| import OAI.Probability.FreeIsing.Energy.Attenuation |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
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| namespace Problem367 |
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| lemma strictMono_tanh_response : StrictMono Real.tanh := by |
| apply strictMono_of_deriv_pos |
| intro x |
| rw [(hasDerivAt_tanh_response x).deriv] |
| exact sub_pos.mpr (Real.tanh_sq_lt_one x) |
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| lemma tanh_abs_response (x : ℝ) : Real.tanh |x| = |Real.tanh x| := by |
| rw [Real.tanh_eq_sinh_div_cosh, Real.tanh_eq_sinh_div_cosh, abs_div, |
| abs_of_pos (Real.cosh_pos x), Real.abs_sinh, Real.cosh_abs] |
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| lemma tanh_one_pos_response : 0 < Real.tanh 1 := by |
| have h := strictMono_tanh_response (by norm_num : (0 : ℝ) < 1) |
| simpa using h |
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| lemma sq_le_cosh_sq_mul_tanh_sq {x M : ℝ} (hx : |x| ≤ M) : |
| x ^ 2 ≤ Real.cosh M ^ 2 * Real.tanh x ^ 2 := by |
| have hsinh : |x| ≤ |Real.sinh x| := by |
| rw [Real.abs_sinh] |
| exact Real.self_le_sinh_iff.mpr (abs_nonneg x) |
| have hcosh : Real.cosh x ≤ Real.cosh M := |
| Real.cosh_le_cosh.mpr (hx.trans (le_abs_self M)) |
| have heq : |Real.sinh x| = Real.cosh x * |Real.tanh x| := by |
| rw [Real.tanh_eq_sinh_div_cosh, abs_div, abs_of_pos (Real.cosh_pos x)] |
| field_simp |
| have h : |x| ≤ Real.cosh M * |Real.tanh x| := by |
| rw [heq] at hsinh |
| exact hsinh.trans (mul_le_mul_of_nonneg_right hcosh (abs_nonneg _)) |
| have hsq := (sq_le_sq₀ (abs_nonneg x) |
| (mul_nonneg (le_of_lt (Real.cosh_pos M)) (abs_nonneg _))).mpr h |
| simpa only [sq_abs, mul_pow] using hsq |
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| lemma tanh_one_sq_le_of_one_le_abs {x : ℝ} (hx : 1 ≤ |x|) : |
| Real.tanh 1 ^ 2 ≤ Real.tanh x ^ 2 := by |
| have h := strictMono_tanh_response.monotone hx |
| rw [tanh_abs_response] at h |
| have hs := (sq_le_sq₀ (le_of_lt tanh_one_pos_response) (abs_nonneg _)).mpr h |
| simpa only [sq_abs] using hs |
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| section Probability |
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| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| variable {G : Ω → ℝ} {beta : ℝ} |
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| lemma integrable_tanh_shift_sq (hG : Measurable G) (z : ℝ) : |
| Integrable (fun w => Real.tanh (z + G w) ^ 2) μ := by |
| have hm : Measurable (fun w => Real.tanh (z + G w) ^ 2) := |
| ((lipschitzWith_tanh_response.continuous.measurable).comp (hG.const_add z)).pow_const 2 |
| apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable |
| filter_upwards [] with w |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact le_of_lt (Real.tanh_sq_lt_one _) |
| |
| lemma variance_le_cosh_sq_mul_tanh_shift_integral |
| (hG : Measurable G) (hBound : ∀ᵐ w ∂μ, |G w| ≤ beta) |
| {z : ℝ} (hz : |z| ≤ beta + 1) : |
| variance G μ ≤ Real.cosh (2 * beta + 1) ^ 2 * |
| ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| have hGI : MemLp G 2 μ := |
| MemLp.of_bound hG.aestronglyMeasurable beta (by simpa only [Real.norm_eq_abs] using hBound) |
| have hshiftI : Integrable (fun w => (z + G w) ^ 2) μ := |
| ((memLp_const z).add hGI).integrable_sq |
