`Bandits.prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

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Minimal Lean file

`prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

LemmaBandits.prob_exists_pullCount_eq_and_sumRewards_mem_le

Details

No docstring.

theorem

Bandits.prob_exists_pullCount_eq_and_sumRewards_mem_le.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  [Nonempty 𝓐] [Countable 𝓐] [MeasurableSingletonClass 𝓐]
  (h : Learning.IsAlgEnvSeq A R alg (Learning.stationaryEnv ν) P)
  (a : 𝓐) (m : ℕ) {B : Set ℝ} (hB : MeasurableSet B) :
  P
      {ω |
        ∃ n,
          Learning.pullCount A a n ω = m ∧
            Learning.sumRewards A R a n ω ∈ B} ≤
    (streamMeasure ν) {ω | ∑ i ∈ Finset.range m, ω i a ∈ B}
Bandits.prob_exists_pullCount_eq_and_sumRewards_mem_le.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  [Nonempty 𝓐] [Countable 𝓐]
  [MeasurableSingletonClass 𝓐]
  (h :
    Learning.IsAlgEnvSeq A R alg
      (Learning.stationaryEnv ν) P)
  (a : 𝓐) (m : ℕ) {B : Set ℝ}
  (hB : MeasurableSet B) :
  P
      {ω |
        ∃ n,
          Learning.pullCount A a n ω = m ∧
            Learning.sumRewards A R 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 [Countable 𝓐] [MeasurableSingletonClass 𝓐]
    (h : IsAlgEnvSeq A R alg (stationaryEnv ν) P) (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 (6)

Body uses (9)

Used by (2)

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Proof

let s := {p : ℕ → 𝓐 → ℕ × ℝ | ∃ n, (p n a).1 = m ∧ (p n a).2 ∈ B}
  have : s = ⋃ n, (fun p ↦ p n a) ⁻¹' ({m} ×ˢ B) := by
    ext p
    simp [s]
  have hs : MeasurableSet s := by measurability
  calc P {ω | ∃ n, pullCount A a n ω = m ∧ sumRewards A R a n ω ∈ B}
    _ = (ArrayModel.arrayMeasure ν) {ω | ∃ n, pullCount (ArrayModel.action alg) a n ω = m ∧
            sumRewards (ArrayModel.action alg) (ArrayModel.reward alg) a n ω ∈ B} :=
        (h.identDistrib_pullCount_sumRewards
          (ArrayModel.isAlgEnvSeq_arrayMeasure alg ν)).measure_mem_eq hs
    _ ≤ _ := ArrayModel.prob_exists_pullCount_eq_and_sumRewards_mem_le a m hB

Dependency graph

Type dependencies (6)

`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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`IsAlgEnvSeq`🔗

StructureLearning.IsAlgEnvSeq

Details

An algorithm-environment sequence: a sequence of actions and feedbacks generated by an algorithm interacting with an environment.

structure

Learning.IsAlgEnvSeq.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨)
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsAlgEnvSeq.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ω)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsAlgEnvSeq
    (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
    (P : Measure Ω) [IsFiniteMeasure P] : Prop where
  /-- The action sequence is measurable. -/
  measurable_action n : Measurable (A n) := by fun_prop
  /-- The feedback sequence is measurable. -/
  measurable_feedback n : Measurable (Y n) := by fun_prop
  /-- The first action has the correct law. -/
  hasLaw_action_zero : HasLaw (fun ω ↦ (A 0 ω)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.ν0 P
  /-- The next action has the correct conditional distribution given the history. -/
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (history A Y n) (alg.policy n) P
  /-- The next feedback has the correct conditional distribution given the history and
  next action. -/
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, A (n + 1) ω))
      (env.feedback n) P

Type uses (3)

Used by (111)

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`stationaryEnv`🔗

DefinitionLearning.stationaryEnv

Details

A stationary environment, in which the distribution of the next feedback depends only on the last action.

def

Learning.stationaryEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (ν : ProbabilityTheory.Kernel 𝓐 𝓨)
  [ProbabilityTheory.IsMarkovKernel ν] : Environment 𝓐 𝓨
Learning.stationaryEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (ν : ProbabilityTheory.Kernel 𝓐 𝓨)
  [ProbabilityTheory.IsMarkovKernel ν] :
  Environment 𝓐 𝓨

Code

def stationaryEnv (ν : Kernel 𝓐 𝓨) [IsMarkovKernel ν] : Environment 𝓐 𝓨 := obliviousEnv fun _ ↦ ν

Type uses (1)

Environment

Body uses (1)

obliviousEnv

Used by (81)

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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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`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

Used by (44)

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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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All dependencies, transitively (3)

`Environment`🔗

StructureLearning.Environment

Details

A stochastic environment.

structure

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

Code

structure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ν0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ν0]

Used by (128)

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`history`🔗

DefinitionLearning.history

Details

History of the algorithm-environment sequence up to time n.

def

Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def history (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) : Iic n → 𝓐 × 𝓨 :=
  fun i ↦ (A i ω, Y i ω)

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`obliviousEnv`🔗

DefinitionLearning.obliviousEnv

Details

An oblivious environment, in which the distribution of the next feedback depends only on the last action, but in a possibly time-dependent manner.

def

Learning.obliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (ν n)] : Environment 𝓐 𝓨
Learning.obliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [∀ (n : ℕ),
      ProbabilityTheory.IsMarkovKernel
        (ν n)] :
  Environment 𝓐 𝓨

Code

def obliviousEnv (ν : ℕ → Kernel 𝓐 𝓨) [∀ n, IsMarkovKernel (ν n)] : Environment 𝓐 𝓨 where
  feedback n := (ν (n + 1)).prodMkLeft _
  ν0 := ν 0

Type uses (1)

Environment

Used by (10)

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Bandits.prob_exists_pullCount_eq_and_sumRewards_mem_le🔗

prob_exists_pullCount_eq_and_sumRewards_mem_le🔗

Algorithm🔗

IsAlgEnvSeq🔗

stationaryEnv🔗

pullCount🔗

sumRewards🔗

streamMeasure🔗

Environment🔗

history🔗

obliviousEnv🔗

`Bandits.prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

`prob_exists_pullCount_eq_and_sumRewards_mem_le`🔗

`Algorithm`🔗

`IsAlgEnvSeq`🔗

`stationaryEnv`🔗

`pullCount`🔗

`sumRewards`🔗

`streamMeasure`🔗

`Environment`🔗

`history`🔗

`obliviousEnv`🔗