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Learning.IsDeterministicAlg.action_ae_all_eq๐Ÿ”—

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action_ae_all_eq๐Ÿ”—

LemmaLearning.IsDeterministicAlg.action_ae_all_eq
Details

No docstring.

๐Ÿ”—theorem
Learning.IsDeterministicAlg.action_ae_all_eq.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {ฮฉ : Type u_3} {mฮฉ : MeasurableSpace ฮฉ}
  {alg : Algorithm ๐“ ๐“จ} {env : Environment ๐“ ๐“จ}
  {P : MeasureTheory.Measure ฮฉ} [MeasureTheory.IsFiniteMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ} [MeasurableEq ๐“]
  [h_det : IsDeterministicAlg alg] (h : IsAlgEnvSeq A Y alg env P) :
  โˆ€แต (ฯ‰ : ฮฉ) โˆ‚P,
    A 0 ฯ‰ = actionZero alg โˆง
      โˆ€ (n : โ„•), A (n + 1) ฯ‰ = nextAction alg n (history A Y n ฯ‰)
Learning.IsDeterministicAlg.action_ae_all_eq.{u_1,
    u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} {ฮฉ : Type u_3}
  {mฮฉ : MeasurableSpace ฮฉ}
  {alg : Algorithm ๐“ ๐“จ}
  {env : Environment ๐“ ๐“จ}
  {P : MeasureTheory.Measure ฮฉ}
  [MeasureTheory.IsFiniteMeasure P]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} {Y : โ„• โ†’ ฮฉ โ†’ ๐“จ}
  [MeasurableEq ๐“]
  [h_det : IsDeterministicAlg alg]
  (h : IsAlgEnvSeq A Y alg env P) :
  โˆ€แต (ฯ‰ : ฮฉ) โˆ‚P,
    A 0 ฯ‰ = actionZero alg โˆง
      โˆ€ (n : โ„•),
        A (n + 1) ฯ‰ =
          nextAction alg n
            (history A Y n ฯ‰)

Code

lemma action_ae_all_eq [MeasurableEq ๐“] [h_det : IsDeterministicAlg alg]
    (h : IsAlgEnvSeq A Y alg env P) :
    โˆ€แต ฯ‰ โˆ‚P, A 0 ฯ‰ = actionZero alg โˆง โˆ€ n, A (n + 1) ฯ‰ = nextAction alg n (history A Y n ฯ‰)
Type uses (7)
Body uses (2)
Used by (1)

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Proof
by
  rw [eventually_and, ae_all_iff]
  exact โŸจaction_zero_ae_eq h, action_ae_eq hโŸฉ

Dependency graph

Type dependencies (7)

Algorithm๐Ÿ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

๐Ÿ”—structure
Learning.Algorithm.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Algorithm (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : โ„•) โ†’ Kernel (Iic n โ†’ ๐“ ร— ๐“จ) ๐“
  /-- The policy is a Markov kernel. -/
  [h_policy : โˆ€ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure ๐“
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
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Environment๐Ÿ”—

StructureLearning.Environment
Details

A stochastic environment.

๐Ÿ”—structure
Learning.Environment.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Environment.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Environment (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : โ„•) โ†’ Kernel ((Iic n โ†’ ๐“ ร— ๐“จ) ร— ๐“) ๐“จ
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : โˆ€ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ฮฝ0 : Kernel ๐“ ๐“จ
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ฮฝ0]
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IsDeterministicAlg๐Ÿ”—

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).

๐Ÿ”—type class
Learning.IsDeterministicAlg.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) : Prop
Learning.IsDeterministicAlg.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) : Prop

Code

class IsDeterministicAlg (alg : Algorithm ๐“ ๐“จ) : Prop where
  exists_action0 : โˆƒ action0, alg.p0 = Measure.dirac action0
  exists_nextAction n : โˆƒ (nextAction : (Iic n โ†’ ๐“ ร— ๐“จ) โ†’ ๐“) (h_meas : Measurable nextAction),
    alg.policy n = Kernel.deterministic nextAction h_meas
Type uses (1)
Used by (14)

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IsAlgEnvSeq๐Ÿ”—

StructureLearning.IsAlgEnvSeq
Details

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

๐Ÿ”—structure
Learning.IsAlgEnvSeq.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ)
  (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ)
  (P : MeasureTheory.Measure ฮฉ) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsAlgEnvSeq.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  {mฮฉ : MeasurableSpace ฮฉ} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (alg : Algorithm ๐“ ๐“จ)
  (env : Environment ๐“ ๐“จ)
  (P : MeasureTheory.Measure ฮฉ)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsAlgEnvSeq
    (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (alg : Algorithm ๐“ ๐“จ) (env : Environment ๐“ ๐“จ)
    (P : Measure ฮฉ) [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 ฯ‰ โ†ฆ (A 0 ฯ‰)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.ฮฝ0 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 ฯ‰ โ†ฆ (history A Y n ฯ‰, A (n + 1) ฯ‰))
      (env.feedback n) P
Type uses (3)
Used by (111)

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actionZero๐Ÿ”—

DefinitionLearning.actionZero
Details

The initial action of a deterministic algorithm.

๐Ÿ”—def
Learning.actionZero.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) [h_det : IsDeterministicAlg alg] : ๐“
Learning.actionZero.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  [h_det : IsDeterministicAlg alg] : ๐“

Code

noncomputable
def actionZero (alg : Algorithm ๐“ ๐“จ) [h_det : IsDeterministicAlg alg] : ๐“ :=
  h_det.exists_action0.choose
Type uses (2)
Used by (12)

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nextAction๐Ÿ”—

DefinitionLearning.nextAction
Details

The next action of a deterministic algorithm after step n.

๐Ÿ”—def
Learning.nextAction.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) [h_det : IsDeterministicAlg alg] (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“
Learning.nextAction.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ)
  [h_det : IsDeterministicAlg alg]
  (n : โ„•) : (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“

Code

noncomputable
def nextAction (alg : Algorithm ๐“ ๐“จ) [h_det : IsDeterministicAlg alg] (n : โ„•) :
    (Iic n โ†’ ๐“ ร— ๐“จ) โ†’ ๐“ :=
  (h_det.exists_nextAction n).choose
Type uses (2)
Used by (9)

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history๐Ÿ”—

DefinitionLearning.history
Details

History of the algorithm-environment sequence up to time n.

๐Ÿ”—def
Learning.history.{u_1, u_2, u_3} {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ
Learning.history.{u_1, u_2, u_3}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {ฮฉ : Type u_3} (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ

Code

def history (A : โ„• โ†’ ฮฉ โ†’ ๐“) (Y : โ„• โ†’ ฮฉ โ†’ ๐“จ) (n : โ„•) (ฯ‰ : ฮฉ) : Iic n โ†’ ๐“ ร— ๐“จ :=
  fun i โ†ฆ (A i ฯ‰, Y i ฯ‰)
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