Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem

`prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

LemmaBandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le

Details

No docstring.

theorem

Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐} [DecidableEq 𝓐] [Countable 𝓐]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐] {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] (a : 𝓐) (m : ℕ) {B : Set ℝ}
  (hB : MeasurableSet B) :
  (arrayMeasure ν)
      {ω |
        ∃ n,
          Learning.pullCount (action alg) a n ω = m ∧
            Learning.sumRewards (action alg) (reward alg) a n ω ∈ B} ≤
    (streamMeasure ν) {ω | ∑ i ∈ Finset.range m, ω i a ∈ B}
Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  [DecidableEq 𝓐] [Countable 𝓐]
  [StandardBorelSpace 𝓐] [Nonempty 𝓐]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  (a : 𝓐) (m : ℕ) {B : Set ℝ}
  (hB : MeasurableSet B) :
  (arrayMeasure ν)
      {ω |
        ∃ n,
          Learning.pullCount (action alg)
                a n ω =
              m ∧
            Learning.sumRewards
                (action alg) (reward alg)
                a n ω ∈
              B} ≤
    (streamMeasure ν)
      {ω |
        ∑ i ∈ Finset.range m, ω i a ∈ B}

Code

lemma prob_exists_pullCount_eq_and_sumRewards_mem_le (a : 𝓐) (m : ℕ) {B : Set ℝ}
    (hB : MeasurableSet B) : 𝔓 {ω | ∃ n, pullCount A a n ω = m ∧ sumRewards A R a n ω ∈ B} ≤
      streamMeasure ν {ω | ∑ i ∈ range m, ω i a ∈ B}

Type uses (9)

Body uses (2)

Used by (1)

prob_exists_pullCount_eq_and_sumRewards_mem_le

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Proof

calc
    _ ≤ 𝔓 {ω | ∑ i ∈ range m, ω.2 i a ∈ B} := by
        apply measure_mono
        intro ω ⟨n, hp, hn⟩
        rwa [sumRewards_eq alg a n ω, hp] at hn
    _ = streamMeasure ν {ω | ∑ i ∈ range m, ω i a ∈ B} :=
        (identDistrib_sum_range_snd a m).measure_mem_eq hB

`Algorithm`🔗

StructureLearning.Algorithm

Details

A stochastic, sequential algorithm.

structure

Learning.Algorithm.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]

Used by (216)

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`probSpace`🔗

DefinitionBandits.ArrayModel.probSpace

Details

Probability space for the array model of stochastic bandits.

def

Bandits.ArrayModel.probSpace.{u_1, u_2} (𝓐 : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)
Bandits.ArrayModel.probSpace.{u_1, u_2}
  (𝓐 : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)

Code

def probSpace : Type _ := (ℕ → I) × (ℕ → 𝓐 → R)

Used by (64)

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`instMeasurableSpaceProbSpace`🔗

InstanceBandits.ArrayModel.instMeasurableSpaceProbSpace

Details

No docstring.

def

Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3, u_4}
  {𝓐 : Type u_3} {R : Type u_4} [MeasurableSpace R] :
  MeasurableSpace (probSpace 𝓐 R)
Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3,
    u_4}
  {𝓐 : Type u_3} {R : Type u_4}
  [MeasurableSpace R] :
  MeasurableSpace (probSpace 𝓐 R)

Code

instance {𝓐 R : Type*} [MeasurableSpace R] : MeasurableSpace (probSpace 𝓐 R)

Type uses (1)

probSpace

Used by (41)

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Proof

inferInstanceAs (MeasurableSpace ((ℕ → I) × (ℕ → 𝓐 → R)))

`arrayMeasure`🔗

DefinitionBandits.ArrayModel.arrayMeasure

Details

Probability measure for the array model of stochastic bandits.

def

Bandits.ArrayModel.arrayMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (probSpace 𝓐 R)
Bandits.ArrayModel.arrayMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (probSpace 𝓐 R)

Code

noncomputable
def arrayMeasure (ν : Kernel 𝓐 R) : Measure (probSpace 𝓐 R) :=
  (Measure.infinitePi fun _ ↦ volume).prod (streamMeasure ν)

Type uses (2)

Body uses (1)

streamMeasure

Used by (29)

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`pullCount`🔗

DefinitionLearning.pullCount

Details

Number of times action a was chosen up to time t (excluding t).

