openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionEstimatorConfidence.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionOracleEstimator | |
| namespace OAI | |
| /-! # Confidence of the complete bounded-word weight estimator -/ | |
| namespace ContingencyTables.CompanionOracleDensity | |
| open CompanionFactorialApplication ForestAssignmentSplit GridBoundary ResidualMixture FirstSuccess | |
| open CommonBasesFPRAS FiniteIndependentAverages | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| lemma average_fintype_irrel {X : Type*} (s t : Fintype X) (f : X → ℝ) : | |
| @uniformAverage X s f=@uniformAverage X t f := by | |
| cases Subsingleton.elim s t | |
| rfl | |
| variable {I J : Type*} [Fintype I] [Fintype J] | |
| def badFactor (A : Instance I J) (C : ℕ) (z : View A → ℕ) | |
| (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (j : ℕ) (delta failure : ℚ) | |
| (v : Grid (I := Fin (dimension A)) K) (i : Option (Fin j)) | |
| (draws : EstimatorInput A C K j delta failure) : Prop := | |
| (delta:ℝ)*targetFactor A C z K hK hW j i < | |
| |(observedFactor A C z K hK hW j delta failure v draws i:ℝ)-targetFactor A C z K hK hW j i| | |
| lemma factor_failure (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) (hd0 : 0 < delta) (hd1 : delta < 1) (hf : 0 < failure) | |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler (dimension A)) | |
| (hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin (dimension A)) K)) | |
| (v : Grid (I := Fin (dimension A)) K) (i : Option (Fin j)) : | |
| uniformAverage (fun draws : EstimatorInput A C K j delta failure => | |
| if badFactor A C z K hK hW j delta failure v i draws then (1:ℝ) else 0) ≤ | |
| (failure:ℝ)/(4*((j:ℝ)+1)) := by | |
| let : Nonempty (Fin j → RatioBlock A C K j delta failure) := ⟨fun _ _ _ _ => false⟩ | |
| let : Nonempty (CorrectionBlock A C K j delta failure) := ⟨fun _ _ => false⟩ | |
| cases i with | |
| | none => | |
| simp only [badFactor,observedFactor,targetFactor,rationalAverage_cast] | |
| convert scheduled_correction_failure A C j j hC hj delta failure hd0 hd1 hf z hz K hK hW hB hPL hCheeger v using 1 | |
| rw [uniformAverage_prod] | |
| dsimp only | |
| convert uniformAverage_const (X := Fin j → RatioBlock A C K j delta failure) | |
| (uniformAverage (fun draws : CorrectionBlock A C K j delta failure => | |
| if (delta:ℝ)*correctionMean A C j z K hK hW < | |
| |sampleAverage (fun bits => (correctionSample A C j z K hK hW (roundingTolerance delta) v | |
| (oracleTime A C K 1 (correctionTheta delta)) (offsetAccuracy A delta) bits:ℝ)) | |
| (sampleCount j delta failure) draws-correctionMean A C j z K hK hW| then 1 else 0)) using 1 | |
| | some a => | |
| simp only [badFactor,observedFactor,targetFactor,rationalAverage_cast] | |
| convert scheduled_ratio_failure A C a.val j hC (by omega) delta failure hd0 hf z hz K hK hB hPL hCheeger v using 1 | |
| rw [uniformAverage_prod] | |
| dsimp only | |
| have hc (x : Fin j → RatioBlock A C K j delta failure) := | |
| uniformAverage_const (X := CorrectionBlock A C K j delta failure) | |
| (if (delta:ℝ)*ratioMean A C a.val z K < | |
| |sampleAverage (fun bits => (ratioSample A C a.val z v (oracleTime A C K 1 delta) bits:ℝ)) | |
| (sampleCount j delta failure) (x a)-ratioMean A C a.val z K| then 1 else 0) | |
| simp_rw [hc] | |
| convert uniformAverage_coordinate (X := fun _ : Fin j => RatioBlock A C K j delta failure) a (fun draws : RatioBlock A C K j delta failure => | |
| if (delta:ℝ)*ratioMean A C a.val z K < | |
| |sampleAverage (fun bits => (ratioSample A C a.val z v (oracleTime A C K 1 delta) bits:ℝ)) | |
| (sampleCount j delta failure) draws-ratioMean A C a.val z K| then 1 else 0) using 1 | |
| exact average_fintype_irrel _ _ _ | |
| /-- All ratio and correction block failures share one explicit budget. | |
| The statement concerns the actual bounded Boolean experiment. -/ | |
| theorem goodEstimate_failure (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) (hd0 : 0 < delta) (hd1 : delta < 1) (hf : 0 < failure) | |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler (dimension A)) | |
