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f205175 | 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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.WeightedCauchy
import OAI.Probability.CriticalPercolation.Modular.InputMass
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Real
open Set Real
open scoped Topology
open scoped BigOperators
open Set
open Set MeasureTheory
open Set MeasureTheory Real
open scoped Interval
open scoped Classical
namespace CriticalPercolation.OrderedQuery
open QueryTree MarkedTree Priority FiniteEntropy FiniteProduct
variable {E I : Type*} [Fintype E] [Fintype I]
abbrev GlobalNode (t : I → QueryTree E) := (o : Order I) × (combined t o.val).bare.Node
noncomputable abbrev globalTag (t : I → QueryTree E) (x : GlobalNode t) : LocalNode t :=
(combined t x.1.val).nodeLabel x.2
omit [Fintype E] in
lemma pooled_weighted_sum (t : I → QueryTree E) (w : Order I → I → ℝ)
(f : GlobalNode t → ℝ) (a : LocalNode t) :
(∑ x : GlobalNode t, if globalTag t x = a then w x.1 (globalTag t x).1 * f x else 0) =
∑ o : Order I, w o a.1 * ∑ g : (combined t o.val).bare.Node,
if (combined t o.val).nodeLabel g = a then f ⟨o,g⟩ else 0 := by
rw [Fintype.sum_sigma]
apply Finset.sum_congr rfl
intro o _
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro g _
by_cases h : globalTag t ⟨o,g⟩ = a
· simp only [h, ite_eq_left]
· simp only [h, ite_false, mul_zero]
lemma weighted_node_square_bound (t : I → QueryTree E) (p : ℝ)
(P : (E → Bool) → ℝ) (hP : ∀ ω, 0 ≤ P ω) (x : GlobalNode t) :
(orderWeight x.1 (globalTag t x).1 * (combined t x.1.val).bare.inputNumerator p P x.2)^2 ≤
(orderSquare x.1 (globalTag t x).1 * (combined t x.1.val).bare.inputMass P x.2) *
(orderMass x.1 * (combined t x.1.val).bare.inputNumerator p P x.2 ^ 2 /
(combined t x.1.val).bare.inputMass P x.2) := by
let M := (combined t x.1.val).bare.inputMass P x.2
let N := (combined t x.1.val).bare.inputNumerator p P x.2
have hM : 0 ≤ M := inputMass_nonneg _ P hP x.2
change (orderWeight x.1 (globalTag t x).1 * N)^2 ≤
(orderSquare x.1 (globalTag t x).1 * M)*(orderMass x.1*N^2/M)
rcases eq_or_lt_of_le hM with hz | hp
· have hN : N = 0 := inputNumerator_eq_zero _ p P hP x.2 hz.symm
simp [hN]
· have hh := mul_le_mul_of_nonneg_right (orderWeight_sq_le x.1 (globalTag t x).1)
(sq_nonneg N)
calc
_ = (orderWeight x.1 (globalTag t x).1)^2 * N^2 := by ring
_ ≤ (orderSquare x.1 (globalTag t x).1 * orderMass x.1)*N^2 := hh
_ = _ := by field_simp
theorem overlap_four_charge (t : I → QueryTree E) (ht : ∀ i, (t i).Valid ∅)
(p : ℝ) (P : (E → Bool) → ℝ) (hP : ∀ ω, 0 ≤ P ω)
(hover : ∀ ω, 0 < P ω → ∀ e, (users t ω e).card ≤ 4) :
(∑ i, (t i).charge (fun r => (r-p)^2) (pushMass (t i).run P)) ≤
16 * ∑ o : Order I, orderMass o *
(combined t o.val).bare.charge (fun r => (r-p)^2)
(pushMass (combined t o.val).bare.run P) := by
let M (x : GlobalNode t) := (combined t x.1.val).bare.inputMass P x.2
let N (x : GlobalNode t) := (combined t x.1.val).bare.inputNumerator p P x.2
