LeanMachineLearning

Bandits.UCB.sum_indicator_good_event_eq_zero🔗

Lemma

From the authors

If C ≥ 8 * c * σ2 * log (n + 1) / gap ν a ^ 2, then arm a is never pulled at a time s < n at which it already has more than C pulls and the means of the best arm and of a lie in their confidence intervals.

Types
  • Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
Given
  • K : ℕNeZero K
  • c : ℝ
  • ν : ProbabilityTheory.Kernel (Fin K) ℝA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel νA kernel is a Markov kernel if every measure in its image is a probability measure.
  • P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsProbabilityMeasure PA measure μ is called a probability measure if μ univ = 1.
  • O : ℕ → Ω → Unit
  • A : ℕ → Ω → Fin K
  • R : ℕ → Ω → ℝ
  • σ2 : NNReal
  • a : Fin K
  • n : ℕ
  • C : ℕ
Assuming
Then
∀ᵐ (ω : Ω) ∂P,
  ∑ s ∈ Finset.range n,
      {s |
            A s ω = a ∧
              C < Learning.pullCount A a s ω ∧
                ∫ (x : ℝ), id x ∂ν (bestArm ν) ≤
                    Learning.empMean A R (bestArm ν) s ω + ucbWidth A (c * ↑σ2) (bestArm ν) s ω ∧
                  Learning.empMean A R (A s ω) s ω - ucbWidth A (c * ↑σ2) (A s ω) s ω ≤
                    ∫ (x : ℝ), id x ∂ν (A s ω)}.indicator
        1 s =
    0
f.Eventually p or ∀ᶠ x in f, p x mean that {x | p x} ∈ f.
Code
lemma sum_indicator_good_event_eq_zero
    (h : IsAlgEnvSeq O A R (ucbAlgorithm K (c * σ2)) (stationaryEnv ν) P)
    (hc : 0 ≤ c) (a : Fin K) (h_gap : 0 < gap ν a) (n C : ℕ)
    (hC : C ≠ 0) (hC' : 8 * c * σ2 * log (n + 1) / gap ν a ^ 2 ≤ C) :
    ∀ᵐ ω ∂P,
    ∑ s ∈ range n, {s | A s ω = a ∧ C < pullCount A a s ω ∧
      (ν (bestArm ν))[id] ≤ empMean A R (bestArm ν) s ω + ucbWidth A (c * σ2) (bestArm ν) s ω ∧
      empMean A R (A s ω) s ω - ucbWidth A (c * σ2) (A s ω) s ω
        ≤ (ν (A s ω))[id]}.indicator 1 s = 0
Proof
by
  have h_ae := forall_ucbIndex_le_ucbIndex_arm h (bestArm ν) (ν := ν) (c := c * σ2)
  have h_gt := time_gt_of_pullCount_gt_one h a (ν := ν) (c := c * σ2)
  filter_upwards [h_ae, h_gt] with ω h_le h_time_ge
  simp only [id_eq, tsub_le_iff_right, sum_eq_zero_iff, mem_range, Set.indicator_apply_eq_zero,
    Set.mem_ofPred_eq, Pi.one_apply, one_ne_zero, imp_false, not_and, not_le]
  intro k hn h_arm hC_lt h_le_best
  by_contra! h_le_arm
  have h := pullCount_le_of_ucbIndex_le (b := A k ω) (by positivity : 0 ≤ c * σ2) h_le_best
    (by simpa) ?_ ?_ ?_
  rotate_left
  · refine h_le _ ?_
    refine (h_time_ge _ ?_).le
    refine lt_of_le_of_lt ?_ hC_lt
    grind
  · rwa [h_arm]
  · rw [h_arm]
    exact zero_le.trans_lt hC_lt
  refine lt_irrefl (8 * c * σ2 * log (n + 1) / gap ν a ^ 2) ?_
  refine hC'.trans_lt (lt_of_lt_of_le ?_ (h.trans ?_))
  · rw [h_arm]
    exact mod_cast hC_lt
  · rw [h_arm]
    simp_rw [← mul_assoc]
    gcongr

New in v4.34.0-rc2-90-gdde3322 (2026-09-16), and its meaning has not changed since.

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: 27 project declarations, 150 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.