openAI-math-dataset / lean /OAI /Probability /FreeIsing /Estimates /BoundedAverageConcentration.lean
| 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 | |