LeanMachineLearning exposition

Learning.measurable_uncurry_empMean_comp๐Ÿ”—

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measurable_uncurry_empMean_comp๐Ÿ”—

LemmaLearning.measurable_uncurry_empMean_comp
Details

No docstring.

๐Ÿ”—theorem
Learning.measurable_uncurry_empMean_comp.{u_1, u_3} {๐“ : Type u_1}
  {ฮฉ : Type u_3} {m๐“ : MeasurableSpace ๐“} {mฮฉ : MeasurableSpace ฮฉ}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} [Countable ๐“]
  [MeasurableSingletonClass ๐“] {R' : โ„• โ†’ ฮฉ โ†’ โ„}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n)) {f : ฮฉ โ†’ ๐“} (hf : Measurable f)
  {g : ฮฉ โ†’ โ„•} (hg : Measurable g) :
  Measurable fun ฯ‰ => empMean A R' (f ฯ‰) (g ฯ‰) ฯ‰
Learning.measurable_uncurry_empMean_comp.{u_1,
    u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  {m๐“ : MeasurableSpace ๐“}
  {mฮฉ : MeasurableSpace ฮฉ} [DecidableEq ๐“]
  {A : โ„• โ†’ ฮฉ โ†’ ๐“} [Countable ๐“]
  [MeasurableSingletonClass ๐“]
  {R' : โ„• โ†’ ฮฉ โ†’ โ„}
  (hA : โˆ€ (n : โ„•), Measurable (A n))
  (hR' : โˆ€ (n : โ„•), Measurable (R' n))
  {f : ฮฉ โ†’ ๐“} (hf : Measurable f)
  {g : ฮฉ โ†’ โ„•} (hg : Measurable g) :
  Measurable fun ฯ‰ =>
    empMean A R' (f ฯ‰) (g ฯ‰) ฯ‰

Code

lemma measurable_uncurry_empMean_comp [Countable ๐“] [MeasurableSingletonClass ๐“] {R' : โ„• โ†’ ฮฉ โ†’ โ„}
    (hA : โˆ€ n, Measurable (A n)) (hR' : โˆ€ n, Measurable (R' n)) {f : ฮฉ โ†’ ๐“} (hf : Measurable f)
    {g : ฮฉ โ†’ โ„•} (hg : Measurable g) :
    Measurable (fun ฯ‰ โ†ฆ empMean A R' (f ฯ‰) (g ฯ‰) ฯ‰)
Type uses (1)
Body uses (4)
Used by (1)

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Proof
by
  unfold empMean
  fun_prop

Dependency graph

Type dependencies (1)

empMean๐Ÿ”—

DefinitionLearning.empMean
Details

Empirical mean reward obtained when pulling action a up to time t (exclusive).

๐Ÿ”—def
Learning.empMean.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•)
  (ฯ‰ : ฮฉ) : โ„
Learning.empMean.{u_1, u_3} {๐“ : Type u_1}
  {ฮฉ : Type u_3} [DecidableEq ๐“]
  (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“)
  (t : โ„•) (ฯ‰ : ฮฉ) : โ„

Code

noncomputable
def empMean (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„ :=
  sumRewards A R' a t ฯ‰ / pullCount A a t ฯ‰
Body uses (2)
Used by (34)

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All dependencies, transitively (2)

sumRewards๐Ÿ”—

DefinitionLearning.sumRewards
Details

Sum of rewards obtained when pulling action a up to time t (exclusive).

๐Ÿ”—def
Learning.sumRewards.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•)
  (ฯ‰ : ฮฉ) : โ„
Learning.sumRewards.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“)
  (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•)
  (ฯ‰ : ฮฉ) : โ„

Code

def sumRewards (A : โ„• โ†’ ฮฉ โ†’ ๐“) (R' : โ„• โ†’ ฮฉ โ†’ โ„) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„ :=
  โˆ‘ s โˆˆ range t, if A s ฯ‰ = a then R' s ฯ‰ else 0
Used by (44)

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pullCount๐Ÿ”—

DefinitionLearning.pullCount
Details

Number of times action a was chosen up to time t (excluding t).

๐Ÿ”—def
Learning.pullCount.{u_1, u_3} {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„•
Learning.pullCount.{u_1, u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“)
  (t : โ„•) (ฯ‰ : ฮฉ) : โ„•

Code

noncomputable
def pullCount (A : โ„• โ†’ ฮฉ โ†’ ๐“) (a : ๐“) (t : โ„•) (ฯ‰ : ฮฉ) : โ„• :=
  #(filter (fun s โ†ฆ A s ฯ‰ = a) (range t))
Used by (146)

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