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97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionOracleEstimator
namespace OAI
/-! # Output range on every execution of the finite-word estimator -/
namespace ContingencyTables.CompanionOracleDensity
open CompanionFactorialApplication ForestAssignmentSplit GridBoundary ResidualMixture FirstSuccess
open CommonBasesFPRAS FiniteIndependentAverages
open scoped BigOperators Classical
noncomputable section
variable {I J : Type*} [Fintype I] [Fintype J]
lemma rationalAverage_bounds {X : Type*} (g : X → ℚ) (N : ℕ) (hN : 0 < N)
(L U : ℝ) (hg : ∀ x,L ≤ (g x:ℝ) ∧ (g x:ℝ) ≤ U) (draws : Fin N → X) :
L ≤ (rationalAverage g N draws:ℝ) ∧ (rationalAverage g N draws:ℝ) ≤ U := by
have hNR : (0:ℝ) < N := by exact_mod_cast hN
have hl := Finset.sum_le_sum (s := Finset.univ) (fun i _ => (hg (draws i)).1)
have hu := Finset.sum_le_sum (s := Finset.univ) (fun i _ => (hg (draws i)).2)
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at hl hu
simp only [rationalAverage,Rat.cast_div,Rat.cast_sum,Rat.cast_natCast]
exact ⟨(le_div_iff₀ hNR).mpr (by linarith),(div_le_iff₀ hNR).mpr (by linarith)⟩
lemma observed_ratio_bounds (A : Instance I J) (C : ℕ) (hC : 0 < C) (z : View A → ℕ)
(hz : BalancedBox.InBox (bounds A) z) (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(hB : (K:ℝ)=binWidth A C) (j : ℕ) (delta failure : ℚ) (hd : 0 < delta) (hf : 0 < failure)
(v : Grid (I := Fin (dimension A)) K) (draws : EstimatorInput A C K j delta failure) (a : Fin j) :
1/2 ≤ (observedFactor A C z K hK hW j delta failure v draws (some a):ℝ) ∧
(observedFactor A C z K hK hW j delta failure v draws (some a):ℝ) ≤ 1 := by
apply rationalAverage_bounds _ _ (sampleCount_positive j delta failure hd hf)
intro bits
rw [ratioSample_cast]
exact ratioObservable_bounds A C (by exact_mod_cast hC) a.val z hz hB.le _
lemma observed_correction_bounds (A : Instance I J) (C : ℕ) (hC : 0 < C) (z : View A → ℕ)
(K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (hB : (K:ℝ)=binWidth A C)
(j : ℕ) (hj : j ≤ rationalHeight A C) (delta failure : ℚ) (hd : 0 < delta) (hd1 : delta < 1)
(hf : 0 < failure) (v : Grid (I := Fin (dimension A)) K)
(draws : EstimatorInput A C K j delta failure) :
1/16 ≤ (observedFactor A C z K hK hW j delta failure v draws none:ℝ) ∧
(observedFactor A C z K hK hW j delta failure v draws none:ℝ) ≤ 8 := by
apply rationalAverage_bounds _ _ (sampleCount_positive j delta failure hd hf)
intro bits
apply computedCorrection_bounds A C j hC hj z K (by omega) hW hB (roundingTolerance delta)
· unfold roundingTolerance
positivity
· unfold roundingTolerance
linarith
/-- These bounds hold for every Boolean input, including executions on
which the empirical averages are statistically inaccurate. -/
theorem estimateValue_range (A : Instance I J) (C : ℕ) (hC : 0 < C) (z : View A → ℕ)
(hz : BalancedBox.InBox (bounds A) z) (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(hB : (K:ℝ)=binWidth A C) (j : ℕ) (hj : j ≤ rationalHeight A C)
(delta failure : ℚ) (hd : 0 < delta) (hd1 : delta < 1) (hf : 0 < failure)
(v : Grid (I := Fin (dimension A)) K) (draws : EstimatorInput A C K j delta failure) :
(assignmentCount A:ℝ)*(1/2:ℝ)^j/16 ≤ (estimateValue A C z K hK hW j delta failure v draws:ℝ) ∧
(estimateValue A C z K hK hW j delta failure v draws:ℝ) ≤ 8*assignmentCount A := by
let r := fun a : Fin j => (observedFactor A C z K hK hW j delta failure v draws (some a):ℝ)
let q := (observedFactor A C z K hK hW j delta failure v draws none:ℝ)
have hr := observed_ratio_bounds A C hC z hz K hK hW hB j delta failure hd hf v draws
have hq := observed_correction_bounds A C hC z K hK hW hB j hj delta failure hd hd1 hf v draws
have hrlow : (1/2:ℝ)^j ≤ ∏ a,r a := by
have hh := Finset.prod_le_prod₀ (s := Finset.univ)
(fun _ _ => by norm_num : ∀ a : Fin j,a∈Finset.univ → (0:ℝ) ≤ 1/2)
(fun a _ => (hr a).1)
simpa only [Finset.prod_const,Finset.card_univ,Fintype.card_fin] using hh
have hrup : (∏ a,r a) ≤ 1 := by
have hh := Finset.prod_le_prod₀ (s := Finset.univ)
(fun a _ => show (0:ℝ) ≤ r a from (by have h := (hr a).1; linarith)) (fun a _ => (hr a).2)
simpa only [Finset.prod_const_one] using hh
have hr0 : (0:ℝ) ≤ ∏ a,r a := Finset.prod_nonneg (fun a _ => by have h := (hr a).1; linarith)
have hq0 : 0 ≤ q := by dsimp only [q]; linarith [hq.1]
have hlow : (1/16:ℝ)*(1/2:ℝ)^j ≤ q*(∏ a,r a) :=
mul_le_mul hq.1 hrlow (by positivity) hq0
have hupp : q*(∏ a,r a) ≤ 8 := by
calc
_ ≤ 8*1 := mul_le_mul hq.2 hrup hr0 (by norm_num)
_ = _ := by norm_num
have hMf : (0:ℝ) ≤ assignmentCount A := Nat.cast_nonneg _
have he : (estimateValue A C z K hK hW j delta failure v draws:ℝ)=
(assignmentCount A:ℝ)*(q*∏ a,r a) := by
simp only [estimateValue,Rat.cast_mul,Rat.cast_natCast,Rat.cast_prod,Fintype.prod_option]
rfl
rw [he]
constructor
· have hh := mul_le_mul_of_nonneg_left hlow hMf
nlinarith
· have hh := mul_le_mul_of_nonneg_left hupp hMf
nlinarith
theorem estimateValue_positive (A : Instance I J) (C : ℕ) (hC : 0 < C) (z : View A → ℕ)
(hz : BalancedBox.InBox (bounds A) z) (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W)
(hB : (K:ℝ)=binWidth A C) (j : ℕ) (hj : j ≤ rationalHeight A C)
(delta failure : ℚ) (hd : 0 < delta) (hd1 : delta < 1) (hf : 0 < failure)
(v : Grid (I := Fin (dimension A)) K) (draws : EstimatorInput A C K j delta failure) :
0 < estimateValue A C z K hK hW j delta failure v draws := by
have hMf : (0:ℝ) < assignmentCount A := by exact_mod_cast assignmentCount_positive A
have hh := (estimateValue_range A C hC z hz K hK hW hB j hj delta failure hd hd1 hf v draws).1
have hp : (0:ℝ) < (assignmentCount A:ℝ)*(1/2:ℝ)^j/16 := by positivity
exact_mod_cast lt_of_lt_of_le hp hh
end
end ContingencyTables.CompanionOracleDensity
end OAI
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