`Bandits.TS.integral_regret_eq_add`🔗

Lemma

No docstring.

theorem

Bandits.TS.integral_regret_eq_add.{u_1, u_2} {K : ℕ} [Nonempty (Fin K)]
  {l u σ2 δ : 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)
  (hm : ∀ (e : 𝓔) (a : Fin K), ∫ (x : ℝ), id x ∂κ (e, a) ∈ Set.Icc l u)
  (n : ℕ) :
  ∫ (x : Ω), Learning.IsBayesAlgEnvSeq.regret κ E A n x ∂P =
    ∫ (x : Ω),
        (fun ω =>
            ∑ t ∈ Finset.range n,
              (Learning.IsBayesAlgEnvSeq.actionMean κ E
                  (Learning.IsBayesAlgEnvSeq.bestAction κ E ω) ω -
                ClippedUCB.ucb A R l u σ2 δ
                  (Learning.IsBayesAlgEnvSeq.bestAction κ E ω) t ω))
          x ∂P +
      ∫ (x : Ω),
        (fun ω =>
            ∑ t ∈ Finset.range n,
              (ClippedUCB.ucb A R l u σ2 δ (A t ω) t ω -
                Learning.IsBayesAlgEnvSeq.actionMean κ E (A t ω) ω))
          x ∂P
Bandits.TS.integral_regret_eq_add.{u_1,
    u_2}
  {K : ℕ} [Nonempty (Fin K)]
  {l u σ2 δ : 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)
  (hm :
    ∀ (e : 𝓔) (a : Fin K),
      ∫ (x : ℝ), id x ∂κ (e, a) ∈
        Set.Icc l u)
  (n : ℕ) :
  ∫ (x : Ω),
      Learning.IsBayesAlgEnvSeq.regret κ E
        A n x ∂P =
    ∫ (x : Ω),
        (fun ω =>
            ∑ t ∈ Finset.range n,
              (Learning.IsBayesAlgEnvSeq.actionMean
                  κ E
                  (Learning.IsBayesAlgEnvSeq.bestAction
                    κ E ω)
                  ω -
                ClippedUCB.ucb A R l u σ2
                  δ
                  (Learning.IsBayesAlgEnvSeq.bestAction
                    κ E ω)
                  t ω))
          x ∂P +
      ∫ (x : Ω),
        (fun ω =>
            ∑ t ∈ Finset.range n,
              (ClippedUCB.ucb A R l u σ2 δ
                  (A t ω) t ω -
                Learning.IsBayesAlgEnvSeq.actionMean
                  κ E (A t ω) ω))
          x ∂P

Code

lemma integral_regret_eq_add (hK : 0 < K) (h : IsBayesAlgEnvSeq Q κ (tsAlgorithm hK Q κ) E A R P)
    (hm : ∀ e a, (κ (e, a))[id] ∈ (Set.Icc l u)) (n : ℕ) :
    P[IsBayesAlgEnvSeq.regret κ E A n] =
      P[fun ω ↦ ∑ t ∈ range n,
        (actionMean κ E (bestAction κ E ω) ω - ucb A R l u σ2 δ (bestAction κ E ω) t ω)] +
      P[fun ω ↦ ∑ t ∈ range n,
        (ucb A R l u σ2 δ (A t ω) t ω - actionMean κ E (A t ω) ω)]

Proof

by
  have hua (t : ℕ) : Integrable (fun ω ↦ ucb A R l u σ2 δ (A t ω) t ω) P :=
    integrable_uncurry_ucb_comp h.measurable_action h.measurable_feedback (h.measurable_action t)
      measurable_const
  have hub (t : ℕ) : Integrable (fun ω ↦ ucb A R l u σ2 δ (bestAction κ E ω) t ω) P :=
    integrable_uncurry_ucb_comp h.measurable_action h.measurable_feedback
      (IsBayesAlgEnvSeq.measurable_bestAction h.measurable_param) measurable_const
  have haa (t : ℕ) : Integrable (fun ω ↦ actionMean κ E (A t ω) ω) P :=
    IsBayesAlgEnvSeq.integrable_uncurry_actionMean_comp h.measurable_param
      (h.measurable_action t) hm
  have hab : Integrable (fun ω ↦ actionMean κ E (bestAction κ E ω) ω) P :=
      IsBayesAlgEnvSeq.integrable_uncurry_actionMean_comp h.measurable_param
        (IsBayesAlgEnvSeq.measurable_bestAction h.measurable_param) hm
  calc
    _  = (∑ t ∈ range n, ∫ ω, actionMean κ E (bestAction κ E ω) ω ∂P) -
            ∑ t ∈ range n, ∫ ω, actionMean κ E (A t ω) ω ∂P := by
        simp_rw [IsBayesAlgEnvSeq.regret_eq_sum_gap, IsBayesAlgEnvSeq.gap_eq_sub]
        rw [integral_finsetSum _ (by fun_prop), ← Finset.sum_sub_distrib]
        simp_rw [integral_sub hab (haa _)]
    _ = ((∑ t ∈ range n, ∫ ω, actionMean κ E (bestAction κ E ω) ω ∂P) -
            ∑ t ∈ range n, ∫ ω, ucb A R l u σ2 δ (bestAction κ E ω) t ω ∂P) +
          ((∑ t ∈ range n, ∫ ω, ucb A R l u σ2 δ (A t ω) t ω ∂P) -
            ∑ t ∈ range n, ∫ ω, actionMean κ E (A t ω) ω ∂P) := by
        simp [integral_ucb_action_eq_integral_ucb_bestAction hK h]
    _ = (∑ t ∈ range n, ∫ ω, actionMean κ E (bestAction κ E ω) ω -
              ucb A R l u σ2 δ (bestAction κ E ω) t ω ∂P) +
          ∑ t ∈ range n, ∫ ω, ucb A R l u σ2 δ (A t ω) t ω -
            actionMean κ E (A t ω) ω ∂P := by
        rw [← Finset.sum_sub_distrib, ← Finset.sum_sub_distrib]
        simp_rw [← integral_sub hab (hub _), ← integral_sub (hua _) (haa _)]
    _ = _ := by
        rw [← integral_finsetSum _ (by fun_prop), ← integral_finsetSum _ (by fun_prop)]

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: 27 project declarations, 140 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_eq_add🔗

`Bandits.TS.integral_regret_eq_add`🔗