Canonical KL bound for the actual standardized RGO

Produce one unique proximal point; Reconcile the square-root-density witness; Retrieve the real density and analytic domains; Apply the noncompact Gaussian function LSI; Identify the finite canonical KL; Retain the exact half coefficient; Exercise actual posterior consumers

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem standardized_rgo_unique_prox_and_kl_le_fisher
    {E S : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [CompleteSpace E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    [MeasurableSpace S] {V : E → ℝ} {κ : ℝ}
    (hκ : 1 ≤ κ) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E, κ⁻¹*‖v‖^2 ≤ fderiv ℝ (fderiv ℝ V) x v v ∧
      fderiv ℝ (fderiv ℝ V) x v v ≤ ‖v‖^2)
    {eta : S → ℝ} {y : S → E} (heta : Measurable eta) (hy : Measurable y)
    (hpos : ∀ s, 0 < eta s) :
    ∃ p : S → E, Measurable p ∧
      (∀ s, p s+eta s • gradient V (p s)=y s) ∧
      (∀ s z, z+eta s • gradient V z=y s → z=p s) ∧
      let rho := fun s u => V (p s+Real.sqrt (eta s) • u)-V (p s)-
        Real.sqrt (eta s)*inner ℝ (gradient V (p s)) u
      let mu := (volume : Measure E).tilted (fun x => -V x)
      let R := fun s => mu.tilted (fun x => -‖x-y s‖^2/(2*eta s))
      let r := fun s => (R s).map (fun x => (Real.sqrt (eta s))⁻¹ • (x-p s))
      let gamma := stdGaussian E
      ∀ s, IsProbabilityMeasure (r s) ∧ r s ≪ gamma ∧
        _root_.InformationTheory.klDiv (r s) gamma ≠ ⊤ ∧
        MemLp (gradient (rho s)) 2 (r s) ∧
        (_root_.InformationTheory.klDiv (r s) gamma).toReal ≤
          (1/2 : ℝ)*(∫ u, ‖gradient (rho s) u‖^2 ∂r s) := by
  obtain ⟨p, hpm, hstat, huniq, hk⟩ :=
    StandardizedRGORelativeEntropy.standardized_rgo_unique_prox_and_finite_entropy
      hκ hV hH heta hy hpos
  obtain ⟨p', hpm', hstat', hd⟩ :=
    StandardizedRGOSqrtDensity.standardized_rgo_sqrt_density_domain
      hκ hV hH heta hy hpos
  have hp : p'=p := funext (fun s => huniq s (p' s) (hstat' s))
  rw [hp] at hd
  let rho := fun s u => V (p s+Real.sqrt (eta s) • u)-V (p s)-
    Real.sqrt (eta s)*inner ℝ (gradient V (p s)) u
  let mu := (volume : Measure E).tilted (fun x => -V x)
  let R := fun s => mu.tilted (fun x => -‖x-y s‖^2/(2*eta s))
  let r := fun s => (R s).map (fun x => (Real.sqrt (eta s))⁻¹ • (x-p s))
  let gamma := stdGaussian E
  let Z := fun s => ∫ u, Real.exp (-rho s u) ∂gamma
  let q := fun s u => Real.exp (-rho s u)/Z s
  let f := fun s u => Real.exp (-rho s u/2)/Real.sqrt (Z s)
  refine ⟨p, hpm, hstat, huniq, ?_⟩
  change ∀ s, IsProbabilityMeasure (r s) ∧ r s ≪ gamma ∧
    _root_.InformationTheory.klDiv (r s) gamma ≠ ⊤ ∧
    MemLp (gradient (rho s)) 2 (r s) ∧
    (_root_.InformationTheory.klDiv (r s) gamma).toReal ≤
      (1/2 : ℝ)*(∫ u, ‖gradient (rho s) u‖^2 ∂r s)
  intro s
  obtain ⟨hZpos, hZle, hrtilt, hrden, hprob, hq, hmass, hfC, hf2,
    hg2, hrho2, hqint, hgf, hglog, he⟩ := hd s
  obtain ⟨hac, hfin, hllr, hlog, hrhoint, hKLq, hKLrho⟩ := hk s
  change (∫ u, q s u ∂gamma)=1 at hmass
  change (∫ u, ‖gradient (f s) u‖^2 ∂gamma) =
    (1/4 : ℝ)*(∫ u, ‖gradient (rho s) u‖^2 ∂r s) at he
  change (_root_.InformationTheory.klDiv (r s) gamma).toReal =
    (∫ u, q s u*Real.log (q s u) ∂gamma) at hKLq
  have hsq : ∀ u, (f s u)^2=q s u := fun u => (hq u).2
  have hphi : Integrable (fun u => (f s u)^2*Real.log ((f s u)^2)) gamma := by
    simpa only [hsq] using hqint
  have hLSI := AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianLogSobolev.gaussian_logSobolev_of_contDiff
    (f s) hfC hf2 hg2 hphi
  change (∫ u, (f s u)^2*Real.log ((f s u)^2) ∂gamma) -
    (∫ u, (f s u)^2 ∂gamma)*Real.log (∫ u, (f s u)^2 ∂gamma) ≤
    2 * ∫ u, ‖gradient (f s) u‖^2 ∂gamma at hLSI
  refine ⟨hprob, hac, hfin, hrho2, ?_⟩
  calc
    _ = ∫ u, q s u*Real.log (q s u) ∂gamma := hKLq
    _ ≤ 2 * ∫ u, ‖gradient (f s) u‖^2 ∂gamma := by
      simpa only [hsq, hmass, Real.log_one, mul_zero, sub_zero] using hLSI
    _ = (1/2 : ℝ)*(∫ u, ‖gradient (rho s) u‖^2 ∂r s) := by
      rw [he]
      ring

end
end AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StandardizedRGOKLFisher

Exact module and namespace context