`Bandits.integral_regret_le_of_forall_integral_pullCount_le`🔗

Lemma

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

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

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-1-g439785b (2026-08-23), the 4th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 7 project declarations, 77 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Bandits.integral_regret_le_of_forall_integral_pullCount_le🔗

`Bandits.integral_regret_le_of_forall_integral_pullCount_le`🔗