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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 | import OAI.Combinatorics.EditApproximation.Geometry.PublishedInputs
namespace OAI
/-!
# Coarse threshold concentration from the published scalar input
The quantitative Bernstein bound used in Section 3 does not require
independence of estimated child values.
The random variables to which Bernstein is applied are fixed true-cost
threshold contributions, conditional on ancestor thresholds.
-/
namespace EditApproximation
open MeasureTheory ProbabilityTheory
open scoped BigOperators
/-- Numerical exponent estimate for the coarse estimator's choice of
deviation. The proof also covers the two regimes in the manuscript at once. -/
theorem coarse_bernstein_exponent {lam V allowance scale Q : ℝ}
(hlam : 0 ≤ lam) (hlamone : lam ≤ 1) (hV : 0 ≤ V)
(ha : 0 < allowance) (hu : 0 < scale) (hQ : Q * scale = allowance) :
lam * Q / 12 ≤ (lam * V + allowance / 2) ^ 2 /
(2 * scale * V + (2 / 3 : ℝ) * scale * (lam * V + allowance / 2)) := by
let z := lam * V + allowance / 2
have hz : 0 < z := by dsimp [z]; positivity
have ha2 : allowance ≤ 2 * z := by dsimp [z]; nlinarith
have hlamV : lam * V ≤ z := by dsimp [z]; linarith
have hfirst : allowance * (lam * V) ≤ allowance * z :=
mul_le_mul_of_nonneg_left hlamV ha.le
have hsecond : lam * (allowance * z) ≤ allowance * z := by
exact (mul_le_mul_of_nonneg_right hlamone (mul_nonneg ha.le hz.le)).trans_eq
(one_mul _)
have hthird : allowance * z ≤ 2 * z ^ 2 := by nlinarith
have hbase : lam * allowance * (2 * V + (2 / 3 : ℝ) * z) ≤ 12 * z ^ 2 := by
nlinarith [sq_nonneg z]
have hdenom : 0 < 2 * scale * V + (2 / 3 : ℝ) * scale * z := by positivity
apply (le_div_iff₀ hdenom).2
change lam * Q / 12 * (2 * scale * V + (2 / 3 : ℝ) * scale * z) ≤ z ^ 2
have heq : lam * Q / 12 * (2 * scale * V + (2 / 3 : ℝ) * scale * z) =
lam * allowance * (2 * V + (2 / 3 : ℝ) * z) / 12 := by
rw [← hQ]
ring
rw [heq]
linarith
/-- Coarse threshold concentration, conditional only on the ancestor
data. Its hypotheses specify the centered *true-cost* random variables,
their moment proxy, and the threshold estimator's small bias. -/
theorem coarse_threshold_concentration {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
(bernstein : FiniteBernsteinInput μ)
{n : ℕ} (X : Fin n → Ω → ℝ) (estimator : Ω → ℝ)
(mean V lam allowance scale Q : ℝ)
(hlam : 0 ≤ lam) (hlamone : lam ≤ 1) (hV : 0 < V)
(ha : 0 < allowance) (hu : 0 < scale) (hQ : Q * scale = allowance)
(hscale : scale ≤ allowance / 2)
(hindep : iIndepFun X μ) (hmeas : ∀ i, Measurable (X i))
(hint : ∀ i, Integrable (X i) μ)
(hmean : ∀ i, (∫ ω, X i ω ∂μ) = 0)
(hbound : ∀ i, ∀ᵐ ω ∂μ, |X i ω| ≤ scale)
(hvariance : (∑ i, ∫ ω, (X i ω) ^ 2 ∂μ) ≤ scale * V)
(hcenter : ∀ ω, estimator ω - mean = ∑ i, X i ω)
(hbias : |mean - V| ≤ scale) :
μ.real {ω | lam * V + allowance < |estimator ω - V|} ≤
2 * Real.exp (-lam * Q / 12) := by
let z := lam * V + allowance / 2
have hz : 0 < z := by dsimp [z]; positivity
have htail := bernstein n X scale (scale * V) z hu (mul_pos hu hV) hz
hindep hmeas hint hmean hbound hvariance
have hsubset : {ω | lam * V + allowance < |estimator ω - V|} ⊆
{ω | z ≤ |∑ i, X i ω|} := by
intro ω hω
have htriangle : |estimator ω - V| ≤ |estimator ω - mean| + |mean - V| := by
calc
|estimator ω - V| = |(estimator ω - mean) + (mean - V)| := by
congr 1
ring
_ ≤ |estimator ω - mean| + |mean - V| := abs_add_le _ _
rw [hcenter ω] at htriangle
change z ≤ |∑ i, X i ω|
dsimp [z]
have hω' : lam * V + allowance < |estimator ω - V| := hω
linarith
apply (measureReal_mono (μ := μ) hsubset).trans
apply htail.trans
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply Real.exp_le_exp.mpr
have hexponent := coarse_bernstein_exponent hlam hlamone hV.le ha hu hQ
change -z ^ 2 / (2 * (scale * V) + (2 / 3 : ℝ) * scale * z) ≤ -lam * Q / 12
simpa only [neg_div, neg_mul, mul_assoc] using neg_le_neg hexponent
end EditApproximation
end OAI
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