`Bandits.UCB.pullCount_arm_le`🔗

Lemma

No docstring.

theorem

Bandits.UCB.pullCount_arm_le.{u_1} {K : ℕ} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ} {Ω : Type u_1}
  {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ} {n : ℕ} {ω : Ω} [Nonempty (Fin K)]
  (hc : 0 ≤ c)
  (h_best :
    ∫ (x : ℝ), id x ∂ν (bestArm ν) ≤
      Learning.empMean A R (bestArm ν) n ω +
        ucbWidth A c (bestArm ν) n ω)
  (h_arm :
    Learning.empMean A R (A n ω) n ω - ucbWidth A c (A n ω) n ω ≤
      ∫ (x : ℝ), id x ∂ν (A n ω))
  (h_le :
    Learning.empMean A R (bestArm ν) n ω +
        ucbWidth A c (bestArm ν) n ω ≤
      Learning.empMean A R (A n ω) n ω + ucbWidth A c (A n ω) n ω)
  (h_gap_pos : 0 < gap ν (A n ω))
  (h_pull_pos : 0 < Learning.pullCount A (A n ω) n ω) :
  ↑(Learning.pullCount A (A n ω) n ω) ≤
    8 * c * Real.log (↑n + 1) / gap ν (A n ω) ^ 2
Bandits.UCB.pullCount_arm_le.{u_1} {K : ℕ}
  {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  {Ω : Type u_1} {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ} {n : ℕ} {ω : Ω}
  [Nonempty (Fin K)] (hc : 0 ≤ c)
  (h_best :
    ∫ (x : ℝ), id x ∂ν (bestArm ν) ≤
      Learning.empMean A R (bestArm ν) n
          ω +
        ucbWidth A c (bestArm ν) n ω)
  (h_arm :
    Learning.empMean A R (A n ω) n ω -
        ucbWidth A c (A n ω) n ω ≤
      ∫ (x : ℝ), id x ∂ν (A n ω))
  (h_le :
    Learning.empMean A R (bestArm ν) n ω +
        ucbWidth A c (bestArm ν) n ω ≤
      Learning.empMean A R (A n ω) n ω +
        ucbWidth A c (A n ω) n ω)
  (h_gap_pos : 0 < gap ν (A n ω))
  (h_pull_pos :
    0 <
      Learning.pullCount A (A n ω) n ω) :
  ↑(Learning.pullCount A (A n ω) n ω) ≤
    8 * c * Real.log (↑n + 1) /
      gap ν (A n ω) ^ 2

Code

lemma pullCount_arm_le [Nonempty (Fin K)] (hc : 0 ≤ c)
    (h_best : (ν (bestArm ν))[id] ≤ empMean A R (bestArm ν) n ω + ucbWidth A c (bestArm ν) n ω)
    (h_arm : empMean A R (A n ω) n ω - ucbWidth A c (A n ω) n ω ≤ (ν (A n ω))[id])
    (h_le : empMean A R (bestArm ν) n ω + ucbWidth A c (bestArm ν) n ω ≤
      empMean A R (A n ω) n ω + ucbWidth A c (A n ω) n ω)
    (h_gap_pos : 0 < gap ν (A n ω)) (h_pull_pos : 0 < pullCount A (A n ω) n ω) :
    pullCount A (A n ω) n ω ≤ 8 * c * log (n + 1) / gap ν (A n ω) ^ 2

Proof

by
  have h_gap_le := gap_arm_le_two_mul_ucbWidth h_best h_arm h_le
  rw [ucbWidth] at h_gap_le
  have h2 : (gap ν (A n ω)) ^ 2 ≤ (2 * √(2 * c * log (n + 1) / pullCount A (A n ω) n ω)) ^ 2 := by
    gcongr
  rw [mul_pow, sq_sqrt] at h2
  · have : (2 : ℝ) ^ 2 = 4 := by norm_num
    rw [this] at h2
    field_simp at h2 ⊢
    grind
  · have : 0 ≤ log (n + 1) := by simp [log_nonneg]
    positivity

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 3th 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: 6 project declarations, 94 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.UCB.pullCount_arm_le🔗

`Bandits.UCB.pullCount_arm_le`🔗