`Learning.randomSampling.tendsto_max₀`🔗

Lemma

The maximum image action converges to the function's global maximum.

theorem

Learning.randomSampling.tendsto_max₀.{u_1, u_3} {𝓐 : Type u_1}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure 𝓐} [MeasureTheory.IsProbabilityMeasure μ]
  {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} [PseudoMetricSpace 𝓐] [SecondCountableTopology 𝓐]
  [OpensMeasurableSpace 𝓐] [MeasureTheory.Measure.IsOpenPosMeasure μ]
  {R : ℕ → Ω → ℝ} {f : 𝓐 → ℝ} (hfc : Continuous f) {a : 𝓐}
  (h : IsAlgEnvSeq A R (randomSampling μ) (evalEnv f ⋯) P)
  (hf_max : ∀ (x : 𝓐), f x ≤ f a) :
  MeasureTheory.TendstoInMeasure P
    (fun n ω => Function.max fun i => f (A (↑i) ω)) Filter.atTop
    fun x => f a
Learning.randomSampling.tendsto_max₀.{u_1,
    u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure 𝓐}
  [MeasureTheory.IsProbabilityMeasure μ]
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {A : ℕ → Ω → 𝓐} [PseudoMetricSpace 𝓐]
  [SecondCountableTopology 𝓐]
  [OpensMeasurableSpace 𝓐]
  [MeasureTheory.Measure.IsOpenPosMeasure
      μ]
  {R : ℕ → Ω → ℝ} {f : 𝓐 → ℝ}
  (hfc : Continuous f) {a : 𝓐}
  (h :
    IsAlgEnvSeq A R (randomSampling μ)
      (evalEnv f ⋯) P)
  (hf_max : ∀ (x : 𝓐), f x ≤ f a) :
  MeasureTheory.TendstoInMeasure P
    (fun n ω =>
      Function.max fun i => f (A (↑i) ω))
    Filter.atTop fun x => f a

Code

lemma tendsto_max₀ (h : IsAlgEnvSeq A R (randomSampling μ) (evalEnv f hfc.measurable) P)
    (hf_max : ∀ x, f x ≤ f a) :
    TendstoInMeasure P (fun n ω ↦ (fun (i : Iic n) ↦ f (A i.1 ω)).max) atTop (fun _ ↦ f a)

Proof

by
  rw [tendstoInMeasure_iff_dist]
  intro ε hε
  refine image_action_tendsto_any hfc h a hε |> tendsto_zero_of_le <| ?_
  intro n
  refine measure_mono ?_
  simp only [Set.ofPred_subset_ofPred]
  intro ω hω
  rw [← argmin_spec]
  set j := argmin (fun (i : Iic n) ↦ dist (f (A i ω)) (f a))
  refine hω.trans ?_
  rw [← argmax_spec]
  set k := argmax (fun (i : Iic n) ↦ f (A i ω))
  have := hf_max (A k ω)
  have : f (A j ω) ≤ f (A k ω) :=
    isMaxOn_argmax (fun (i : Iic n) ↦ f (A i ω)) j
  simp [Real.dist_eq]
  grind

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: 10 project declarations, 87 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.randomSampling.tendsto_max₀🔗

`Learning.randomSampling.tendsto_max₀`🔗