def

Learning.pullCount.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ
Learning.pullCount.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (t : ℕ) (ω : Ω) : ℕ

Code

noncomputable
def pullCount (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ :=
  #(filter (fun s ↦ A s ω = a) (range t))

Used by (146)

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`action`🔗

DefinitionBandits.ArrayModel.action

Details

Action taken at time n in the array model.

def

Bandits.ArrayModel.action.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐] (alg : Learning.Algorithm 𝓐 R)
  (n : ℕ) (ω : probSpace 𝓐 R) : 𝓐
Bandits.ArrayModel.action.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : ℕ)
  (ω : probSpace 𝓐 R) : 𝓐

Code

noncomputable
def action [DecidableEq 𝓐] (alg : Algorithm 𝓐 R) (n : ℕ) (ω : probSpace 𝓐 R) : 𝓐 :=
  (hist alg ω n ⟨n, by simp⟩).1

Type uses (2)

Body uses (1)

hist

Used by (43)

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`sumRewards`🔗

DefinitionLearning.sumRewards

Details

Sum of rewards obtained when pulling action a up to time t (exclusive).

def

Learning.sumRewards.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (R' : ℕ → Ω → ℝ) (a : 𝓐) (t : ℕ)
  (ω : Ω) : ℝ
Learning.sumRewards.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐)
  (R' : ℕ → Ω → ℝ) (a : 𝓐) (t : ℕ)
  (ω : Ω) : ℝ

Code

def sumRewards (A : ℕ → Ω → 𝓐) (R' : ℕ → Ω → ℝ) (a : 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  ∑ s ∈ range t, if A s ω = a then R' s ω else 0

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`reward`🔗

DefinitionBandits.ArrayModel.reward

Details

Reward received at time n in the array model.

def

Bandits.ArrayModel.reward.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐] (alg : Learning.Algorithm 𝓐 R)
  (n : ℕ) (ω : probSpace 𝓐 R) : R
Bandits.ArrayModel.reward.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : ℕ)
  (ω : probSpace 𝓐 R) : R

Code

noncomputable
def reward [DecidableEq 𝓐] (alg : Algorithm 𝓐 R) (n : ℕ) (ω : probSpace 𝓐 R) : R :=
  (hist alg ω n ⟨n, by simp⟩).2

Type uses (2)

Body uses (1)

hist

Used by (24)

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`streamMeasure`🔗

DefinitionBandits.streamMeasure

Details

Measure of an infinite stream of rewards from each action.

def

Bandits.streamMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) : MeasureTheory.Measure (ℕ → 𝓐 → R)
Bandits.streamMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (ℕ → 𝓐 → R)

Code

noncomputable
def streamMeasure (ν : Kernel 𝓐 R) : Measure (ℕ → 𝓐 → R) :=
  Measure.infinitePi fun _ ↦ Measure.infinitePi ν

Used by (56)

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`instIsProbabilityMeasureP0`🔗

InstanceLearning.instIsProbabilityMeasureP0

Details

No docstring.

theorem

Learning.instIsProbabilityMeasureP0.{u_1, u_2} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm 𝓐 𝓨) : IsProbabilityMeasure alg.p0

Type uses (1)

Algorithm

Used by (13)

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Proof

alg.hp0

`initAlgFunction`🔗

DefinitionBandits.ArrayModel.initAlgFunction

Details

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

def

Bandits.ArrayModel.initAlgFunction.{u_1, u_2} {𝓐 : Type u_1}
  {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  [Nonempty 𝓐] [StandardBorelSpace 𝓐] (alg : Learning.Algorithm 𝓐 R) :
  ↑unitInterval → 𝓐
Bandits.ArrayModel.initAlgFunction.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐]
  (alg : Learning.Algorithm 𝓐 R) :
  ↑unitInterval → 𝓐

Code

noncomputable
def initAlgFunction (alg : Algorithm 𝓐 R) : I → 𝓐 :=
  (Measure.exists_measurable_map_eq alg.p0).choose

Type uses (1)

Algorithm

Body uses (1)

instIsProbabilityMeasureP0

Used by (12)

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`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy

Details

No docstring.

theorem

Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨} (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (alg : Algorithm 𝓐 𝓨) (n : ℕ) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm 𝓐 𝓨) (n : ℕ) : IsMarkovKernel (alg.policy n)

