LeanMachineLearning exposition

Bandits.ArrayModel.measurable_pullCount_add_one_truePast๐Ÿ”—

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Minimal Lean file

measurable_pullCount_add_one_truePast๐Ÿ”—

LemmaBandits.ArrayModel.measurable_pullCount_add_one_truePast
Details

No docstring.

๐Ÿ”—theorem
Bandits.ArrayModel.measurable_pullCount_add_one_truePast.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“] [StandardBorelSpace ๐“]
  [DecidableEq ๐“] [Nonempty R] [Countable ๐“]
  (alg : Learning.Algorithm ๐“ R) (a : ๐“) (n : โ„•) :
  Measurable (Learning.pullCount (action alg) a (n + 1))
Bandits.ArrayModel.measurable_pullCount_add_one_truePast.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  [Nonempty R] [Countable ๐“]
  (alg : Learning.Algorithm ๐“ R) (a : ๐“)
  (n : โ„•) :
  Measurable
    (Learning.pullCount (action alg) a
      (n + 1))

Code

lemma measurable_pullCount_add_one_truePast [Countable ๐“] (alg : Algorithm ๐“ R) (a : ๐“) (n : โ„•) :
    Measurable[MeasurableSpace.comap (truePast alg a n) inferInstance]
      (pullCount (action alg) a (n + 1))
Type uses (6)
Body uses (5)

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Proof
by
  change Measurable[MeasurableSpace.comap (truePast alg a n) inferInstance]
    (fun ฯ‰ โ†ฆ pullCount (action alg) a (n + 1) ฯ‰)
  simp_rw [pullCount_eq_sum]
  refine measurable_sum _ fun i hi โ†ฆ Measurable.ite ?_ (by fun_prop) (by fun_prop)
  refine (measurableSet_singleton _).preimage ?_
  have h_meas := measurable_hist_truePast alg a n
  simp_rw [hist_eq _ _ n, @measurable_pi_iff] at h_meas
  exact (h_meas โŸจi, by grindโŸฉ).fst

Dependency graph

Type dependencies (6)

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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probSpace๐Ÿ”—

DefinitionBandits.ArrayModel.probSpace
Details

Probability space for the array model of stochastic bandits.

๐Ÿ”—def
Bandits.ArrayModel.probSpace.{u_1, u_2} (๐“ : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)
Bandits.ArrayModel.probSpace.{u_1, u_2}
  (๐“ : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)

Code

def probSpace : Type _ := (โ„• โ†’ I) ร— (โ„• โ†’ ๐“ โ†’ R)
Used by (64)

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truePast๐Ÿ”—

DefinitionBandits.ArrayModel.truePast
Details

All random variables in the space, except for the unseen rewards for action a after time n.

๐Ÿ”—def
Bandits.ArrayModel.truePast.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“] [Nonempty R]
  (alg : Learning.Algorithm ๐“ R) (a : ๐“) (n : โ„•) (ฯ‰ : probSpace ๐“ R) :
  probSpace ๐“ R
Bandits.ArrayModel.truePast.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  [Nonempty R]
  (alg : Learning.Algorithm ๐“ R) (a : ๐“)
  (n : โ„•) (ฯ‰ : probSpace ๐“ R) :
  probSpace ๐“ R

Code

noncomputable
def truePast (alg : Algorithm ๐“ R) (a : ๐“) (n : โ„•) (ฯ‰ : probSpace ๐“ R) :
    probSpace ๐“ R :=
  (ฯ‰.1, fun i b โ†ฆ if b = a then if pullCount (action alg) a (n + 1) ฯ‰ โ‰  0 then
      ฯ‰.2 (min i ((pullCount (action alg) a (n + 1) ฯ‰) - 1)) a else Nonempty.some inferInstance
    else ฯ‰.2 i b)
Type uses (2)
Body uses (2)
Used by (6)

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instMeasurableSpaceProbSpace๐Ÿ”—

InstanceBandits.ArrayModel.instMeasurableSpaceProbSpace
Details

No docstring.

๐Ÿ”—def
Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3, u_4}
  {๐“ : Type u_3} {R : Type u_4} [MeasurableSpace R] :
  MeasurableSpace (probSpace ๐“ R)
Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3,
    u_4}
  {๐“ : Type u_3} {R : Type u_4}
  [MeasurableSpace R] :
  MeasurableSpace (probSpace ๐“ R)

Code

instance {๐“ R : Type*} [MeasurableSpace R] : MeasurableSpace (probSpace ๐“ R)
Type uses (1)
Used by (41)

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Proof
inferInstanceAs (MeasurableSpace ((โ„• โ†’ I) ร— (โ„• โ†’ ๐“ โ†’ R)))

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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action๐Ÿ”—

DefinitionBandits.ArrayModel.action
Details

Action taken at time n in the array model.

