`Learning.IsBayesAlgEnvSeq.prob_empMean_bestAction_sub_actionMean_le_le`🔗

Lemma

No docstring.

theorem

Learning.IsBayesAlgEnvSeq.prob_empMean_bestAction_sub_actionMean_le_le.{u_1,
    u_2}
  {𝓔 : Type u_1} {Ω : Type u_2} [MeasurableSpace 𝓔] [MeasurableSpace Ω]
  {K : ℕ} [Nonempty (Fin K)] {Q : MeasureTheory.Measure 𝓔}
  {κ : ProbabilityTheory.Kernel (𝓔 × Fin K) ℝ}
  [ProbabilityTheory.IsMarkovKernel κ] {alg : Algorithm (Fin K) ℝ}
  {E : Ω → 𝓔} {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  (h : IsBayesAlgEnvSeq Q κ alg E A R P) {σ2 : NNReal} (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)))
  {δ : ℝ} (hδ : 0 < δ) (n : ℕ) :
  P
      {ω |
        ∃ t < n,
          pullCount A (bestAction κ E ω) t ω ≠ 0 ∧
            empMean A R (bestAction κ E ω) t ω -
                actionMean κ E (bestAction κ E ω) ω ≤
              -√(2 * ↑σ2 * Real.log (1 / δ) /
                    ↑(pullCount A (bestAction κ E ω) t ω))} ≤
    ENNReal.ofReal ((↑n - 1) * δ)
Learning.IsBayesAlgEnvSeq.prob_empMean_bestAction_sub_actionMean_le_le.{u_1,
    u_2}
  {𝓔 : Type u_1} {Ω : Type u_2}
  [MeasurableSpace 𝓔] [MeasurableSpace Ω]
  {K : ℕ} [Nonempty (Fin K)]
  {Q : MeasureTheory.Measure 𝓔}
  {κ :
    ProbabilityTheory.Kernel (𝓔 × Fin K)
      ℝ}
  [ProbabilityTheory.IsMarkovKernel κ]
  {alg : Algorithm (Fin K) ℝ} {E : Ω → 𝓔}
  {A : ℕ → Ω → Fin K} {R : ℕ → Ω → ℝ}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  (h : IsBayesAlgEnvSeq Q κ alg E A R P)
  {σ2 : NNReal} (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)))
  {δ : ℝ} (hδ : 0 < δ) (n : ℕ) :
  P
      {ω |
        ∃ t < n,
          pullCount A (bestAction κ E ω) t
                ω ≠
              0 ∧
            empMean A R (bestAction κ E ω)
                  t ω -
                actionMean κ E
                  (bestAction κ E ω) ω ≤
              -√(2 * ↑σ2 *
                      Real.log (1 / δ) /
                    ↑(pullCount A
                        (bestAction κ E ω)
                        t ω))} ≤
    ENNReal.ofReal ((↑n - 1) * δ)

Code

lemma prob_empMean_bestAction_sub_actionMean_le_le (h : IsBayesAlgEnvSeq Q κ alg E A R P)
    {σ2 : ℝ≥0} (hσ2 : 0 < σ2)
    (hs : ∀ e a, HasSubgaussianMGF (fun x ↦ x - (κ (e, a))[id]) σ2 (κ (e, a)))
    {δ : ℝ} (hδ : 0 < δ) (n : ℕ) :
    P {ω | ∃ t < n, pullCount A (bestAction κ E ω) t ω ≠ 0 ∧
        empMean A R (bestAction κ E ω) t ω - actionMean κ E (bestAction κ E ω) ω ≤
          -√(2 * σ2 * Real.log (1 / δ) / (pullCount A (bestAction κ E ω) t ω))}
      ≤ ENNReal.ofReal ((n - 1) * δ)

Proof

by
  have := h.measurable_param
  have := h.measurable_action
  have := h.measurable_feedback
  let S := {(e, τ) | ∃ t < n, pullCount IT.action (bestAction κ id e) t τ ≠ 0 ∧
    sumRewards IT.action IT.feedback (bestAction κ id e) t τ -
        pullCount IT.action (bestAction κ id e) t τ * actionMean κ id (bestAction κ id e) e ≤
          -√(2 * pullCount IT.action (bestAction κ id e) t τ * σ2 * Real.log (1 / δ))}
  calc
    _ ≤ (P.map (fun ω ↦ (E ω, trajectory A R ω))) S := by
        rw [Measure.map_apply (by fun_prop) (by measurability)]
        apply measure_mono
        intro ω ⟨t, ht, hpc, hle⟩
        rw [empMean] at hle
        exact ⟨t, ht, hpc, sub_le_neg_sqrt_two_mul hpc hle⟩
    _ = (P.map E ⊗ₘ condDistrib (trajectory A R) E P) S := by
        rw [← compProd_map_condDistrib (by fun_prop)]
    _ = ∫⁻ e, condDistrib (trajectory A R) E P e (Prod.mk e ⁻¹' S) ∂(P.map E) :=
        Measure.compProd_apply (by measurability)
    _ ≤ ∫⁻ e, ENNReal.ofReal ((n - 1) * δ) ∂(P.map E) := by
        apply lintegral_mono_ae
        rw [h.hasLaw_env.map_eq]
        filter_upwards [h.ae_IsAlgEnvSeq] with e he
        exact Bandits.prob_sumRewards_sub_pullCount_mul_le_le (ν := κ.sectR e) hσ2 (hs e _) he
          hδ
    _ = ENNReal.ofReal ((n - 1) * δ) := by
      simp [Measure.map_apply h.measurable_param]

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: 11 project declarations, 128 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.

Learning.IsBayesAlgEnvSeq.prob_empMean_bestAction_sub_actionMean_le_le🔗

`Learning.IsBayesAlgEnvSeq.prob_empMean_bestAction_sub_actionMean_le_le`🔗