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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Combinatorics.ContingencyTables.Dense.DenseBoundedTrial
import OAI.Combinatorics.ContingencyTables.Dense.DenseIdealRejection
import OAI.Combinatorics.ContingencyTables.Transport.RejectionLaw
namespace OAI
/-! # The bounded dense sampler's actual finite output law -/
namespace ContingencyTables.GridBoundary
open scoped BigOperators Classical
open ResidualMixture FirstSuccess
noncomputable section
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
variable {q : ℕ}
def trialSuccess (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j)
(t : DenseTrial R P d q) : Option (Table R P) :=
if ht : TrialAccepted R P d e t then some (trialTable R P d e htotal t ht) else none
omit [DecidableEq I] [DecidableEq J] in
theorem trialSuccess_some_iff (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j)
(e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j)
(t : DenseTrial R P d q) (X : Table R P) :
trialSuccess R P d e htotal t = some X ↔ t = tableTrial R P d hd hR hP e X := by
constructor
· intro h
unfold trialSuccess at h
split_ifs at h with ht
· have he := Option.some.inj h
rw [← he, tableTrial_trialTable]
· rintro rfl
simp only [trialSuccess, dite_eq_left (tableTrial_accepted R P d hd hR hP e X),
trialTable_tableTrial]
theorem trialSuccess_ideal_mass (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j)
(e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) (X : Table R P) :
successMass (rationalTrialLaw R P d hd hR hP e) (trialSuccess R P d e htotal) X =
rationalTableMass R P d hd hR hP e X := by
unfold successMass
simp_rw [trialSuccess_some_iff R P d hd hR hP e htotal]
simp only [Finset.sum_ite_eq', Finset.mem_univ, ite_true]
rfl
theorem trialSuccess_ideal_failure (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j)
(e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) :
failureMass (rationalTrialLaw R P d hd hR hP e) (trialSuccess R P d e htotal) =
1 - ∑ X, rationalTableMass R P d hd hR hP e X := by
have h := success_failure_total (rationalTrialLaw R P d hd hR hP e) (trialSuccess R P d e htotal)
simp_rw [trialSuccess_ideal_mass R P d hd hR hP e htotal] at h
linarith
def denseBoundedLaw (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ)
(hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j)
(hfree : Fintype.card (I × J) ≤ d) (e : I × J ≃ Fin q)
(htotal : ∑ i, R i = ∑ j, P j) (h : ℕ)
(a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) : RationalLaw (Table R P) :=
retryLaw (boundedTrialLaw R P d hd hR hP hfree e h a)
(trialSuccess R P d e htotal) fallback (d ^ 4 * (h + 1))
/-- The finite sampler always returns an actual table and is within the
requested error of uniform, assuming finite-box interpolation and finite
Cheeger inequalities. -/
theorem denseBoundedLaw_variation (R : Option I → ℕ) (P : Option J → ℕ)
(d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j)
(hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none)
(hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d)
(hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j)
(e : I × J ≃ Fin q) (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin q) (4 * d ^ 8)))
(h : ℕ) (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) :
let _ : Nonempty (Table R P) := ⟨fallback⟩
(variation (denseBoundedLaw R P d hd hR hP hfree e htotal h a fallback)
(uniformLaw (α := Table R P)) : ℝ) ≤ (2 : ℝ) ^ (-(h : ℝ)) := by
let : Nonempty (Table R P) := ⟨fallback⟩
let p := boundedTrialLaw R P d hd hR hP hfree e h a
let q₀ := rationalTrialLaw R P d hd hR hP e
let f := trialSuccess R P d e htotal
let n := d ^ 4 * (h + 1)
have hg : successMass q₀ f = rationalTableMass R P d hd hR hP e :=
funext (trialSuccess_ideal_mass R P d hd hR hP e htotal)
have hf : failureMass q₀ f = 1 - ∑ X, rationalTableMass R P d hd hR hP e X :=
trialSuccess_ideal_failure R P d hd hR hP e htotal
have hi : (variation (retryLaw q₀ f fallback n) (uniformLaw (α := Table R P)) : ℝ) ≤
(63 / 64 : ℝ) ^ n := by
have hi₀ := dense_ideal_rejection_variation R P d hd hR hP hRmax hPmax hdimR hdimP
hfree htotal e n fallback
unfold variation
change (((∑ X : Table R P,
|(retryLaw q₀ f fallback n).mass X - uniformMass|) / 2 : ℚ) : ℝ) ≤ _
simp_rw [retryLaw_mass]
rw [hg, hf]
exact hi₀
have hv := retryLaw_variation p q₀ f fallback n
have hvc : (variation (retryLaw p f fallback n) (retryLaw q₀ f fallback n) : ℝ) ≤
(n : ℝ) * (variation p q₀ : ℝ) := by exact_mod_cast hv
have ht := boundedTrialLaw_variation R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal
e hPL hCheeger h a
have hm := mul_le_mul_of_nonneg_left ht (show (0 : ℝ) ≤ n by positivity)
have htri := variation_triangle (retryLaw p f fallback n) (retryLaw q₀ f fallback n)
(uniformLaw (α := Table R P))
have htric : (variation (retryLaw p f fallback n) (uniformLaw (α := Table R P)) : ℝ) ≤
(variation (retryLaw p f fallback n) (retryLaw q₀ f fallback n) : ℝ) +
(variation (retryLaw q₀ f fallback n) (uniformLaw (α := Table R P)) : ℝ) := by exact_mod_cast htri
change (variation (retryLaw p f fallback n) (uniformLaw (α := Table R P)) : ℝ) ≤ _
apply (htric.trans (add_le_add (hvc.trans hm) hi)).trans
have he : (n : ℝ) * ((1 + (d : ℝ)) * (2 : ℝ) ^ (-(4 * ((h : ℝ) + d)))) =
((d ^ 4 * (h + 1) * (1 + d) : ℕ) : ℝ) * (2 : ℝ) ^ (-(4 * ((h : ℝ) + d))) := by
dsimp [n]
push_cast
ring
rw [he, add_comm]
exact dense_total_error_budget d h hd
end
end ContingencyTables.GridBoundary
end OAI
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