File size: 5,261 Bytes
e63eb7f | 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 | import Mathlib
namespace OAI
/-!
# Accuracy assembly for the edit-distance approximation scheme
The probability and integer-conversion bounds of Section 8 of
*An Almost-Linear Approximation Scheme for Edit Distance*
(OpenAI, September 24, 2026) follow from the refinement estimate and
simultaneous seed guarantee using Markov's inequality.
-/
namespace EditApproximation
open MeasureTheory
/-- The implementation clamps to the input length and rounds down. -/
noncomputable def integerAnswer (N : ℕ) (value : ℝ) : ℕ :=
⌊max 0 (min (N : ℝ) value)⌋₊
/-- Clamping and rounding down preserve a one-sided approximation of an
integer distance. This does not assert that an estimator has that bound. -/
theorem integerAnswer_spec {N D : ℕ} {value ε : ℝ}
(hDN : D ≤ N) (hε : 0 ≤ ε)
(hlower : (D : ℝ) ≤ value) (hupper : value ≤ (1 + ε) * D) :
D ≤ integerAnswer N value ∧
(integerAnswer N value : ℝ) ≤ (1 + ε) * D := by
have hD : (0 : ℝ) ≤ D := Nat.cast_nonneg D
have hmin : (D : ℝ) ≤ min (N : ℝ) value :=
le_min (by exact_mod_cast hDN) hlower
have hnonneg : 0 ≤ max 0 (min (N : ℝ) value) := le_max_left _ _
constructor
· exact (Nat.le_floor_iff hnonneg).2 (hmin.trans (le_max_right _ _))
· apply (Nat.floor_le hnonneg).trans
exact max_le (by positivity) ((min_le_right _ _).trans hupper)
@[simp] theorem integerAnswer_zero (N : ℕ) : integerAnswer N 0 = 0 := by
simp [integerAnswer, Nat.cast_nonneg]
/-- The positive excess on executions having a good seed. -/
noncomputable def goodExcess {Ω : Type*} (G : Set Ω) (value : Ω → ℝ) (D : ℕ) :
Ω → ℝ := G.indicator (fun ω => max (value ω - D) 0)
theorem goodExcess_nonneg {Ω : Type*} (G : Set Ω) (value : Ω → ℝ) (D : ℕ)
(ω : Ω) : 0 ≤ goodExcess G value D ω := by
classical
by_cases hω : ω ∈ G <;> simp [goodExcess, hω]
/-- Markov's inequality converts the positive-excess expectation bound of
Section 8 into an approximation tail bound. -/
theorem goodExcess_tail {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ]
{G : Set Ω} {value : Ω → ℝ} {D : ℕ} {ℓ ε : ℝ}
(hD : 0 < D) (hℓ : 0 < ℓ) (hε : 0 < ε)
(hint : Integrable (goodExcess G value D) μ)
(hmoment : (∫ ω, goodExcess G value D ω ∂μ) ≤ (D : ℝ) / ℓ) :
μ.real {ω | ε * D ≤ goodExcess G value D ω} ≤ 1 / (ℓ * ε) := by
have hDr : (0 : ℝ) < D := by exact_mod_cast hD
have hmarkov := mul_meas_ge_le_integral_of_nonneg
(ae_of_all μ (goodExcess_nonneg G value D)) hint (ε * D)
have htail : μ.real {ω | ε * D ≤ goodExcess G value D ω} ≤
((D : ℝ) / ℓ) / (ε * D) :=
(le_div_iff₀ (mul_pos hε hDr)).2 (by
simpa only [mul_comm] using hmarkov.trans hmoment)
have hratio : ((D : ℝ) / ℓ) / (ε * D) = 1 / (ℓ * ε) := by
field_simp
rwa [hratio] at htail
/-- Final probability assembly, with the two genuinely unresolved
algorithmic guarantees exposed: a good-seed event and its excess moment.
The stronger success probability `4/5` implies the paper's `2/3`. -/
theorem final_answer_accuracy {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{G : Set Ω} {value : Ω → ℝ} {N D : ℕ} {ℓ ε : ℝ}
(hvalue : Measurable value) (hDN : D ≤ N) (hD : 0 < D)
(hℓ : 0 < ℓ) (hε : 0 < ε) (haccuracy : 10 ≤ ℓ * ε)
(hseed : μ.real Gᶜ ≤ 1 / 10)
(hlower : ∀ ω ∈ G, (D : ℝ) ≤ value ω)
(hint : Integrable (goodExcess G value D) μ)
(hmoment : (∫ ω, goodExcess G value D ω ∂μ) ≤ (D : ℝ) / ℓ) :
(4 / 5 : ℝ) ≤ μ.real {ω | D ≤ integerAnswer N (value ω) ∧
(integerAnswer N (value ω) : ℝ) ≤ (1 + ε) * D} := by
classical
let S : Set Ω := {ω | (D : ℝ) ≤ value ω ∧ value ω ≤ (1 + ε) * D}
let B : Set Ω := {ω | ε * D ≤ goodExcess G value D ω}
have hS : MeasurableSet S :=
(measurableSet_le measurable_const hvalue).inter
(measurableSet_le hvalue measurable_const)
have hbad : Sᶜ ⊆ Gᶜ ∪ B := by
intro ω hω
by_cases hG : ω ∈ G
· right
have hupper : (1 + ε) * D < value ω := by
by_contra hn
exact hω ⟨hlower ω hG, le_of_not_gt hn⟩
change ε * D ≤ goodExcess G value D ω
simp only [goodExcess, Set.indicator_of_mem hG]
apply le_trans _ (le_max_left _ _)
nlinarith
· exact Or.inl hG
have htail : μ.real B ≤ 1 / (ℓ * ε) :=
goodExcess_tail μ hD hℓ hε hint hmoment
have hratio : 1 / (ℓ * ε) ≤ (1 / 10 : ℝ) := by
exact one_div_le_one_div_of_le (by norm_num) haccuracy
have hfail : μ.real Sᶜ ≤ (1 / 5 : ℝ) := calc
μ.real Sᶜ ≤ μ.real (Gᶜ ∪ B) := measureReal_mono hbad
_ ≤ μ.real Gᶜ + μ.real B := measureReal_union_le (μ := μ) _ _
_ ≤ 1 / 5 := by linarith
have hmass : μ.real S + μ.real Sᶜ = 1 := by
simpa using measureReal_add_measureReal_compl (μ := μ) hS
have hsuccess : (4 / 5 : ℝ) ≤ μ.real S := by linarith
apply hsuccess.trans
exact measureReal_mono (μ := μ) (fun ω hω =>
integerAnswer_spec hDN hε.le hω.1 hω.2)
end EditApproximation
end OAI
|