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Learning.sumRewards_eq_pullCount_mul_empMean๐Ÿ”—

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sumRewards_eq_pullCount_mul_empMean๐Ÿ”—

LemmaLearning.sumRewards_eq_pullCount_mul_empMean
Details

No docstring.

๐Ÿ”—theorem
Learning.sumRewards_eq_pullCount_mul_empMean.{u_1, u_3} {๐“ : Type u_1}
  {ฮฉ : Type u_3} [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {a : ๐“} {t : โ„•}
  {R' : โ„• โ†’ ฮฉ โ†’ โ„} {ฯ‰ : ฮฉ} (h_pull : pullCount A a t ฯ‰ โ‰  0) :
  sumRewards A R' a t ฯ‰ = โ†‘(pullCount A a t ฯ‰) * empMean A R' a t ฯ‰
Learning.sumRewards_eq_pullCount_mul_empMean.{u_1,
    u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {a : ๐“}
  {t : โ„•} {R' : โ„• โ†’ ฮฉ โ†’ โ„} {ฯ‰ : ฮฉ}
  (h_pull : pullCount A a t ฯ‰ โ‰  0) :
  sumRewards A R' a t ฯ‰ =
    โ†‘(pullCount A a t ฯ‰) *
      empMean A R' a t ฯ‰

Code

lemma sumRewards_eq_pullCount_mul_empMean {R' : โ„• โ†’ ฮฉ โ†’ โ„} {ฯ‰ : ฮฉ}
    (h_pull : pullCount A a t ฯ‰ โ‰  0) :
    sumRewards A R' a t ฯ‰ = pullCount A a t ฯ‰ * empMean A R' a t ฯ‰
Type uses (3)
Used by (1)

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Proof
by unfold empMean; field_simp

Dependency graph

Type dependencies (3)

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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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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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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