LeanMachineLearning exposition

Bandits.ArrayModel.hasLaw_action_zero๐Ÿ”—

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Minimal Lean file

hasLaw_action_zero๐Ÿ”—

LemmaBandits.ArrayModel.hasLaw_action_zero
Details

No docstring.

๐Ÿ”—theorem
Bandits.ArrayModel.hasLaw_action_zero.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  [Nonempty ๐“] [StandardBorelSpace ๐“] [DecidableEq ๐“] [Countable ๐“]
  (alg : Learning.Algorithm ๐“ R) (ฮฝ : ProbabilityTheory.Kernel ๐“ R)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] :
  ProbabilityTheory.HasLaw (action alg 0) (Learning.Algorithm.p0 alg)
    (arrayMeasure ฮฝ)
Bandits.ArrayModel.hasLaw_action_zero.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] [DecidableEq ๐“]
  [Countable ๐“]
  (alg : Learning.Algorithm ๐“ R)
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R)
  [ProbabilityTheory.IsMarkovKernel ฮฝ] :
  ProbabilityTheory.HasLaw (action alg 0)
    (Learning.Algorithm.p0 alg)
    (arrayMeasure ฮฝ)

Code

lemma hasLaw_action_zero (alg : Algorithm ๐“ R) (ฮฝ : Kernel ๐“ R) [IsMarkovKernel ฮฝ] :
    HasLaw (action alg 0) alg.p0 (arrayMeasure ฮฝ) where
  map_eq
Type uses (5)
Body uses (6)
Used by (1)

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Proof
by
    calc (arrayMeasure ฮฝ).map (fun ฯ‰ โ†ฆ initAlgFunction alg (ฯ‰.1 0))
    _ = ((arrayMeasure ฮฝ).fst.map (Function.eval 0)).map (initAlgFunction alg) := by
      rw [Measure.fst, Measure.map_map (by fun_prop) (by fun_prop),
        Measure.map_map (by fun_prop) (by fun_prop)]
      rfl
    _ = (volume : Measure I).map (initAlgFunction alg) := by
      simp only [arrayMeasure, Measure.fst_prod]
      rw [(measurePreserving_eval_infinitePi (fun _ โ†ฆ volume) 0).map_eq]
    _ = alg.p0 := initAlgFunction_map alg

Dependency graph

Type dependencies (5)

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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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)))

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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arrayMeasure๐Ÿ”—

DefinitionBandits.ArrayModel.arrayMeasure
Details

Probability measure for the array model of stochastic bandits.

๐Ÿ”—def
Bandits.ArrayModel.arrayMeasure.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R) :
  MeasureTheory.Measure (probSpace ๐“ R)
Bandits.ArrayModel.arrayMeasure.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R) :
  MeasureTheory.Measure (probSpace ๐“ R)

Code

noncomputable
def arrayMeasure (ฮฝ : Kernel ๐“ R) : Measure (probSpace ๐“ R) :=
  (Measure.infinitePi fun _ โ†ฆ volume).prod (streamMeasure ฮฝ)
Type uses (2)
Body uses (1)
Used by (29)

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All dependencies, transitively (7)

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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streamMeasure๐Ÿ”—

DefinitionBandits.streamMeasure
Details

Measure of an infinite stream of rewards from each action.

๐Ÿ”—def
Bandits.streamMeasure.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R) : MeasureTheory.Measure (โ„• โ†’ ๐“ โ†’ R)
Bandits.streamMeasure.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R}
  (ฮฝ : ProbabilityTheory.Kernel ๐“ R) :
  MeasureTheory.Measure (โ„• โ†’ ๐“ โ†’ R)

Code

noncomputable
def streamMeasure (ฮฝ : Kernel ๐“ R) : Measure (โ„• โ†’ ๐“ โ†’ R) :=
  Measure.infinitePi fun _ โ†ฆ Measure.infinitePi ฮฝ
Used by (56)

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