openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactRarePublicBound.lean
| 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 | |