`Bandits.integral_regret_le_of_forall_integral_pullCount_le`🔗

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

`integral_regret_le_of_forall_integral_pullCount_le`🔗

LemmaBandits.integral_regret_le_of_forall_integral_pullCount_le

Details

To bound the expected regret, it suffices to bound the expected number of pulls for each action with positive gap.

theorem

Bandits.integral_regret_le_of_forall_integral_pullCount_le.{u_1, u_2}
  {𝓐 : Type u_1} {Ω : Type u_2} [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω} {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {n : ℕ} [StandardBorelSpace 𝓐]
  [Fintype 𝓐] {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {alg : Learning.Algorithm 𝓐 ℝ}
  {env : Learning.Environment 𝓐 ℝ} {B : 𝓐 → ℝ}
  (h : Learning.IsAlgEnvSeq A R alg env P)
  (h_le :
    ∀ (a : 𝓐),
      gap ν a ≠ 0 → ∫ (ω : Ω), ↑(Learning.pullCount A a n ω) ∂P ≤ B a) :
  ∫ (x : Ω), regret ν A n x ∂P ≤ ∑ a, gap ν a * B a
Bandits.integral_regret_le_of_forall_integral_pullCount_le.{u_1,
    u_2}
  {𝓐 : Type u_1} {Ω : Type u_2}
  [DecidableEq 𝓐] {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ} {n : ℕ}
  [StandardBorelSpace 𝓐] [Fintype 𝓐]
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {alg : Learning.Algorithm 𝓐 ℝ}
  {env : Learning.Environment 𝓐 ℝ}
  {B : 𝓐 → ℝ}
  (h : Learning.IsAlgEnvSeq A R alg env P)
  (h_le :
    ∀ (a : 𝓐),
      gap ν a ≠ 0 →
        ∫ (ω : Ω),
            ↑(Learning.pullCount A a n
                ω) ∂P ≤
          B a) :
  ∫ (x : Ω), regret ν A n x ∂P ≤
    ∑ a, gap ν a * B a

Code

lemma integral_regret_le_of_forall_integral_pullCount_le
    [StandardBorelSpace 𝓐] [Fintype 𝓐] {P : Measure Ω} [IsProbabilityMeasure P]
    {alg : Algorithm 𝓐 ℝ} {env : Environment 𝓐 ℝ} {B : 𝓐 → ℝ}
    (h : IsAlgEnvSeq A R alg env P)
    (h_le : ∀ a, gap ν a ≠ 0 → ∫ ω, (pullCount A a n ω : ℝ) ∂P ≤ B a) :
    P[regret ν A n] ≤ ∑ a, gap ν a * B a

Type uses (6)

Body uses (2)

Used by (2)

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Proof

by
  have hA := h.measurable_action
  rw [integral_regret_eq_sum_gap_mul_integral_pullCount hA]
  gcongr 1 with a
  by_cases h_gap : gap ν a = 0
  · simp [h_gap]
  gcongr
  · exact gap_nonneg
  · exact h_le a h_gap

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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`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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`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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`gap`🔗

DefinitionBandits.gap

Details

Gap of an action a: difference between the highest mean of the actions and the mean of a.

def

Bandits.gap.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ) (a : 𝓐) : ℝ
Bandits.gap.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (a : 𝓐) : ℝ

Code

noncomputable
def gap (ν : Kernel 𝓐 ℝ) (a : 𝓐) : ℝ := (⨆ i, (ν i)[id]) - (ν a)[id]

Used by (27)

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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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`regret`🔗

DefinitionBandits.regret

Details

Regret of a sequence of pulls k : ℕ → 𝓐 at time t for the reward kernel ν ; Kernel 𝓐 ℝ.

def

Bandits.regret.{u_1, u_2} {𝓐 : Type u_1} {Ω : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ
Bandits.regret.{u_1, u_2} {𝓐 : Type u_1}
  {Ω : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ

Code

noncomputable
def regret (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  t * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]

Used by (11)

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All dependencies, transitively (1)

`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)

Used by (72)

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Bandits.integral_regret_le_of_forall_integral_pullCount_le🔗

integral_regret_le_of_forall_integral_pullCount_le🔗

Algorithm🔗

Environment🔗

IsAlgEnvSeq🔗

gap🔗

pullCount🔗

regret🔗

history🔗

`Bandits.integral_regret_le_of_forall_integral_pullCount_le`🔗

`integral_regret_le_of_forall_integral_pullCount_le`🔗

`Algorithm`🔗

`Environment`🔗

`IsAlgEnvSeq`🔗

`gap`🔗

`pullCount`🔗

`regret`🔗

`history`🔗