openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionEstimatorAmplification.lean
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionEstimatorWords | |
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteMedianConfidence | |
| namespace OAI | |
| /-! # Logarithmic confidence amplification of the actual weight estimator -/ | |
| namespace ContingencyTables.CompanionOracleDensity | |
| open CompanionFactorialApplication ForestAssignmentSplit GridBoundary ResidualMixture FirstSuccess | |
| open CommonBasesFPRAS FiniteMedianConfidence | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {I J : Type*} [Fintype I] [Fintype J] | |
| /-- Each complete estimator uses constant confidence; the requested | |
| failure probability changes only the number of independent medians. -/ | |
| def medianEstimatorBits (A : Instance I J) (C K j : ℕ) (epsilon delta : ℚ) : ℕ := | |
| repetitionCount delta * estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1 | |
| def medianEstimate (A : Instance I J) (C : ℕ) (z : View A → ℕ) | |
| (K : ℕ) (hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (j : ℕ) (epsilon delta : ℚ) | |
| (v : Grid (I := Fin (dimension A)) K) | |
| (bits : Fin (medianEstimatorBits A C K j epsilon delta) → Bool) : ℚ := | |
| rationalMedian (List.ofFn (fun i : Fin (repetitionCount delta) => | |
| wordEstimate A C z K hK hW j (epsilon/2) 1 v | |
| (oracleWalkWordEquiv (repetitionCount delta) | |
| (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1) bits i))) | |
| theorem medianEstimate_success (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 delta : ℚ) (he0 : 0 < epsilon) (he1 : epsilon < 1) (hd : 0 < delta) | |
| (hPL : PublishedInputs.FiniteBoxPrekopaLeindler (dimension A)) | |
| (hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin (dimension A)) K)) | |
| (v : Grid (I := Fin (dimension A)) K) : | |
| 1-(delta:ℝ) ≤ uniformAverage (fun bits : Fin (medianEstimatorBits A C K j epsilon delta) → Bool => | |
| if (1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ≤ | |
| (medianEstimate A C z K hK hW j epsilon delta v bits:ℝ) ∧ | |
| (medianEstimate A C z K hK hW j epsilon delta v bits:ℝ) ≤ | |
| (1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z | |
| then (1:ℝ) else 0) := by | |
| have heh0 : 0 < epsilon/2 := by positivity | |
| have heh1 : epsilon/2 < 1 := by linarith | |
| have ht : 0 < h A ((j:ℝ)*C/height A C) z := h_positive A _ _ | |
| have heR : (0:ℝ) < epsilon := by exact_mod_cast he0 | |
| have hbase := wordEstimate_failure A C hC z hz K hK hW hB j hj (epsilon/2) 1 heh0 heh1 | |
| (by norm_num) hPL hCheeger v | |
| have hbad : uniformAverage (fun bits : Fin (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1) → Bool => | |
| if (wordEstimate A C z K hK hW j (epsilon/2) 1 v bits:ℝ) < | |
| (1-(epsilon:ℝ)/2)*h A ((j:ℝ)*C/height A C) z ∨ | |
| (1+(epsilon:ℝ)/2)*h A ((j:ℝ)*C/height A C) z < | |
| (wordEstimate A C z K hK hW j (epsilon/2) 1 v bits:ℝ) | |
| then (1:ℝ) else 0) ≤ 5/16 := by | |
| have hh : (1:ℝ)/4 ≤ 5/16 := by norm_num | |
| apply le_trans ?_ hh | |
| simpa only [Rat.cast_div,Rat.cast_ofNat,Rat.cast_one] using hbase | |
| have hm := median_confidence_real | |
| (wordEstimate A C z K hK hW j (epsilon/2) 1 v) | |
| ((1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z) | |
| ((1-(epsilon:ℝ)/2)*h A ((j:ℝ)*C/height A C) z) | |
| ((1+(epsilon:ℝ)/2)*h A ((j:ℝ)*C/height A C) z) | |
| ((1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z) | |
| (by nlinarith) (by nlinarith) delta hd hbad | |
| let test := fun xs : Fin (repetitionCount delta) → | |
| (Fin (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1) → Bool) => | |
| if (1-(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z ≤ | |
| (rationalMedian (List.ofFn (fun i => wordEstimate A C z K hK hW j (epsilon/2) 1 v (xs i))):ℝ) ∧ | |
| (rationalMedian (List.ofFn (fun i => wordEstimate A C z K hK hW j (epsilon/2) 1 v (xs i))):ℝ) ≤ | |
| (1+(epsilon:ℝ))*h A ((j:ℝ)*C/height A C) z | |
| then (1:ℝ) else 0 | |
| have he := uniformAverage_equiv | |
| (oracleWalkWordEquiv (repetitionCount delta) | |
| (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1)) test | |
| change 1-(delta:ℝ) ≤ uniformAverage (fun bits => test | |
| (oracleWalkWordEquiv (repetitionCount delta) | |
| (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1) bits)) | |
| exact hm.trans_eq he.symm | |
| /-- Amplification also preserves the proved range on every word, | |
| including statistically unsuccessful words. -/ | |
| theorem medianEstimate_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) | |
| (epsilon delta : ℚ) (he0 : 0 < epsilon) (he1 : epsilon < 1) | |
| (v : Grid (I := Fin (dimension A)) K) | |
| (bits : Fin (medianEstimatorBits A C K j epsilon delta) → Bool) : | |
| (assignmentCount A:ℝ)*(1/2:ℝ)^j/16 ≤ | |
| (medianEstimate A C z K hK hW j epsilon delta v bits:ℝ) ∧ | |
| (medianEstimate A C z K hK hW j epsilon delta v bits:ℝ) ≤ 8*assignmentCount A := by | |
| let samples := List.ofFn (fun i : Fin (repetitionCount delta) => | |
| wordEstimate A C z K hK hW j (epsilon/2) 1 v | |
| (oracleWalkWordEquiv (repetitionCount delta) | |
| (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1) bits i)) | |
| have hmem := rationalMedian_mem samples (by | |
| dsimp only [samples] | |
| rw [List.length_ofFn] | |
| exact repetitionCount_pos delta) | |
| obtain ⟨i,hi⟩ := List.mem_ofFn.mp hmem | |
| have hr := wordEstimate_range A C hC z hz K hK hW hB j hj (epsilon/2) 1 | |
| (by positivity) (by linarith) (by norm_num) v | |
| (oracleWalkWordEquiv (repetitionCount delta) | |
| (estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1) bits i) | |
| change _ ≤ (rationalMedian samples:ℝ) ∧ (rationalMedian samples:ℝ) ≤ _ | |
| rw [←hi] | |
| exact hr | |
| /-- The exact bit budget grows linearly with the binary confidence | |
| parameter, rather than with its reciprocal. -/ | |
| lemma medianEstimatorBits_eq (A : Instance I J) (C K j : ℕ) (epsilon delta : ℚ) : | |
| medianEstimatorBits A C K j epsilon delta = | |
| 64*(delta.den.bits.length+1) * | |
| estimatorWordBits A C K j (innerTolerance (epsilon/2) j) 1 := rfl | |
| end | |
| end ContingencyTables.CompanionOracleDensity | |
| end OAI | |