`Bandits.UCB.forall_arm_prop`🔗

Lemma

No docstring.

theorem

Bandits.UCB.forall_arm_prop.{u_1} {K : ℕ} {hK : 0 < K} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {Ω : Type u_1}
  {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ}
  (h :
    Learning.IsAlgEnvSeq A R (ucbAlgorithm hK c)
      (Learning.stationaryEnv ν) P) :
  ∀ᵐ (h : Ω) ∂P,
    (∀ n < K, A n h = ⟨n % K, ⋯⟩) ∧
      ∀ (n : ℕ),
        K ≤ n →
          ∀ (a : Fin K),
            Learning.empMean A R a n h + ucbWidth A c a n h ≤
              Learning.empMean A R (A n h) n h +
                ucbWidth A c (A n h) n h
Bandits.UCB.forall_arm_prop.{u_1} {K : ℕ}
  {hK : 0 < K} {c : ℝ}
  {ν : ProbabilityTheory.Kernel (Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {Ω : Type u_1} {mΩ : MeasurableSpace Ω}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ}
  (h :
    Learning.IsAlgEnvSeq A R
      (ucbAlgorithm hK c)
      (Learning.stationaryEnv ν) P) :
  ∀ᵐ (h : Ω) ∂P,
    (∀ n < K, A n h = ⟨n % K, ⋯⟩) ∧
      ∀ (n : ℕ),
        K ≤ n →
          ∀ (a : Fin K),
            Learning.empMean A R a n h +
                ucbWidth A c a n h ≤
              Learning.empMean A R (A n h)
                  n h +
                ucbWidth A c (A n h) n h

Code

lemma forall_arm_prop
    (h : IsAlgEnvSeq A R (ucbAlgorithm hK c) (stationaryEnv ν) P) :
    ∀ᵐ h ∂P,
      (∀ n < K, A n h = ⟨n % K, Nat.mod_lt _ hK⟩) ∧
      (∀ n, K ≤ n → ∀ a, empMean A R a n h + ucbWidth A c a n h ≤
        empMean A R (A n h) n h + ucbWidth A c (A n h) n h)

Proof

by
  simp only [eventually_and]
  constructor
  · exact forall_arm_eq_mod_of_lt h
  · simp_rw [ae_all_iff]
    intro n hn a
    have h_ae := forall_ucbIndex_le_ucbIndex_arm h a
    simp_rw [ae_all_iff] at h_ae
    exact h_ae n hn

Actions: Source · Open Issue

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 5th 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: 21 project declarations, 117 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.forall_arm_prop🔗

`Bandits.UCB.forall_arm_prop`🔗