| import OAI.Combinatorics.EditApproximation.Geometry.PublishedInputs |
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| namespace OAI |
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| /-! |
| # Coarse threshold concentration from the published scalar input |
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| 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. |
| -/ |
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| namespace EditApproximation |
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| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
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| / |
| 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 |
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| / |
| 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 |
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| end EditApproximation |
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| end OAI |
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