File size: 3,486 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactRareMeasureBound
import OAI.Combinatorics.ContingencyTables.Sampling.ExactRareCost
namespace OAI
/-! # Absorbing the actual public correction cost by its branch probability -/
namespace ContingencyTables.SmallGraphProfiles
open MatchingFPRAS MatchingFPRAS.TreeTyped ProfilePrograms DensePrograms CompletionCounts Polynomial
open scoped BigOperators Classical
noncomputable section
lemma exact_scale_absorbed {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
(p : Polynomial ℕ) (A : ℕ) :
p.eval (A*exactBoxScale r c) ≤ 2^(correctionPrecision r c)*
(p.eval A*2^((40*p.natDegree)^2)) := by
apply RareBranchRuntime.polynomial_tabulation_bound
change ((dimensionAllowance (I := Fin m) (J := Fin n))^3*smallBinaryLength r c)^2 ≤
(dimensionAllowance (I := Fin m) (J := Fin n))^6*(smallBinaryLength r c)^2
rw [mul_pow,←pow_mul]
theorem exact_public_correction_bound (print : Polynomial ℕ) : ∃ p : Polynomial ℕ,
∀ (m n : ℕ) (r : Fin m → ℕ) (c : Fin n → ℕ), (∑ i,r i)=∑ j,c j →
let B := print.eval (ExactPrintCost.matrixSize (n := n) r)
ExactPublicCost.budget r c 1 B B+1 ≤
2^(correctionPrecision r c)*p.eval (Algorithms.samplingSize n r 0) := by
obtain ⟨p,hp⟩ := ExactRareCost.public_polynomial
obtain ⟨q,a,hq⟩ := exact_rare_measure_bound
obtain ⟨g,hg⟩ := exact_gate_measure_polynomial
let H := a+g+91*X^2+print.comp (25*X^3)+1
let P := p.comp (X+q)
refine ⟨C (2^((40*P.natDegree)^2))*(P.comp H),?_⟩
intro m n r c htotal
let B := print.eval (ExactPrintCost.matrixSize (n := n) r)
let S := Algorithms.samplingSize n r 0
let E := exactBoxScale r c
let W := H.eval S*E
have hE : 1 ≤ E := exactBoxScale_pos r c
have hHW : H.eval S ≤ W := by simpa only [Nat.mul_one] using Nat.mul_le_mul_left (H.eval S) hE
have ha : a.eval S ≤ H.eval S := by simp only [H,eval_add,eval_one]; omega
have hrare : exactRareMeasure (List.ofFn r,List.ofFn c) ≤ q.eval W :=
(hq m n r c htotal).trans (polynomial_eval_monotone q (Nat.mul_le_mul_right E ha))
have hscalar : weight (Algorithms.encodeMargins r c)+
ExactGateCost.measure (List.ofFn r,List.ofFn c)+B ≤ H.eval S := by
have hw := exact_public_weight r c htotal
have hgate := hg m n r c htotal
have hprint : print.eval (ExactPrintCost.matrixSize (n := n) r) ≤ print.eval (25*S^3) :=
polynomial_eval_monotone print (exact_matrix_size (n := n) r)
simp only [H,eval_add,eval_mul,eval_ofNat,eval_pow,eval_X,eval_comp,eval_one]
unfold B S at *
omega
have hmeasure : ExactRareCost.publicMeasure (Algorithms.encodeMargins r c) B ≤ (X+q).eval W := by
change weight (Algorithms.encodeMargins r c)+
(ExactGateCost.measure (ProfilePrograms.exactParsedMargins (Algorithms.encodeMargins r c))+
exactRareMeasure (ProfilePrograms.exactParsedMargins (Algorithms.encodeMargins r c)))+B ≤ _
rw [exactParsedMargins_spec]
simp only [eval_add,eval_X]
omega
have hcost : ExactPublicCost.budget r c 1 B B+1 ≤ P.eval W := by
have ht := (hp (Algorithms.encodeMargins r c) B).trans (polynomial_eval_monotone p hmeasure)
simpa only [ExactPublicCost.budget,ExactTablePrefix.input,P,eval_comp] using ht
have habsorb := exact_scale_absorbed r c P (H.eval S)
apply hcost.trans (habsorb.trans_eq ?_)
simp only [eval_mul,eval_C,eval_comp]
ring
end
end ContingencyTables.SmallGraphProfiles
end OAI
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