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Learning.sumRewards_eq_sumRewards'๐Ÿ”—

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sumRewards_eq_sumRewards'๐Ÿ”—

LemmaLearning.sumRewards_eq_sumRewards'
Details

No docstring.

๐Ÿ”—theorem
Learning.sumRewards_eq_sumRewards'.{u_1, u_3} {๐“ : Type u_1}
  {ฮฉ : Type u_3} [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {a : ๐“}
  {R' : โ„• โ†’ ฮฉ โ†’ โ„} {n : โ„•} {ฯ‰ : ฮฉ} (hn : n โ‰  0) :
  sumRewards A R' a n ฯ‰ =
    sumRewards' (n - 1) (fun i => (A (โ†‘i) ฯ‰, R' (โ†‘i) ฯ‰)) a
Learning.sumRewards_eq_sumRewards'.{u_1,
    u_3}
  {๐“ : Type u_1} {ฮฉ : Type u_3}
  [DecidableEq ๐“] {A : โ„• โ†’ ฮฉ โ†’ ๐“} {a : ๐“}
  {R' : โ„• โ†’ ฮฉ โ†’ โ„} {n : โ„•} {ฯ‰ : ฮฉ}
  (hn : n โ‰  0) :
  sumRewards A R' a n ฯ‰ =
    sumRewards' (n - 1)
      (fun i => (A (โ†‘i) ฯ‰, R' (โ†‘i) ฯ‰)) a

Code

lemma sumRewards_eq_sumRewards' {R' : โ„• โ†’ ฮฉ โ†’ โ„} {n : โ„•} {ฯ‰ : ฮฉ} (hn : n โ‰  0) :
    sumRewards A R' a n ฯ‰ = sumRewards' (n - 1) (fun i โ†ฆ (A i ฯ‰, R' i ฯ‰)) a
Type uses (2)
Body uses (1)
Used by (1)

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Proof
by
  cases n with
  | zero => exact absurd rfl hn
  | succ n => simp [sumRewards_add_one_eq_sumRewards']

Dependency graph

Type dependencies (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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sumRewards'๐Ÿ”—

DefinitionLearning.sumRewards'
Details

Sum of rewards of arm a up to (and including) time n.

๐Ÿ”—def
Learning.sumRewards'.{u_1} {๐“ : Type u_1} [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) : โ„
Learning.sumRewards'.{u_1} {๐“ : Type u_1}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) :
  โ„

Code

noncomputable
def sumRewards' (n : โ„•) (h : Iic n โ†’ ๐“ ร— โ„) (a : ๐“) :=
  โˆ‘ s, if (h s).1 = a then (h s).2 else 0
Used by (9)

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