LeanMachineLearning exposition

Bandits.ucbAlgorithm๐Ÿ”—

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Minimal Lean file

ucbAlgorithm๐Ÿ”—

DefinitionBandits.ucbAlgorithm
Details

The UCB algorithm.

๐Ÿ”—def
Bandits.ucbAlgorithm {K : โ„•} (hK : 0 < K) (c : โ„) :
  Learning.Algorithm (Fin K) โ„
Bandits.ucbAlgorithm {K : โ„•} (hK : 0 < K)
  (c : โ„) : Learning.Algorithm (Fin K) โ„

Code

noncomputable
def ucbAlgorithm (hK : 0 < K) (c : โ„) : Algorithm (Fin K) โ„ :=
  detAlgorithm (UCB.nextArm hK c) (by fun_prop) โŸจ0, hKโŸฉ
Type uses (1)
Body uses (3)
Used by (16)

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Dependency graph

Type dependencies (1)

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]
Used by (216)

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All dependencies, transitively (11)

detAlgorithm๐Ÿ”—

DefinitionLearning.detAlgorithm
Details

A deterministic algorithm, which chooses the action given by the function nextAction.

๐Ÿ”—def
Learning.detAlgorithm.{u_1, u_2} {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (nextA : (n : โ„•) โ†’ (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“)
  (h_next : โˆ€ (n : โ„•), Measurable (nextA n)) (action0 : ๐“) :
  Algorithm ๐“ ๐“จ
Learning.detAlgorithm.{u_1, u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (nextA :
    (n : โ„•) โ†’
      (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— ๐“จ) โ†’ ๐“)
  (h_next :
    โˆ€ (n : โ„•), Measurable (nextA n))
  (action0 : ๐“) : Algorithm ๐“ ๐“จ

Code

noncomputable
def detAlgorithm (nextA : (n : โ„•) โ†’ (Iic n โ†’ ๐“ ร— ๐“จ) โ†’ ๐“)
    (h_next : โˆ€ n, Measurable (nextA n)) (action0 : ๐“) :
    Algorithm ๐“ ๐“จ where
  policy n := Kernel.deterministic (nextA n) (h_next n)
  p0 := Measure.dirac action0
Type uses (1)
Used by (15)

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nextAction๐Ÿ”—

DefinitionLearning.RoundRobin.nextAction
Details

Action chosen by the Round-Robin algorithm at time n + 1. This is action (n + 1) % K.

๐Ÿ”—def
Learning.RoundRobin.nextAction {K : โ„•} (hK : 0 < K) (n : โ„•) : Fin K
Learning.RoundRobin.nextAction {K : โ„•}
  (hK : 0 < K) (n : โ„•) : Fin K

Code

noncomputable
def RoundRobin.nextAction (hK : 0 < K) (n : โ„•) : Fin K := โŸจ(n + 1) % K, Nat.mod_lt _ hKโŸฉ
Used by (14)

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max๐Ÿ”—

DefinitionFunction.max
Details

The maximum value of a tuple.

๐Ÿ”—def
Function.max.{u_1, u_2} {ฮน : Type u_1} {ฮฑ : Type u_2} [LinearOrder ฮฑ]
  [Fintype ฮน] [Nonempty ฮน] (f : ฮน โ†’ ฮฑ) : ฮฑ
Function.max.{u_1, u_2} {ฮน : Type u_1}
  {ฮฑ : Type u_2} [LinearOrder ฮฑ]
  [Fintype ฮน] [Nonempty ฮน] (f : ฮน โ†’ ฮฑ) : ฮฑ

Code

abbrev max : ฮฑ := univ.sup' univ_nonempty f
Used by (8)

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exists_argmax๐Ÿ”—

Lemmaexists_argmax
Details

No docstring.

