`Bandits.TS.integral_regret_le`🔗

Theorem

If Thompson sampling has the correct prior over environments and every environment has K actions, each of which has a corresponding reward between l and u that is sub-Gaussian with variance proxy σ2 after its mean is subtracted, then the Bayesian regret at time n is at most (2 * K + 1) * (u - l) + 8 * √(σ2 * K * n * Real.log n).

theorem

Bandits.TS.integral_regret_le.{u_1, u_2} {K : ℕ} [Nonempty (Fin K)]
  {l u σ2 : ℝ} {Ω : Type u_1} [MeasurableSpace Ω] {𝓔 : Type u_2}
  [MeasurableSpace 𝓔] [StandardBorelSpace 𝓔] [Nonempty 𝓔]
  {Q : MeasureTheory.Measure 𝓔} [MeasureTheory.IsProbabilityMeasure Q]
  {κ : ProbabilityTheory.Kernel (𝓔 × Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel κ] {E : Ω → 𝓔} {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] (hK : 0 < K)
  (h : Learning.IsBayesAlgEnvSeq Q κ (tsAlgorithm hK Q κ) E A R P)
  (hlu : l ≤ u)
  (hm : ∀ (e : 𝓔) (a : Fin K), ∫ (x : ℝ), id x ∂κ (e, a) ∈ Set.Icc l u)
  (hσ2 : 0 < σ2)
  (hs :
    ∀ (e : 𝓔) (a : Fin K),
      ProbabilityTheory.HasSubgaussianMGF
        (fun x => x - ∫ (x : ℝ), id x ∂κ (e, a)) ⟨σ2, ⋯⟩ (κ (e, a)))
  (n : ℕ) :
  ∫ (x : Ω), Learning.IsBayesAlgEnvSeq.regret κ E A n x ∂P ≤
    (2 * ↑K + 1) * (u - l) + 8 * √(σ2 * ↑K * ↑n * Real.log ↑n)
Bandits.TS.integral_regret_le.{u_1, u_2}
  {K : ℕ} [Nonempty (Fin K)] {l u σ2 : ℝ}
  {Ω : Type u_1} [MeasurableSpace Ω]
  {𝓔 : Type u_2} [MeasurableSpace 𝓔]
  [StandardBorelSpace 𝓔] [Nonempty 𝓔]
  {Q : MeasureTheory.Measure 𝓔}
  [MeasureTheory.IsProbabilityMeasure Q]
  {κ :
    ProbabilityTheory.Kernel (𝓔 × Fin K)
      ℝ}
  [ProbabilityTheory.IsMarkovKernel κ]
  {E : Ω → 𝓔} {A : ℕ → Ω → Fin K}
  {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  (hK : 0 < K)
  (h :
    Learning.IsBayesAlgEnvSeq Q κ
      (tsAlgorithm hK Q κ) E A R P)
  (hlu : l ≤ u)
  (hm :
    ∀ (e : 𝓔) (a : Fin K),
      ∫ (x : ℝ), id x ∂κ (e, a) ∈
        Set.Icc l u)
  (hσ2 : 0 < σ2)
  (hs :
    ∀ (e : 𝓔) (a : Fin K),
      ProbabilityTheory.HasSubgaussianMGF
        (fun x =>
          x - ∫ (x : ℝ), id x ∂κ (e, a))
        ⟨σ2, ⋯⟩ (κ (e, a)))
  (n : ℕ) :
  ∫ (x : Ω),
      Learning.IsBayesAlgEnvSeq.regret κ E
        A n x ∂P ≤
    (2 * ↑K + 1) * (u - l) +
      8 * √(σ2 * ↑K * ↑n * Real.log ↑n)

Code

theorem integral_regret_le (hK : 0 < K) (h : IsBayesAlgEnvSeq Q κ (tsAlgorithm hK Q κ) E A R P)
    (hlu : l ≤ u) (hm : ∀ e a, (κ (e, a))[id] ∈ (Set.Icc l u)) (hσ2 : 0 < σ2)
    (hs : ∀ e a, HasSubgaussianMGF (fun x ↦ x - (κ (e, a))[id]) ⟨σ2, hσ2.le⟩ (κ (e, a))) (n : ℕ) :
    P[IsBayesAlgEnvSeq.regret κ E A n]
      ≤ (2 * K + 1) * (u - l) + 8 * √(σ2 * K * n * Real.log n)

Proof

by
  by_cases hn : n = 0
  · simp [hn, IsBayesAlgEnvSeq.regret, Bandits.regret]
    nlinarith
  have hδ : (0 : ℝ) < 1 / n ^ 2 := by positivity
  calc P[IsBayesAlgEnvSeq.regret κ E A n]
      = _ :=
        integral_regret_eq_add hK h hm n
    _ ≤ _ :=
        add_le_add
          (integral_sum_range_actionMean_bestAction_sub_ucb_bestAction_le h hlu hm hσ2 hs hδ n)
          (integral_sum_range_ucb_action_sub_actionMean_action_le h hlu hm hσ2 hs hδ n)
    _ = K * (u - l) + (K + 1) * (u - l) * ((n - 1) / n)
          + 4 * √((2 : ℝ) ^ 2 * (σ2 * K * n * Real.log n)) := by
        field_simp
        rw [Real.log_pow]
        ring_nf
    _ = K * (u - l) + (K + 1) * (u - l) * ((n - 1) / n) + 8 * √(σ2 * K * n * Real.log n) := by
        rw [Real.sqrt_mul (by positivity), Real.sqrt_sq (by norm_num)]
        ring
    _ ≤ K * (u - l) + (K + 1) * (u - l) * 1 + 8 * √(σ2 * K * n * Real.log n) := by -- loose
        have : 0 ≤ u - l := sub_nonneg.2 hlu
        gcongr
        rw [div_le_one (by positivity)]
        linarith
    _ = _ := by
        ring

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: 23 project declarations, 126 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.TS.integral_regret_le🔗

`Bandits.TS.integral_regret_le`🔗