let z (x : GlobalNode t) := orderWeight x.1 (globalTag t x).1 * N x
let u (x : GlobalNode t) := orderSquare x.1 (globalTag t x).1 * M x
let v (x : GlobalNode t) := orderMass x.1 * (N x)^2 / M x
have h := pooled_two_weight_bound (globalTag t) z u v
(fun a : LocalNode t => (t a.1).inputMass P a.2)
(fun a : LocalNode t => (t a.1).inputNumerator p P a.2)
(C := 16)
(fun x => mul_nonneg (orderSquare_nonneg _ _) (inputMass_nonneg _ P hP _))
(fun x => div_nonneg (mul_nonneg (orderMass_nonneg _) (sq_nonneg _))
(inputMass_nonneg _ P hP _))
(weighted_node_square_bound t p P hP)
(fun a => inputMass_nonneg _ P hP _) (by norm_num) ?_ ?_
· convert h using 1
· rw [Fintype.sum_sigma]
apply Finset.sum_congr rfl
intro i _
exact charge_eq_input_sum (t i) p P
· rw [Fintype.sum_sigma]
congr 1
apply Finset.sum_congr rfl
intro o _
rw [charge_eq_input_sum, Finset.mul_sum]
apply Finset.sum_congr rfl
intro g _
dsimp [v,N,M]
ring
· intro a
exact (weighted_pooled_numerator t ht p P hP hover a.1 a.2).symm.trans
(pooled_weighted_sum t orderWeight N a).symm
· intro a
exact (pooled_weighted_sum t orderSquare M a).trans_le
(weighted_pooled_mass_le t ht P hP a.1 a.2)
theorem overlap_four_entropy {p : ℝ} (hp : p ∈ Set.Ioo 0 1)
(t : I → QueryTree E) (ht : ∀ i, (t i).Valid ∅)
(P : (E → Bool) → ℝ) (hP : ∀ ω, 0 ≤ P ω) (hMass : ∑ ω, P ω = 1)
(hover : ∀ ω, 0 < P ω → ∀ e, (users t ω e).card ≤ 4) :
(∑ i, divergence (pushMass (t i).run P) ((t i).reference p)) ≤
16/(2*p*(1-p)) * divergence P (density (bitWeight p)) := by
have hc := overlap_four_charge t ht p P hP hover
have hb : 2*(∑ o : Order I, orderMass o *
(combined t o.val).bare.charge (fun r => (r-p)^2)
(pushMass (combined t o.val).bare.run P)) ≤ divergence P (density (bitWeight p)) := by
calc
_ = ∑ o : Order I, orderMass o * (2*
(combined t o.val).bare.charge (fun r => (r-p)^2)
(pushMass (combined t o.val).bare.run P)) := by
rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro o _; ring
_ ≤ ∑ o : Order I, orderMass o * divergence P (density (bitWeight p)) := by
apply Finset.sum_le_sum
intro o _
exact mul_le_mul_of_nonneg_left
(quadratic_bias_budget hp _ (combined_valid t o.val) P hP hMass) (orderMass_nonneg o)
_ = _ := by rw [← Finset.sum_mul, sum_orderMass, one_mul]
have hlocal : (∑ i, (t i).charge (fun r => (r-p)^2) (pushMass (t i).run P)) ≤
8 * divergence P (density (bitWeight p)) := by linarith
calc
_ ≤ ∑ i, (p*(1-p))⁻¹ * (t i).charge (fun r => (r-p)^2) (pushMass (t i).run P) :=
Finset.sum_le_sum (fun i _ => transcript_divergence_upper hp (t i) P hP hMass)
_ = (p*(1-p))⁻¹ * ∑ i, (t i).charge (fun r => (r-p)^2) (pushMass (t i).run P) :=
(Finset.mul_sum _ _ _).symm
_ ≤ (p*(1-p))⁻¹ * (8 * divergence P (density (bitWeight p))) :=
mul_le_mul_of_nonneg_left hlocal (inv_nonneg.mpr (mul_nonneg hp.1.le (sub_nonneg.mpr hp.2.le)))
_ = _ := by field_simp; ring
end CriticalPercolation.OrderedQuery
end
end
end OAI
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