๐Ÿ”—theorem
exists_argmax.{u_1, u_2} {ฮน : Type u_1} {ฮฑ : Type u_2} [LinearOrder ฮฑ]
  [Fintype ฮน] [Nonempty ฮน] (f : ฮน โ†’ ฮฑ) : โˆƒ i, f i = Function.max f
exists_argmax.{u_1, u_2} {ฮน : Type u_1}
  {ฮฑ : Type u_2} [LinearOrder ฮฑ]
  [Fintype ฮน] [Nonempty ฮน] (f : ฮน โ†’ ฮฑ) :
  โˆƒ i, f i = Function.max f

Code

lemma exists_argmax : โˆƒ i, f i = f.max
Type uses (1)
Used by (3)

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Proof
by
  obtain โŸจi, -, hiโŸฉ := Finset.exists_mem_eq_sup' (by simp : Finset.univ.Nonempty) f
  exact โŸจi, hi.symmโŸฉ

argmax๐Ÿ”—

Definitionargmax
Details

The index of the maximum value of a tuple.

๐Ÿ”—def
argmax.{u_1, u_2} {ฮน : Type u_1} {ฮฑ : Type u_2} [LinearOrder ฮฑ]
  [Fintype ฮน] [Nonempty ฮน] (f : ฮน โ†’ ฮฑ) : ฮน
argmax.{u_1, u_2} {ฮน : Type u_1}
  {ฮฑ : Type u_2} [LinearOrder ฮฑ]
  [Fintype ฮน] [Nonempty ฮน] (f : ฮน โ†’ ฮฑ) : ฮน

Code

noncomputable def argmax := (exists_argmax f).choose
Body uses (2)
Used by (17)

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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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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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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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ucbWidth'๐Ÿ”—

DefinitionBandits.ucbWidth'
Details

The exploration bonus of the UCB algorithm, which corresponds to the width of a confidence interval.

๐Ÿ”—def
Bandits.ucbWidth' {K : โ„•} (c : โ„) (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ Fin K ร— โ„) (a : Fin K) : โ„
Bandits.ucbWidth' {K : โ„•} (c : โ„) (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ Fin K ร— โ„)
  (a : Fin K) : โ„

Code

noncomputable def ucbWidth' (c : โ„) (n : โ„•) (h : Iic n โ†’ Fin K ร— โ„) (a : Fin K) : โ„ :=
  โˆš(2 * c * log (n + 2) / pullCount' n h a)
Body uses (1)
Used by (6)

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nextArm๐Ÿ”—

DefinitionBandits.UCB.nextArm
Details

Arm pulled by the UCB algorithm at time n + 1.

๐Ÿ”—def
Bandits.UCB.nextArm {K : โ„•} (hK : 0 < K) (c : โ„) (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ Fin K ร— โ„) : Fin K
Bandits.UCB.nextArm {K : โ„•} (hK : 0 < K)
  (c : โ„) (n : โ„•)
  (h : โ†ฅ(Finset.Iic n) โ†’ Fin K ร— โ„) :
  Fin K

Code

noncomputable
def UCB.nextArm (hK : 0 < K) (c : โ„) (n : โ„•) (h : Iic n โ†’ Fin K ร— โ„) : Fin K :=
  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
  if n < K - 1 then RoundRobin.nextAction hK n else
  argmax (fun a โ†ฆ empMean' n h a + ucbWidth' c n h a)
Body uses (4)
Used by (7)

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measurable_nextArm๐Ÿ”—

LemmaBandits.UCB.measurable_nextArm
Details

No docstring.

๐Ÿ”—theorem
Bandits.UCB.measurable_nextArm {K : โ„•} (hK : 0 < K) (c : โ„) (n : โ„•) :
  Measurable (nextArm hK c n)
Bandits.UCB.measurable_nextArm {K : โ„•}
  (hK : 0 < K) (c : โ„) (n : โ„•) :
  Measurable (nextArm hK c n)

Code

lemma UCB.measurable_nextArm (hK : 0 < K) (c : โ„) (n : โ„•) : Measurable (nextArm hK c n)
Type uses (1)
Body uses (9)
Used by (3)

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Proof
by
  refine Measurable.ite (by simp) (by fun_prop) ?_
  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK
  unfold ucbWidth'
  fun_prop