| have htanI := integrable_tanh_shift_sq (μ := μ) hG z |
| calc |
| variance G μ = variance (fun w => z + G w) μ := |
| (variance_const_add hG.aestronglyMeasurable z).symm |
| _ ≤ ∫ w, (z + G w) ^ 2 ∂μ := |
| variance_le_expectation_sq (hG.const_add z).aestronglyMeasurable |
| _ ≤ ∫ w, Real.cosh (2 * beta + 1) ^ 2 * Real.tanh (z + G w) ^ 2 ∂μ := by |
| apply integral_mono_ae hshiftI (htanI.const_mul _) |
| filter_upwards [hBound] with w hw |
| apply sq_le_cosh_sq_mul_tanh_sq |
| calc |
| |z + G w| ≤ |z| + |G w| := abs_add_le _ _ |
| _ ≤ (beta + 1) + beta := add_le_add hz hw |
| _ = 2 * beta + 1 := by ring |
| _ = Real.cosh (2 * beta + 1) ^ 2 * |
| ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := integral_const_mul _ _ |
| |
| lemma variance_le_beta_sq_of_abs_le |
| (hG : Measurable G) (hBound : ∀ᵐ w ∂μ, |G w| ≤ beta) : |
| variance G μ ≤ beta ^ 2 := by |
| have h := variance_le_sq_of_bounded (hBound.mono fun _ hw => abs_le.mp hw) |
| hG.aemeasurable |
| convert! h using 1 |
| ring |
| |
| lemma variance_le_tanh_shift_integral_of_far |
| (hG : Measurable G) (hBound : ∀ᵐ w ∂μ, |G w| ≤ beta) |
| {z : ℝ} (hz : beta + 1 < |z|) : |
| variance G μ ≤ (beta ^ 2 / Real.tanh 1 ^ 2) * |
| ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| have htpos : 0 < Real.tanh 1 ^ 2 := sq_pos_of_pos tanh_one_pos_response |
| have htanI := integrable_tanh_shift_sq (μ := μ) hG z |
| have hlow : Real.tanh 1 ^ 2 ≤ ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| calc |
| Real.tanh 1 ^ 2 = ∫ _w : Ω, Real.tanh 1 ^ 2 ∂μ := by simp |
| _ ≤ ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| apply integral_mono_ae (integrable_const _) htanI |
| filter_upwards [hBound] with w hw |
| apply tanh_one_sq_le_of_one_le_abs |
| have htri : |z| ≤ |z + G w| + |G w| := by |
| calc |
| |z| = |(z + G w) + (-G w)| := by congr 1; ring |
| _ ≤ |z + G w| + |-G w| := abs_add_le _ _ |
| _ = |z + G w| + |G w| := by rw [abs_neg] |
| linarith |
| calc |
| variance G μ ≤ beta ^ 2 := variance_le_beta_sq_of_abs_le hG hBound |
| _ = (beta ^ 2 / Real.tanh 1 ^ 2) * Real.tanh 1 ^ 2 := by |
| rw [div_mul_cancel₀ _ (ne_of_gt htpos)] |
| _ ≤ (beta ^ 2 / Real.tanh 1 ^ 2) * |
| ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := |
| mul_le_mul_of_nonneg_left hlow (div_nonneg (sq_nonneg _) (le_of_lt htpos)) |
| |
| lemma variance_le_uniform_tanh_shift_integral |
| (hG : Measurable G) (hBound : ∀ᵐ w ∂μ, |G w| ≤ beta) (z : ℝ) : |
| variance G μ ≤ (Real.cosh (2 * beta + 1) ^ 2 + beta ^ 2 / Real.tanh 1 ^ 2) * |
| ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| have hnonneg : 0 ≤ ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := |
| integral_nonneg fun _ => sq_nonneg _ |
| by_cases hz : |z| ≤ beta + 1 |
| · exact (variance_le_cosh_sq_mul_tanh_shift_integral hG hBound hz).trans |
| (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (by positivity)) hnonneg) |
| · exact (variance_le_tanh_shift_integral_of_far hG hBound (lt_of_not_ge hz)).trans |
| (mul_le_mul_of_nonneg_right (le_add_of_nonneg_left (sq_nonneg _)) hnonneg) |
| |
| lemma exists_pos_uniform_shifted_tanh_variance (beta : ℝ) : |
| ∃ c : ℝ, 0 < c ∧ ∀ (G : Ω → ℝ), Measurable G → |
| (∀ᵐ w ∂μ, |G w| ≤ beta) → ∀ z : ℝ, |
| c * variance G μ ≤ ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| let C : ℝ := Real.cosh (2 * beta + 1) ^ 2 + beta ^ 2 / Real.tanh 1 ^ 2 |
