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a2ee19a | 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import Mathlib
import OAI.Probability.Perceptron.Control.StepControlTail
import OAI.Probability.Perceptron.Control.QuantileControlContinuity
import OAI.Probability.Perceptron.Variational.QuantileUpperTail
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SphericalPerceptronFreeEnergy
lemma sampledQuantile_tail_order (n : ℕ) {q r : Time→Time} (hq : Monotone q) (hr : Monotone r)
(ht : ∀ t : Time, (∫ u in Ici t, (q u:ℝ) ∂timeLaw) ≤ ∫ u in Ici t, (r u:ℝ) ∂timeLaw)
(k : ℕ) : 0 ≤ ∑ i : Fin (n+1), if k ≤ i.val then
(1/(n+1:ℝ))*((uniformQuantileUpper n r i:ℝ)-uniformQuantileLower n q i) else 0 := by
by_cases hk : k<n+1
· let j : Fin (n+1) := ⟨k,hk⟩
have h1 : (∫ u in Ici (uniformQuantileGrid n j.castSucc), (sampleQuantileLower n q u:ℝ) ∂timeLaw) ≤
∫ u in Ici (uniformQuantileGrid n j.castSucc), (sampleQuantileUpper n r u:ℝ) ∂timeLaw := by
calc
_ ≤ ∫ u in Ici (uniformQuantileGrid n j.castSucc), (q u:ℝ) ∂timeLaw := by
apply integral_mono (unit_valued_integrable _ (sampleQuantileLower_mono n hq).measurable)
(unit_valued_integrable _ hq.measurable)
exact fun u => (sampleQuantile_bracket n hq u).1
_ ≤ ∫ u in Ici (uniformQuantileGrid n j.castSucc), (r u:ℝ) ∂timeLaw := ht _
_ ≤ _ := by
apply integral_mono (unit_valued_integrable _ hr.measurable)
(unit_valued_integrable _ (sampleQuantileUpper_mono n hr).measurable)
exact fun u => (sampleQuantile_bracket n hr u).2
change (∫ u in Ici (uniformQuantileGrid n j.castSucc),
(uniformQuantileLower n q (uniformQuantileCell n u):ℝ) ∂timeLaw) ≤
∫ u in Ici (uniformQuantileGrid n j.castSucc),
(uniformQuantileUpper n r (uniformQuantileCell n u):ℝ) ∂timeLaw at h1
rw [uniformQuantileCell_tail_integral n (fun i => (uniformQuantileLower n q i:ℝ)) j,
uniformQuantileCell_tail_integral n (fun i => (uniformQuantileUpper n r i:ℝ)) j] at h1
have he : (∑ i : Fin (n+1), if k ≤ i.val then
(1/(n+1:ℝ))*((uniformQuantileUpper n r i:ℝ)-uniformQuantileLower n q i) else 0)=
(∑ i : Fin (n+1), if j ≤ i then (1/(n+1:ℝ))*(uniformQuantileUpper n r i:ℝ) else 0)-
(∑ i : Fin (n+1), if j ≤ i then (1/(n+1:ℝ))*(uniformQuantileLower n q i:ℝ) else 0) := by
rw [← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i _
change (if k ≤ i.val then _ else _)=(if k ≤ i.val then _ else _)-(if k ≤ i.val then _ else _)
split_ifs <;> ring
rw [he]
exact sub_nonneg.mpr h1
· have hz : ∀ i : Fin (n+1), ¬k ≤ i.val := fun i => by have := i.isLt; omega
simp only [hz,ite_false,Finset.sum_const_zero,le_refl]
lemma sampleQuantile_control_error (P : Measure BrownianPath) [IsProbabilityMeasure P]
(g : Jet3) (q : Time→Time) (hq : Monotone q) (n : ℕ) :
|controlValue P g.f (quantileTrial (sampleQuantileLower n q))-controlValue P g.f (quantileTrial q)| ≤
(3/2:ℝ)*(‖g.d1‖₊:ℝ)^2*(1/(n+1:ℝ)) ∧
|controlValue P g.f (quantileTrial (sampleQuantileUpper n q))-controlValue P g.f (quantileTrial q)| ≤
(3/2:ℝ)*(‖g.d1‖₊:ℝ)^2*(1/(n+1:ℝ)) := by
constructor
· exact (controlValue_quantile_abs_sub_le P g _ _ (sampleQuantileLower_mono n hq).measurable hq.measurable).trans
(mul_le_mul_of_nonneg_left (sampleQuantileLower_L1 n hq) (by positivity))
· exact (controlValue_quantile_abs_sub_le P g _ _ (sampleQuantileUpper_mono n hq).measurable hq.measurable).trans
(mul_le_mul_of_nonneg_left (sampleQuantileUpper_L1 n hq) (by positivity))
theorem sourceQuantile_control_tail_order (P : Measure BrownianPath) [IsProbabilityMeasure P]
(hB : IsBrownianReal brownianEval P) (g : Jet3) (q r : Time→Time)
(hq : Monotone q) (hr : Monotone r)
(ht : ∀ t : Time, (∫ u in Ici t, (q u:ℝ) ∂timeLaw) ≤ ∫ u in Ici t, (r u:ℝ) ∂timeLaw) :
controlValue P g.f (quantileTrial r) ≤ controlValue P g.f (quantileTrial q) := by
have hi (n : ℕ) : controlValue P g.f (quantileTrial (sampleQuantileUpper n r)) ≤
controlValue P g.f (quantileTrial (sampleQuantileLower n q)) := by
unfold sampleQuantileUpper sampleQuantileLower
rw [sampleQuantile_trial,sampleQuantile_trial]
exact sourceWeightedTrial_control_tail_order n _ _ _ (uniformQuantileWeights_nonneg n)
(uniformQuantileWeights_sum n) (uniformQuantileLower_mono n hq) (uniformQuantileUpper_mono n hr)
(sampledQuantile_tail_order n hq hr ht) g P hB
have hz : Tendsto (fun n : ℕ => (3/2:ℝ)*(‖g.d1‖₊:ℝ)^2*(1/(n+1:ℝ))) atTop (𝓝 0) := by
simpa using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ)).const_mul ((3/2:ℝ)*(‖g.d1‖₊:ℝ)^2)
have hl : Tendsto (fun n => controlValue P g.f (quantileTrial (sampleQuantileLower n q))) atTop
(𝓝 (controlValue P g.f (quantileTrial q))) := by
rw [tendsto_iff_dist_tendsto_zero]
apply squeeze_zero (fun _ => dist_nonneg) (fun n => ?_) hz
simpa only [Real.dist_eq] using (sampleQuantile_control_error P g q hq n).1
have hu : Tendsto (fun n => controlValue P g.f (quantileTrial (sampleQuantileUpper n r))) atTop
(𝓝 (controlValue P g.f (quantileTrial r))) := by
rw [tendsto_iff_dist_tendsto_zero]
apply squeeze_zero (fun _ => dist_nonneg) (fun n => ?_) hz
simpa only [Real.dist_eq] using (sampleQuantile_control_error P g r hr n).2
exact le_of_tendsto_of_tendsto hu hl (Eventually.of_forall hi)
end SphericalPerceptronFreeEnergy
end
end OAI
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