| (hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin (dimension A)) K)) | |
| (v : Grid (I := Fin (dimension A)) K) : | |
| uniformAverage (fun draws : EstimatorInput A C K j delta failure => | |
| if ¬GoodEstimate A C z K hK hW j delta failure v draws then (1:ℝ) else 0) ≤ (failure:ℝ)/4 := by | |
| have he (draws : EstimatorInput A C K j delta failure) : | |
| (¬GoodEstimate A C z K hK hW j delta failure v draws) ↔ | |
| ∃ i,badFactor A C z K hK hW j delta failure v i draws := by | |
| simp only [GoodEstimate,badFactor,not_forall,not_le] | |
| simp_rw [he] | |
| calc | |
| _ ≤ ∑ i : Option (Fin j),uniformAverage (fun draws : EstimatorInput A C K j delta failure => | |
| if badFactor A C z K hK hW j delta failure v i draws then (1:ℝ) else 0) := | |
| uniform_union_bound _ | |
| _ ≤ ∑ _i : Option (Fin j),(failure:ℝ)/(4*((j:ℝ)+1)) := | |
| Finset.sum_le_sum (fun i _ => factor_failure A C hC z hz K hK hW hB j hj delta failure | |
| hd0 hd1 hf hPL hCheeger v i) | |
| _ = _ := by | |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_option,Fintype.card_fin,nsmul_eq_mul, | |
| Nat.cast_add,Nat.cast_one] | |
| field_simp | |
| lemma innerTolerance_pos (epsilon : ℚ) (j : ℕ) (he : 0 < epsilon) : | |
| 0 < innerTolerance epsilon j := by unfold innerTolerance; positivity | |
| lemma innerTolerance_lt_one (epsilon : ℚ) (j : ℕ) (he : epsilon < 1) : | |
| innerTolerance epsilon j < 1 := by | |
| unfold innerTolerance | |
| rw [div_lt_iff₀ (by positivity : (0:ℚ)<200*(j+1))] | |
| have hj : (0:ℚ) ≤ j := Nat.cast_nonneg _ | |
| nlinarith | |
| /-- A concrete rational estimator with bounded independent Boolean input | |
| has the required relative-error guarantee for the original h-value. -/ | |
| theorem estimateValue_failure (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) | |
| (epsilon failure : ℚ) (he0 : 0 < epsilon) (he1 : epsilon < 1) (hf : 0 < failure) | |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler (dimension A)) | |
| (hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin (dimension A)) K)) | |
| (v : Grid (I := Fin (dimension A)) K) : | |
| uniformAverage (fun draws : EstimatorInput A C K j (innerTolerance epsilon j) failure => | |
| if (estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ) < | |
| (1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ∨ | |
| (1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z < | |
| (estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ) | |
| then (1:ℝ) else 0) ≤ (failure:ℝ)/4 := by | |
| have ht : 0 ≤ h A ((j:ℝ)*C/height A C) z := (h_positive A _ _).le | |
| have heR : (0:ℝ) < epsilon := by exact_mod_cast he0 | |
| have hp (draws : EstimatorInput A C K j (innerTolerance epsilon j) failure) : | |
| (if (estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ) < | |
| (1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ∨ | |
| (1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z < | |
| (estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ) | |
| then (1:ℝ) else 0) ≤ | |
| if ¬GoodEstimate A C z K hK hW j (innerTolerance epsilon j) failure v draws then 1 else 0 := by | |
| by_cases hg : GoodEstimate A C z K hK hW j (innerTolerance epsilon j) failure v draws | |
| · have ha := goodEstimate_accuracy A C hC z hz K hK hW hB j hj epsilon failure he0 he1 v draws hg | |
| have hn : ¬((estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ) < | |
| (1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ∨ | |
| (1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z < | |
| (estimateValue A C z K hK hW j (innerTolerance epsilon j) failure v draws:ℝ)) := by | |
| rintro (hb|hb) <;> nlinarith | |
| simp only [hn,hg,not_true_eq_false,ite_false,le_refl] | |
| · simp only [hg,not_false_eq_true,ite_true] | |
| split_ifs <;> norm_num | |
| exact (uniformAverage_mono _ _ hp).trans | |
| (goodEstimate_failure A C hC z hz K hK hW hB j hj (innerTolerance epsilon j) failure | |
| (innerTolerance_pos epsilon j he0) (innerTolerance_lt_one epsilon j he1) hf hPL hCheeger v) | |
| end | |
| end ContingencyTables.CompanionOracleDensity | |
| end OAI | |