| have hC : 0 < C := by |
| exact add_pos_of_pos_of_nonneg (sq_pos_of_pos (Real.cosh_pos _)) (by positivity) |
| refine ⟨C⁻¹, inv_pos.mpr hC, ?_⟩ |
| intro G hG hBound z |
| have h := variance_le_uniform_tanh_shift_integral hG hBound z |
| calc |
| C⁻¹ * variance G μ ≤ C⁻¹ * (C * ∫ w, Real.tanh (z + G w) ^ 2 ∂μ) := |
| mul_le_mul_of_nonneg_left h (le_of_lt (inv_pos.mpr hC)) |
| _ = ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| rw [← mul_assoc, inv_mul_cancel₀ (ne_of_gt hC), one_mul] |
| |
| lemma continuous_responseAttenuation {theta : ℝ} (hθ : theta ^ 2 ≤ 1) : |
| Continuous (responseAttenuation theta) := by |
| have ht := lipschitzWith_tanh_response.continuous |
| exact (continuous_const.sub (ht.pow 2)).div |
| (continuous_const.sub (continuous_const.mul (ht.pow 2))) |
| (fun z => ne_of_gt (attenuation_denominator_pos hθ z)) |
| |
| lemma integral_responseAttenuation_shift_sq_le {theta z : ℝ} |
| (hθ : theta ^ 2 ≤ 1) (hG : Measurable G) : |
| (∫ w, responseAttenuation theta (z + G w) ^ 2 ∂μ) ≤ |
| 1 - (1 - theta ^ 2) * ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| have hm : Measurable (fun w => responseAttenuation theta (z + G w) ^ 2) := |
| ((continuous_responseAttenuation hθ).measurable.comp (hG.const_add z)).pow_const 2 |
| have ha_le : ∀ w, responseAttenuation theta (z + G w) ^ 2 ≤ |
| responseAttenuation theta (z + G w) := by |
| intro w |
| have hp := responseAttenuation_pos hθ (z + G w) |
| have hle := responseAttenuation_le_one hθ (z + G w) |
| nlinarith |
| have hInt : Integrable (fun w => responseAttenuation theta (z + G w) ^ 2) μ := by |
| apply (integrable_const (1 : ℝ)).mono' hm.aestronglyMeasurable |
| filter_upwards [] with w |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact (ha_le w).trans (responseAttenuation_le_one hθ (z + G w)) |
| calc |
| (∫ w, responseAttenuation theta (z + G w) ^ 2 ∂μ) ≤ |
| ∫ w, 1 - (1 - theta ^ 2) * Real.tanh (z + G w) ^ 2 ∂μ := by |
| apply integral_mono hInt |
| ((integrable_const _).sub ((integrable_tanh_shift_sq hG z).const_mul _)) |
| intro w |
| exact (ha_le w).trans (responseAttenuation_le_one_sub hθ (z + G w)) |
| _ = 1 - (1 - theta ^ 2) * ∫ w, Real.tanh (z + G w) ^ 2 ∂μ := by |
| rw [integral_sub (integrable_const _) |
| ((integrable_tanh_shift_sq hG z).const_mul _), integral_const_mul] |
| simp |
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| lemma exists_pos_uniform_shifted_attenuation_loss {theta : ℝ} (hθ : theta ^ 2 < 1) |
| (beta : ℝ) : |
| ∃ c : ℝ, 0 < c ∧ ∀ (G : Ω → ℝ), Measurable G → |
| (∀ᵐ w ∂μ, |G w| ≤ beta) → ∀ z : ℝ, |
| (∫ w, responseAttenuation theta (z + G w) ^ 2 ∂μ) ≤ 1 - c * variance G μ := by |
| obtain ⟨c, hc, hcomp⟩ := exists_pos_uniform_shifted_tanh_variance (μ := μ) beta |
| refine ⟨(1 - theta ^ 2) * c, mul_pos (sub_pos.mpr hθ) hc, ?_⟩ |
| intro G hG hBound z |
| have h := mul_le_mul_of_nonneg_left (hcomp G hG hBound z) |
| (sub_nonneg.mpr (le_of_lt hθ)) |
| have hi := integral_responseAttenuation_shift_sq_le (μ := μ) (le_of_lt hθ) hG (z := z) |
| nlinarith |
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| end Probability |
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| end Problem367 |
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| end |
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| end OAI |
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