| import OAI.Probability.InvariantIsing.Magnetic.RestrictedFieldOrder |
| import Mathlib.Analysis.Convex.Integral |
|
|
| namespace OAI |
|
|
| /-! One Gaussian backward step propagates a terminal loss by expectation |
| under its canonical tilted transition. No lower bound on its positive |
| exponent is required. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
|
|
| namespace InvariantIsing |
|
|
| lemma integral_le_log_integral_exp {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) [IsProbabilityMeasure μ] (F : X → ℝ) |
| (hF : Integrable F μ) (he : Integrable (fun x => Real.exp (F x)) μ) : |
| (∫ x, F x ∂μ) ≤ Real.log (∫ x, Real.exp (F x) ∂μ) := by |
| have hj := convexOn_exp.map_integral_le Real.continuous_exp.continuousOn |
| isClosed_univ (ae_of_all μ (fun _ => Set.mem_univ _)) hF he |
| have hl := Real.log_le_log (Real.exp_pos (∫ x, F x ∂μ)) hj |
| rwa [Real.log_exp] at hl |
|
|
| theorem logMean_loss_le_tilted_mean {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) [IsProbabilityMeasure μ] (F G : X → ℝ) |
| {b : ℝ} (hb : 0 < b) |
| (hF : Integrable (fun x => Real.exp (b * F x)) μ) |
| (hG : Integrable (fun x => Real.exp (b * G x)) μ) |
| (hD : Integrable (fun x => F x - G x) (μ.tilted (fun x => b * F x))) : |
| logMean b μ F - logMean b μ G ≤ |
| ∫ x, F x - G x ∂μ.tilted (fun x => b * F x) := by |
| let ν := μ.tilted (fun x => b * F x) |
| let : IsProbabilityMeasure ν := isProbabilityMeasure_tilted hF |
| have hE : Integrable (fun x => Real.exp (-b * (F x - G x))) ν := by |
| apply (integrable_tilted_iff hF _).mpr |
| convert hG using 1 |
| funext x |
| rw [smul_eq_mul, ← Real.exp_add] |
| congr 1 |
| ring |
| have hi := integral_le_log_integral_exp ν (fun x => -b * (F x - G x)) |
| (hD.const_mul (-b)) hE |
| have he : (∫ x, Real.exp (-b * (F x - G x)) ∂ν) = |
| (∫ x, Real.exp (b * G x) ∂μ) / ∫ x, Real.exp (b * F x) ∂μ := by |
| rw [integral_exp_tilted] |
| congr 2 |
| funext x |
| congr 1 |
| change b * F x + -b * (F x - G x) = b * G x |
| ring |
| rw [integral_const_mul, he, Real.log_div (integral_exp_pos μ G hG).ne' |
| (integral_exp_pos μ F hF).ne'] at hi |
| change Real.log (∫ x, Real.exp (b * F x) ∂μ) / b - |
| Real.log (∫ x, Real.exp (b * G x) ∂μ) / b ≤ _ |
| rw [← sub_div] |
| apply (div_le_iff₀ hb).mpr |
| dsimp only [ν] at hi |
| linarith |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