Type uses (1)

Algorithm

Used by (14)

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Proof

alg.h_policy n

`algFunction`🔗

DefinitionBandits.ArrayModel.algFunction

Details

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

def

Bandits.ArrayModel.algFunction.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] (alg : Learning.Algorithm 𝓐 R) (n : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × R) → ↑unitInterval → 𝓐
Bandits.ArrayModel.algFunction.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐]
  (alg : Learning.Algorithm 𝓐 R) (n : ℕ) :
  (↥(Finset.Iic n) → 𝓐 × R) →
    ↑unitInterval → 𝓐

Code

noncomputable
def algFunction (alg : Algorithm 𝓐 R) (n : ℕ) :
    (Iic n → 𝓐 × R) → I → 𝓐 :=
  (Kernel.exists_measurable_map_eq_unitInterval (alg.policy n)).choose

Type uses (1)

Algorithm

Body uses (1)

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy

Used by (17)

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`pullCount'`🔗

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.

def

Learning.pullCount'.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  [DecidableEq 𝓐] (n : ℕ) (h : ↥(Finset.Iic n) → 𝓐 × R) (a : 𝓐) : ℕ
Learning.pullCount'.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  [DecidableEq 𝓐] (n : ℕ)
  (h : ↥(Finset.Iic n) → 𝓐 × R) (a : 𝓐) :
  ℕ

Code

noncomputable
def pullCount' (n : ℕ) (h : Iic n → 𝓐 × R) (a : 𝓐) := #{s | (h s).1 = a}

Used by (29)

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`hist`🔗

DefinitionBandits.ArrayModel.hist

Details

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

def

Bandits.ArrayModel.hist.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐] (alg : Learning.Algorithm 𝓐 R)
  (ω : probSpace 𝓐 R) (n : ℕ) : ↥(Finset.Iic n) → 𝓐 × R
Bandits.ArrayModel.hist.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} [Nonempty 𝓐]
  [StandardBorelSpace 𝓐] [DecidableEq 𝓐]
  (alg : Learning.Algorithm 𝓐 R)
  (ω : probSpace 𝓐 R) (n : ℕ) :
  ↥(Finset.Iic n) → 𝓐 × R

Code

noncomputable
def hist [DecidableEq 𝓐] (alg : Algorithm 𝓐 R) (ω : probSpace 𝓐 R) : (n : ℕ) → Iic n → 𝓐 × R
| 0 => fun _ ↦ (initAlgFunction alg (ω.1 0), ω.2 0 (initAlgFunction alg (ω.1 0)))
| n + 1 =>
  let hn : Iic n → 𝓐 × R := hist alg ω n
  let a : 𝓐 := algFunction alg n hn (ω.1 (n + 1))
  fun i ↦ if hin : i ≤ n then hn ⟨i, by simp [hin]⟩ else (a, ω.2 (pullCount' n hn a) a)

Type uses (2)

Body uses (3)

Used by (30)

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LeanMachineLearning exposition

`Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

`prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

`Algorithm`🔗

`probSpace`🔗

`instMeasurableSpaceProbSpace`🔗

`arrayMeasure`🔗

`pullCount`🔗

`action`🔗

`sumRewards`🔗

`reward`🔗

`streamMeasure`🔗

`instIsProbabilityMeasureP0`🔗

`initAlgFunction`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`algFunction`🔗

`pullCount'`🔗

`hist`🔗

Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le🔗

prob_exists_pullCount_eq_and_sumRewards_mem_le🔗

Algorithm🔗

probSpace🔗

instMeasurableSpaceProbSpace🔗

arrayMeasure🔗

pullCount🔗

action🔗

sumRewards🔗

reward🔗

streamMeasure🔗

instIsProbabilityMeasureP0🔗

initAlgFunction🔗

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy🔗

algFunction🔗

pullCount'🔗

hist🔗

`Bandits.ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

`prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

`Algorithm`🔗

`probSpace`🔗

`instMeasurableSpaceProbSpace`🔗

`arrayMeasure`🔗

`pullCount`🔗

`action`🔗

`sumRewards`🔗

`reward`🔗

`streamMeasure`🔗

`instIsProbabilityMeasureP0`🔗

`initAlgFunction`🔗

`instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy`🔗

`algFunction`🔗

`pullCount'`🔗

`hist`🔗