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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 | import OAI.Combinatorics.EditApproximation.Probability.BernsteinMGF
namespace OAI
/-! Finite scalar Bernstein concentration for independent
centered bounded real sums with a summed second-moment upper bound. -/
namespace EditApproximation
open MeasureTheory ProbabilityTheory
open scoped BigOperators
private theorem bernstein_chernoff_exponent (b v d : ℝ) (hb : 0 < b)
(hv : 0 < v) (hd : 0 < d) :
let t := d / (v + b * d / 3);
-t * d + t ^ 2 * v / (2 * (1 - t * b / 3)) =
-d ^ 2 / (2 * v + (2 / 3 : ℝ) * b * d) := by
let D := v + b * d / 3
have hD : D ≠ 0 := ne_of_gt (by dsimp only [D]; positivity)
have hunit : 1 - (d / D) * b / 3 = v / D := by
field_simp [hD]
dsimp only [D]
ring
change -(d / D) * d + (d / D) ^ 2 * v / (2 * (1 - (d / D) * b / 3)) = _
rw [hunit]
have hfinal : 2 * v + (2 / 3 : ℝ) * b * d = 2 * D := by dsimp only [D]; ring
rw [hfinal]
field_simp [hD, hv.ne']
ring
theorem finite_bernstein_upper {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ]
{n : ℕ} (X : Fin n → Ω → ℝ) (b v d : ℝ)
(hb : 0 < b) (hv : 0 < v) (hd : 0 < d)
(hind : iIndepFun X μ) (hm : ∀ i, Measurable (X i))
(hi : ∀ i, Integrable (X i) μ) (hmean : ∀ i, (∫ ω, X i ω ∂μ) = 0)
(hbound : ∀ i, ∀ᵐ ω ∂μ, |X i ω| ≤ b)
(hvar : (∑ i, ∫ ω, (X i ω) ^ 2 ∂μ) ≤ v) :
μ.real {ω | d ≤ ∑ i, X i ω} ≤
Real.exp (-d ^ 2 / (2 * v + (2 / 3 : ℝ) * b * d)) := by
let t := d / (v + b * d / 3)
have hden : 0 < v + b * d / 3 := by positivity
have ht : 0 ≤ t := (div_pos hd hden).le
have htb : t * b < 3 := by
dsimp only [t]
rw [div_mul_eq_mul_div, div_lt_iff₀ hden]
nlinarith only [hv]
have heach (i : Fin n) : Integrable (fun ω => Real.exp (t * X i ω)) μ :=
integrable_exp_mul_of_le t b ht (hm i).aemeasurable
((hbound i).mono fun ω hω => (le_abs_self _).trans hω)
have hsum : (∑ i, X i) = fun ω => ∑ i, X i ω := by
ext ω
simp only [Finset.sum_apply]
have hsumint := hind.integrable_exp_mul_sum hm (s := Finset.univ) (fun i _ => heach i)
rw [hsum] at hsumint
have htail := measure_ge_le_exp_mul_mgf (μ := μ) (X := fun ω => ∑ i, X i ω) d ht hsumint
have hmgf := bounded_centered_sum_mgf μ X b v t hb ht htb hind hm hi hmean hbound hvar
calc
_ ≤ Real.exp (-t * d) * mgf (fun ω => ∑ i, X i ω) μ t := htail
_ ≤ Real.exp (-t * d) * Real.exp (t ^ 2 * v / (2 * (1 - t * b / 3))) :=
mul_le_mul_of_nonneg_left hmgf (Real.exp_nonneg _)
_ = _ := by
rw [← Real.exp_add]
congr 1
exact bernstein_chernoff_exponent b v d hb hv hd
/-- Two-sided Bernstein concentration for independent centered bounded sums. -/
theorem finiteBernsteinInput {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] : FiniteBernsteinInput μ := by
intro n X b v d hb hv hd hind hm hi hmean hbound hvar
have hplus := finite_bernstein_upper μ X b v d hb hv hd hind hm hi hmean hbound hvar
have hnegind : iIndepFun (fun i ω => -X i ω) μ :=
hind.comp (fun _ x => -x) (fun _ => measurable_neg)
have hnegmean (i) : (∫ ω, -X i ω ∂μ) = 0 := by rw [integral_neg, hmean, neg_zero]
have hnegvar : (∑ i, ∫ ω, (-X i ω) ^ 2 ∂μ) ≤ v := by simpa only [neg_sq] using hvar
have hminus := finite_bernstein_upper μ (fun i ω => -X i ω) b v d hb hv hd hnegind
(fun i => (hm i).neg) (fun i => (hi i).neg) hnegmean
(fun i => (hbound i).mono fun ω hω => by simpa only [abs_neg] using hω) hnegvar
have hevent : {ω | d ≤ |∑ i, X i ω|} =
{ω | d ≤ ∑ i, X i ω} ∪ {ω | d ≤ ∑ i, -X i ω} := by
ext ω
simp only [Set.mem_ofPred_eq, Set.mem_union, Finset.sum_neg_distrib]
rw [le_abs]
rw [hevent]
exact (measureReal_union_le _ _).trans (by linarith only [hplus, hminus])
end EditApproximation
end OAI
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