| 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 |
|
|
