eq_eval_action

Details

No docstring.

theorem

Learning.forall_feedback_evalEnv_ae_eq_eval_action.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ω : Type u_3} {mΩ : MeasurableSpace Ω}
  {alg : Algorithm 𝓐 𝓨} {f : 𝓐 → 𝓨} {hf : Measurable f}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨} [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  (h : IsAlgEnvSeq A Y alg (evalEnv f hf) P) :
  ∀ᵐ (ω : Ω) ∂P, ∀ (n : ℕ), Y n ω = f (A n ω)
Learning.forall_feedback_evalEnv_ae_eq_eval_action.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} {Ω : Type u_3}
  {mΩ : MeasurableSpace Ω}
  {alg : Algorithm 𝓐 𝓨} {f : 𝓐 → 𝓨}
  {hf : Measurable f}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} {Y : ℕ → Ω → 𝓨}
  [StandardBorelSpace 𝓨] [Nonempty 𝓨]
  (h :
    IsAlgEnvSeq A Y alg (evalEnv f hf)
      P) :
  ∀ᵐ (ω : Ω) ∂P,
    ∀ (n : ℕ), Y n ω = f (A n ω)

Code

lemma forall_feedback_evalEnv_ae_eq_eval_action [StandardBorelSpace 𝓨] [Nonempty 𝓨]
    (h : IsAlgEnvSeq A Y alg (evalEnv f hf) P) :
    ∀ᵐ ω ∂P, ∀ n, Y n ω = f (A n ω)

Type uses (3)

Body uses (1)

forall_feedback_onlineEvalEnv_ae_eq_eval_action

Used by (1)

feedback_evalEnv_ae_eq_eval_action_comp

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Proof

forall_feedback_onlineEvalEnv_ae_eq_eval_action h

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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`IsAlgEnvSeq`🔗

StructureLearning.IsAlgEnvSeq

Details

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

structure

Learning.IsAlgEnvSeq.{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] : Prop
Learning.IsAlgEnvSeq.{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] : Prop

Code

structure IsAlgEnvSeq
    (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
    (P : Measure Ω) [IsFiniteMeasure P] : 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 :
    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 :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, A (n + 1) ω))
      (env.feedback n) P

Type uses (3)

Used by (111)

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`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)

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All dependencies, transitively (4)

`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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`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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`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)

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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)

Environment

Body uses (1)

obliviousEnv

Used by (11)

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Learning.forall_feedback_evalEnv_ae_eq_eval_action🔗

forall_feedback_evalEnv_ae_eq_eval_action🔗

Algorithm🔗

IsAlgEnvSeq🔗

evalEnv🔗

Environment🔗

history🔗

obliviousEnv🔗

onlineEvalEnv🔗

`Learning.forall_feedback_evalEnv_ae_eq_eval_action`🔗

`forall_feedback_evalEnv_ae_eq_eval_action`🔗

`Algorithm`🔗

`IsAlgEnvSeq`🔗

`evalEnv`🔗

`Environment`🔗

`history`🔗

`obliviousEnv`🔗

`onlineEvalEnv`🔗