File size: 2,381 Bytes
c4cdbbd | 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 | 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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