| import OAI.Probability.InvariantIsing.Pressure.RandomCoefficientGG |
| import OAI.Probability.InvariantIsing.Arrays.OverlapBounds |
| import OAI.Probability.IsingPerceptron.MonomialGgAtMinimum |
| import OAI.Probability.IsingPerceptron.BoundedTiltAverage |
|
|
| namespace OAI |
|
|
| /-! The deterministic spectral-diagonal perturbation estimate at a parameter minimum. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
|
|
| namespace InvariantIsing |
|
|
| lemma bounded_nonnegative_cgf_bounds {X : Type*} [MeasurableSpace X] |
| (ν : Measure X) [IsProbabilityMeasure ν] (Y : X → ℝ) (hY : Measurable Y) |
| (hbound : ∀ x, Y x ∈ Set.Icc (0 : ℝ) 1) {t : ℝ} (ht : 0 ≤ t) : |
| 0 ≤ cgf Y ν t ∧ cgf Y ν t ≤ t := by |
| have he : Integrable (fun x => Real.exp (t * Y x)) ν := |
| exp_mul_integrable_of_bound ν hY |
| (fun x => by rw [abs_of_nonneg (hbound x).1]; exact (hbound x).2) t |
| have hlo : (1 : ℝ) ≤ ∫ x, Real.exp (t * Y x) ∂ν := by |
| calc |
| _ = ∫ _ : X, (1 : ℝ) ∂ν := by simp |
| _ ≤ _ := integral_mono (integrable_const _) he (fun x => |
| Real.one_le_exp_iff.mpr (mul_nonneg ht (hbound x).1)) |
| have hhi : (∫ x, Real.exp (t * Y x) ∂ν) ≤ Real.exp t := by |
| calc |
| _ ≤ ∫ _ : X, Real.exp t ∂ν := integral_mono he (integrable_const _) |
| (fun x => Real.exp_le_exp.mpr ((mul_le_mul_of_nonneg_left (hbound x).2 ht).trans_eq (mul_one _))) |
| _ = _ := by simp |
| constructor |
| · exact Real.log_nonneg hlo |
| · change Real.log (∫ x, Real.exp (t * Y x) ∂ν) ≤ t |
| exact (Real.log_le_log (by linarith) hhi).trans_eq (Real.log_exp t) |
|
|
| lemma integrable_bounded_random_cgf {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| (P : Measure Ω) [IsProbabilityMeasure P] (ν : Ω → Measure X) (hν : Measurable ν) |
| [∀ ω, IsProbabilityMeasure (ν ω)] (Y : Ω × X → ℝ) (hY : Measurable Y) |
| (B : ℝ) (hB : ∀ ω x, |Y (ω, x)| ≤ B) (t : ℝ) : |
| Integrable (fun ω => cgf (fun x => Y (ω, x)) (ν ω) t) P := by |
| have hm : Measurable (fun ω => cgf (fun x => Y (ω, x)) (ν ω) t) := by |
| simpa only [cgf, mgf, referencePartition] using |
| (measurable_random_referencePartition hν (hY.const_mul t)).log |
| apply Integrable.of_bound hm.aestronglyMeasurable (|t| * B) |
| exact ae_of_all _ fun ω => by |
| have hYm : Measurable (fun x => t * Y (ω, x)) := |
| (hY.comp (measurable_const.prodMk measurable_id)).const_mul t |
| have hb : ∀ x, |t * Y (ω, x)| ≤ |t| * B := fun x => by |
| rw [abs_mul] |
| exact mul_le_mul_of_nonneg_left (hB ω x) (abs_nonneg _) |
| have h := logMean_bounds (ν ω) hYm (b := 1) (by norm_num) hb |
| simpa only [cgf, mgf, logMean, one_mul, div_one, Real.norm_eq_abs] using h |
|
|
| /-- Unlike the Gaussian perturbation, a diagonal coordinate is a bounded |
| observable of the random rotation. Its exact tilted fluctuation still follows |
| from convexity, the deterministic minimum, and three pointwise concentration |
| bounds. -/ |
| theorem boundedObservable_energy_at_minimum {Ω X : Type*} |
| [MeasurableSpace Ω] [MeasurableSpace X] |
| (P : Measure Ω) [IsProbabilityMeasure P] (ν : Ω → Measure X) (hν : Measurable ν) |
| [∀ ω, IsProbabilityMeasure (ν ω)] (Y : Ω × X → ℝ) (hY : Measurable Y) |
| (B : ℝ) (hB : ∀ ω x, |Y (ω, x)| ≤ B) |
| {v c a s δ K : ℝ} (hs : 0 < s) (hK : 0 < K) |
| (hmin : ∀ t ∈ Set.Icc (v - s) (v + s), |
| -(∫ ω, cgf (fun x => Y (ω, x)) (ν ω) v ∂P) + c * (v - a) ^ 2 ≤ |
| -(∫ ω, cgf (fun x => Y (ω, x)) (ν ω) t ∂P) + c * (t - a) ^ 2) |
| (hconc : ∀ t ∈ ({v - s, v, v + s} : Set ℝ), |
| (∫ ω, |cgf (fun x => Y (ω, x)) (ν ω) t - |
| (∫ ω', cgf (fun x => Y (ω', x)) (ν ω') t ∂P)| ∂P) ≤ δ) : |
| let E := fun ω => ∫ x, |Y (ω, x) - 2 * c * (v - a)| |
| ∂(ν ω).tilted (fun x => v * Y (ω, x)) |
| Integrable E P ∧ (∫ ω, E ω ∂P) ≤ (2 * c + K ^ 2) / (2 * K) + c * s + 4 * δ / s := by |
| apply expected_tilted_energy_at_minimum hν hY hs hK |
| · exact ae_of_all _ fun ω t => exp_mul_integrable_of_bound (ν ω) |
| (hY.comp (measurable_const.prodMk measurable_id)) (hB ω) t |
| · exact integrable_bounded_random_cgf P ν hν Y hY B hB |
| · exact hmin |
| · exact hconc |
| |
| end InvariantIsing |
| |
| end |
| |
| end OAI |
| |
