LeanMachineLearning

Learning.randomSampling.tendsto_min🔗

Lemma

The minimum reward converges to the function's global minimum.

🔗theorem
Learning.randomSampling.tendsto_min.{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_min : ∀ (x : 𝓐), f a ≤ f x) :
  MeasureTheory.TendstoInMeasure P
    (fun n ω => Function.min fun i => R (↑i) ω) Filter.atTop fun x =>
    f a
Learning.randomSampling.tendsto_min.{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_min : ∀ (x : 𝓐), f a ≤ f x) :
  MeasureTheory.TendstoInMeasure P
    (fun n ω =>
      Function.min fun i => R (↑i) ω)
    Filter.atTop fun x => f a

Code

lemma tendsto_min (h : IsAlgEnvSeq A R (randomSampling μ) (evalEnv f hfc.measurable) P)
    (hf_min : ∀ x, f a ≤ f x) :
    TendstoInMeasure P (fun n ω ↦ (fun (i : Iic n) ↦ R i.1 ω).min) atTop (fun _ ↦ f a)
Proof
by
  refine TendstoInMeasure.congr_left (fun n ↦ ?_) <| tendsto_min₀ hfc h hf_min
  filter_upwards [feedback_evalEnv_ae_eq_eval_action_comp h Function.min] with ω hω
  rw [← hω]

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.