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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
namespace BoundedAverageConcentration
variable {Ω ι : Type*} [MeasurableSpace Ω]
variable {μ : Measure Ω} [IsProbabilityMeasure μ]
theorem measureReal_average_ge_le
{X : ι → Ω → ℝ} (h_indep : iIndepFun X μ)
(s : Finset ι) (hs : s.Nonempty) {τ : ℝ} (hτ : 0 ≤ τ)
(h_meas : ∀ i ∈ s, AEMeasurable (X i) μ)
(h_bdd : ∀ i ∈ s, ∀ᵐ ω ∂μ, X i ω ∈ Set.Icc (0 : ℝ) 1)
(h_mean : ∀ i ∈ s, (∫ ω, X i ω ∂μ) ≤ τ / 2) :
μ.real {ω | τ ≤ (∑ i ∈ s, X i ω) / (s.card : ℝ)} ≤
Real.exp (-(s.card : ℝ) * τ ^ 2 / 2) := by
classical
have hcard : (0 : ℝ) < s.card := by exact_mod_cast hs.card_pos
have hcenter : iIndepFun (fun i ω => X i ω - ∫ x, X i x ∂μ) μ := by
exact h_indep.comp (fun i x => x - ∫ ω, X i ω ∂μ)
(fun _ => measurable_id.sub_const _)
have hsub : ∀ i ∈ s,
HasSubgaussianMGF (fun ω => X i ω - ∫ x, X i x ∂μ) (1 / 4 : ℝ≥0) μ := by
intro i hi
convert hasSubgaussianMGF_of_mem_Icc (h_meas i hi) (h_bdd i hi) using 1
norm_num
have hsum_mean : (∑ i ∈ s, ∫ x, X i x ∂μ) ≤ (s.card : ℝ) * τ / 2 := by
calc
(∑ i ∈ s, ∫ x, X i x ∂μ) ≤ ∑ _i ∈ s, τ / 2 :=
Finset.sum_le_sum h_mean
_ = (s.card : ℝ) * τ / 2 := by simp [mul_div_assoc]
have hsubset : {ω | τ ≤ (∑ i ∈ s, X i ω) / (s.card : ℝ)} ⊆
{ω | (s.card : ℝ) * τ / 2 ≤ ∑ i ∈ s, (X i ω - ∫ x, X i x ∂μ)} := by
intro ω hω
have hw : τ * (s.card : ℝ) ≤ ∑ i ∈ s, X i ω :=
(le_div_iff₀ hcard).mp hω
simp only [Set.mem_ofPred_eq, Finset.sum_sub_distrib]
nlinarith
have hhoeff := HasSubgaussianMGF.measure_sum_ge_le_of_iIndepFun hcenter hsub
(show 0 ≤ (s.card : ℝ) * τ / 2 by positivity)
have hexp :
-((s.card : ℝ) * τ / 2) ^ 2 /
(2 * ((∑ _i ∈ s, (1 / 4 : ℝ≥0) : ℝ≥0) : ℝ)) =
-(s.card : ℝ) * τ ^ 2 / 2 := by
simp only [Finset.sum_const, nsmul_eq_mul]
push_cast
field_simp [ne_of_gt hcard]
ring
exact (measureReal_mono hsubset).trans (hexp ▸ hhoeff)
theorem measure_average_ge_le
{X : ι → Ω → ℝ} (h_indep : iIndepFun X μ)
(s : Finset ι) (hs : s.Nonempty) {τ : ℝ} (hτ : 0 ≤ τ)
(h_meas : ∀ i ∈ s, AEMeasurable (X i) μ)
(h_bdd : ∀ i ∈ s, ∀ᵐ ω ∂μ, X i ω ∈ Set.Icc (0 : ℝ) 1)
(h_mean : ∀ i ∈ s, (∫ ω, X i ω ∂μ) ≤ τ / 2) :
μ {ω | τ ≤ (∑ i ∈ s, X i ω) / (s.card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(s.card : ℝ) * τ ^ 2 / 2)) := by
rw [← ofReal_measureReal]
exact ENNReal.ofReal_le_ofReal
(measureReal_average_ge_le h_indep s hs hτ h_meas h_bdd h_mean)
theorem exponential_card_bound
{N b ℓ q : ℝ} (hb : 0 < b) (hℓ : 0 ≤ ℓ)
(hq : q ≤ Real.log b / 8)
(hN : b ^ (ℓ / 2 - 1) ≤ N) :
Real.exp (-N * (Real.exp (-q * ℓ)) ^ 2 / 2) ≤
Real.exp (-(b ^ (ℓ / 4)) / (2 * b)) := by
have hpow : 0 < b ^ (ℓ / 2 - 1) := Real.rpow_pos_of_pos hb _
have hexp : Real.exp (-(Real.log b) * ℓ / 4) ≤ (Real.exp (-q * ℓ)) ^ 2 := by
rw [← Real.exp_nat_mul]
apply Real.exp_le_exp.mpr
