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c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | import OAI.Probability.InvariantIsing.Cavity.CavityRegularizedKernel
import OAI.Probability.InvariantIsing.Cavity.CavityCutoffNormalizer
namespace OAI
/-! A quantitative denominator-removal bound for the restricted weights.
The prior tail controls the exceptional small-normalizer event. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
namespace InvariantIsing
theorem cavity_bounded_regularization_mean_error {Ω X : Type*}
[MeasurableSpace Ω] [MeasurableSpace X]
(P : Measure Ω) [IsProbabilityMeasure P]
(ν : Ω → Measure X) (hν : Measurable ν) [∀ ω, IsProbabilityMeasure (ν ω)]
(w : Ω × X → ℝ) (hw : Measurable w) {r : ℕ}
(F : Ω × (Fin r → X) → ℝ) (hF : Measurable F)
(t : Ω → ℝ) (ht : Measurable t) (hti : Integrable t P) (ht0 : ∀ ω, 0 ≤ t ω)
{M B δ a : ℝ} (hM : 0 ≤ M) (hB : 0 ≤ B) (hδ : 0 ≤ δ) (ha : 0 < a)
(hwb : ∀ ω x, w (ω,x) ∈ Icc 0 M) (hFb : ∀ ω σ, |F (ω,σ)| ≤ B)
(hlow : ∀ ω, t ω ≤ 1/2 → a ≤ cavityWeightNormalizer (ν ω) (fun x => w (ω,x))) :
|(∫ ω, cavityRegularizedReplicaMean (ν ω) (fun x => w (ω,x))
(fun σ => F (ω,σ)) δ ∂P) -
∫ ω, cavityWeightedReplicaMean (ν ω) (fun x => w (ω,x))
(fun σ => F (ω,σ)) ∂P| ≤ B * r * δ / a + 4 * B * ∫ ω, t ω ∂P := by
let Z ω := cavityWeightNormalizer (ν ω) (fun x => w (ω,x))
let A ω := cavityWeightNumerator (ν ω) (fun x => w (ω,x)) (fun σ => F (ω,σ))
have hZ : Measurable Z := measurable_cavityWeightNormalizer ν hν w hw
have hA : Measurable A := measurable_cavityWeightNumerator ν hν w hw F hF
have hZ0 ω : 0 ≤ Z ω :=
(cavityWeightNormalizer_mem (ν ω) _ (hw.comp measurable_prodMk_left) le_rfl (hwb ω)).1
have hAZ ω : |A ω| ≤ B * Z ω^r :=
cavityWeightNumerator_abs_le_normalizer (ν ω) _ (hw.comp measurable_prodMk_left)
_ (hF.comp measurable_prodMk_left) hM hB (hwb ω) (hFb ω)
have hi : Integrable (fun ω => A ω / Z ω^r) P :=
integrable_of_measurable_abs_le (hA.div (hZ.pow_const r))
(fun ω => cavity_normalized_numerator_bound (hZ0 ω) hB r (hAZ ω))
have hiδ : Integrable (fun ω => A ω / (Z ω + δ)^r) P :=
integrable_of_measurable_abs_le (hA.div ((hZ.add_const δ).pow_const r))
(fun ω => cavity_regularized_numerator_bound (hZ0 ω) hB hδ r (hAZ ω))
have he := cavity_normalizer_regularization_mean_error P A Z hA hZ hB hδ ha r hZ0 hAZ
have hp := cavity_small_normalizer_probability P Z t hZ ht hti ht0 hlow
change |(∫ ω, A ω / (Z ω + δ)^r ∂P) - ∫ ω, A ω / Z ω^r ∂P| ≤ _
rw [abs_sub_comm, ← integral_sub hi hiδ]
exact he.trans (by nlinarith)
end InvariantIsing
end
end OAI
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