LeanMachineLearning exposition

Bandits.ClippedUCB.ucb'๐Ÿ”—

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Minimal Lean file

ucb'๐Ÿ”—

DefinitionBandits.ClippedUCB.ucb'
Details

Clipped upper confidence bound (history-based version).

๐Ÿ”—def
Bandits.ClippedUCB.ucb' {K : โ„•} (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ Fin K ร— โ„) (l u ฯƒ2 ฮด : โ„) (a : Fin K) : โ„
Bandits.ClippedUCB.ucb' {K : โ„•} (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ Fin K ร— โ„)
  (l u ฯƒ2 ฮด : โ„) (a : Fin K) : โ„

Code

noncomputable
def ucb' (n : โ„•) (h : Iic n โ†’ Fin K ร— โ„) (l u ฯƒ2 ฮด : โ„) (a : Fin K) : โ„ :=
  if pullCount' n h a = 0 then u
  else max l (min u (empMean' n h a + โˆš(2 * ฯƒ2 * Real.log (1 / ฮด) / (pullCount' n h a))))
Body uses (2)
Used by (3)

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Dependency graph

All dependencies, transitively (3)

pullCount'๐Ÿ”—

DefinitionLearning.pullCount'
Details

Number of pulls of arm a up to (and including) time n. This is the number of entries in h in which the arm is a.

๐Ÿ”—def
Learning.pullCount'.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•) (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) : โ„•
Learning.pullCount'.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) (a : ๐“) :
  โ„•

Code

noncomputable
def pullCount' (n : โ„•) (h : Iic n โ†’ ๐“ ร— R) (a : ๐“) := #{s | (h s).1 = a}
Used by (29)

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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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empMean'๐Ÿ”—

DefinitionLearning.empMean'
Details

Empirical mean of arm a at time n.

๐Ÿ”—def
Learning.empMean'.{u_1} {๐“ : Type u_1} [DecidableEq ๐“] (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— โ„) (a : ๐“) : โ„
Learning.empMean'.{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 empMean' (n : โ„•) (h : Iic n โ†’ ๐“ ร— โ„) (a : ๐“) :=
  (sumRewards' n h a) / (pullCount' n h a)
Body uses (2)
Used by (18)

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