๐Ÿ”—def
Bandits.ArrayModel.action.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“] (alg : Learning.Algorithm ๐“ R)
  (n : โ„•) (ฯ‰ : probSpace ๐“ R) : ๐“
Bandits.ArrayModel.action.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•)
  (ฯ‰ : probSpace ๐“ R) : ๐“

Code

noncomputable
def action [DecidableEq ๐“] (alg : Algorithm ๐“ R) (n : โ„•) (ฯ‰ : probSpace ๐“ R) : ๐“ :=
  (hist alg ฯ‰ n โŸจn, by simpโŸฉ).1
Type uses (2)
Body uses (1)
Used by (43)

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All dependencies, transitively (6)

instIsProbabilityMeasureP0๐Ÿ”—

InstanceLearning.instIsProbabilityMeasureP0
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsProbabilityMeasureP0.{u_1, u_2} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm ๐“ ๐“จ) : IsProbabilityMeasure alg.p0
Type uses (1)
Used by (13)

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Proof
alg.hp0

initAlgFunction๐Ÿ”—

DefinitionBandits.ArrayModel.initAlgFunction
Details

The initial action is the image of a uniform random variable by this function.

๐Ÿ”—def
Bandits.ArrayModel.initAlgFunction.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  [Nonempty ๐“] [StandardBorelSpace ๐“] (alg : Learning.Algorithm ๐“ R) :
  โ†‘unitInterval โ†’ ๐“
Bandits.ArrayModel.initAlgFunction.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“]
  (alg : Learning.Algorithm ๐“ R) :
  โ†‘unitInterval โ†’ ๐“

Code

noncomputable
def initAlgFunction (alg : Algorithm ๐“ R) : I โ†’ ๐“ :=
  (Measure.exists_measurable_map_eq alg.p0).choose
Type uses (1)
Body uses (1)
Used by (12)

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instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy๐Ÿ”—

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (alg : Algorithm ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm ๐“ ๐“จ) (n : โ„•) : IsMarkovKernel (alg.policy n)
Type uses (1)
Used by (14)

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Proof
alg.h_policy n

algFunction๐Ÿ”—

DefinitionBandits.ArrayModel.algFunction
Details

The next action is the image of the history and a uniform random variable by this function.

๐Ÿ”—def
Bandits.ArrayModel.algFunction.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) โ†’ โ†‘unitInterval โ†’ ๐“
Bandits.ArrayModel.algFunction.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) โ†’
    โ†‘unitInterval โ†’ ๐“

Code

noncomputable
def algFunction (alg : Algorithm ๐“ R) (n : โ„•) :
    (Iic n โ†’ ๐“ ร— R) โ†’ I โ†’ ๐“ :=
  (Kernel.exists_measurable_map_eq_unitInterval (alg.policy n)).choose
Type uses (1)
Body uses (1)
Used by (17)

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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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hist๐Ÿ”—

DefinitionBandits.ArrayModel.hist
Details

History of actions and rewards up to time n in the array model.

๐Ÿ”—def
Bandits.ArrayModel.hist.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“] (alg : Learning.Algorithm ๐“ R)
  (ฯ‰ : probSpace ๐“ R) (n : โ„•) : โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R
Bandits.ArrayModel.hist.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  (alg : Learning.Algorithm ๐“ R)
  (ฯ‰ : probSpace ๐“ R) (n : โ„•) :
  โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R

Code

noncomputable
def hist [DecidableEq ๐“] (alg : Algorithm ๐“ R) (ฯ‰ : probSpace ๐“ R) : (n : โ„•) โ†’ Iic n โ†’ ๐“ ร— R
| 0 => fun _ โ†ฆ (initAlgFunction alg (ฯ‰.1 0), ฯ‰.2 0 (initAlgFunction alg (ฯ‰.1 0)))
| n + 1 =>
  let hn : Iic n โ†’ ๐“ ร— R := hist alg ฯ‰ n
  let a : ๐“ := algFunction alg n hn (ฯ‰.1 (n + 1))
  fun i โ†ฆ if hin : i โ‰ค n then hn โŸจi, by simp [hin]โŸฉ else (a, ฯ‰.2 (pullCount' n hn a) a)
Type uses (2)
Body uses (3)
Used by (30)

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