nlinarith [mul_le_mul_of_nonneg_right hq hℓ]
have hprod : b ^ (ℓ / 4) / b ≤ N * (Real.exp (-q * ℓ)) ^ 2 := by
calc
b ^ (ℓ / 4) / b =
b ^ (ℓ / 2 - 1) * Real.exp (-(Real.log b) * ℓ / 4) := by
rw [Real.rpow_def_of_pos hb, Real.rpow_def_of_pos hb,
← Real.exp_log hb, ← Real.exp_sub, ← Real.exp_add]
simp only [Real.log_exp]
congr 1
ring
_ ≤ N * (Real.exp (-q * ℓ)) ^ 2 :=
mul_le_mul hN hexp (Real.exp_pos _).le (hpow.le.trans hN)
apply Real.exp_le_exp.mpr
have := div_le_div_of_nonneg_right hprod (show (0 : ℝ) ≤ 2 by norm_num)
simpa [div_div, neg_div, mul_comm] using neg_le_neg this
theorem measureReal_average_ge_le_double_exp
{X : ι → Ω → ℝ} (h_indep : iIndepFun X μ)
(s : Finset ι) (hs : s.Nonempty) {b ℓ q : ℝ}
(hb : 0 < b) (hℓ : 0 ≤ ℓ) (hq : q ≤ Real.log b / 8)
(hcard : b ^ (ℓ / 2 - 1) ≤ (s.card : ℝ))
(h_meas : ∀ i ∈ s, AEMeasurable (X i) μ)
(h_bdd : ∀ i ∈ s, ∀ᵐ ω ∂μ, X i ω ∈ Set.Icc (0 : ℝ) 1)
(h_mean : ∀ i ∈ s, (∫ ω, X i ω ∂μ) ≤ Real.exp (-q * ℓ) / 2) :
μ.real {ω | Real.exp (-q * ℓ) ≤ (∑ i ∈ s, X i ω) / (s.card : ℝ)} ≤
Real.exp (-(b ^ (ℓ / 4)) / (2 * b)) := by
exact (measureReal_average_ge_le h_indep s hs (Real.exp_pos _).le
h_meas h_bdd h_mean).trans (exponential_card_bound hb hℓ hq hcard)
theorem measure_average_ge_le_of_kernel
{Y : Type*} [MeasurableSpace Y] (ν : Measure Y) [IsProbabilityMeasure ν]
(κ : Kernel Y Ω) [IsMarkovKernel κ]
{X : ι → Ω → ℝ} (s : Finset ι) (hs : s.Nonempty)
{τ : ℝ} (hτ : 0 ≤ τ)
(h_meas : ∀ i ∈ s, Measurable (X i))
(h_indep : ∀ᵐ y ∂ν, iIndepFun X (κ y))
(h_bdd : ∀ᵐ y ∂ν, ∀ i ∈ s, ∀ᵐ ω ∂κ y, X i ω ∈ Set.Icc (0 : ℝ) 1)
(h_mean : ∀ᵐ y ∂ν, ∀ i ∈ s, (∫ ω, X i ω ∂κ y) ≤ τ / 2) :
(κ ∘ₘ ν) {ω | τ ≤ (∑ i ∈ s, X i ω) / (s.card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(s.card : ℝ) * τ ^ 2 / 2)) := by
have hsum : Measurable (fun ω => ∑ i ∈ s, X i ω) :=
Finset.measurable_fun_sum s h_meas
rw [Measure.bind_apply (measurableSet_le measurable_const (hsum.div_const _))
κ.aemeasurable]
calc
(∫⁻ y, κ y {ω | τ ≤ (∑ i ∈ s, X i ω) / (s.card : ℝ)} ∂ν) ≤
∫⁻ _y, ENNReal.ofReal (Real.exp (-(s.card : ℝ) * τ ^ 2 / 2)) ∂ν := by
apply lintegral_mono_ae
filter_upwards [h_indep, h_bdd, h_mean] with y hy_indep hy_bdd hy_mean
exact measure_average_ge_le hy_indep s hs hτ
(fun i hi => (h_meas i hi).aemeasurable) hy_bdd hy_mean
_ = ENNReal.ofReal (Real.exp (-(s.card : ℝ) * τ ^ 2 / 2)) := by simp
theorem attenuation_le_half_threshold
{c η x : ℝ} (hx : 0 ≤ x) (hη : η ≤ c / 4)
(hlength : 4 * Real.log 2 ≤ c * x) :
Real.exp (-c * x / 2) ≤ Real.exp (-η * x) / 2 := by
apply (le_div_iff₀ (show (0 : ℝ) < 2 by norm_num)).mpr
conv_lhs => rhs; rw [← Real.exp_log (show (0 : ℝ) < 2 by norm_num)]
rw [← Real.exp_add]
apply Real.exp_le_exp.mpr
nlinarith [mul_le_mul_of_nonneg_right hη hx]
end BoundedAverageConcentration
end Problem367
end
end OAI
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