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a2ee19a | 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import Mathlib
import OAI.Probability.Perceptron.Model
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped Topology BigOperators
namespace SphericalPerceptronFreeEnergy
lemma exists_slow_diagonal {b : ℕ→ℕ→ℝ} (hb : ∀ n p, 0≤b n p)
(hlim : ∀ p, Tendsto (fun n => b n p) atTop (𝓝 0)) :
∃ J : ℕ→ℕ, Monotone J ∧ (∀ n, J n≤n) ∧ Tendsto J atTop atTop ∧
Tendsto (fun n => ∑ p∈Finset.range (J n), b n p) atTop (𝓝 0) := by
classical
let P := fun n k => ∀ m≥n, (∑ p∈Finset.range k, b m p)≤1/(k+1:ℝ)
have hP0 (n) : P n 0 := by intro m hm; simp
let J := fun n => Nat.findGreatest (P n) n
have hJ (n) : J n≤n := Nat.findGreatest_le _
have hJP (n) : P n (J n) := Nat.findGreatest_spec (Nat.zero_le n) (hP0 n)
have hJmono : Monotone J := by
intro m n hmn
exact Nat.findGreatest_mono (fun k hk l hl => hk l (hmn.trans hl)) hmn
have hPJ (k) : ∀ᶠ n in atTop, k≤J n := by
have hs : Tendsto (fun n => ∑ p∈Finset.range k, b n p) atTop (𝓝 (0:ℝ)) := by
simpa using tendsto_finsetSum (Finset.range k) (fun p _ => hlim p)
have he := hs.eventually (gt_mem_nhds (by positivity : (0:ℝ)<1/(k+1:ℝ)))
obtain ⟨N,hN⟩ := eventually_atTop.1 he
filter_upwards [eventually_ge_atTop (max N k)] with n hn
exact Nat.le_findGreatest ((le_max_right N k).trans hn)
(fun m hm => (hN m ((le_max_left N k).trans (hn.trans hm))).le)
have hJlim : Tendsto J atTop atTop := tendsto_atTop.2 hPJ
refine ⟨J,hJmono,hJ,hJlim,?_⟩
apply squeeze_zero (fun n => Finset.sum_nonneg (fun p _ => hb n p))
(fun n => hJP n n le_rfl)
have hh : Tendsto (fun n => (J n:ℝ)+1) atTop atTop :=
(tendsto_natCast_atTop_atTop.comp hJlim).atTop_add tendsto_const_nhds
simpa only [one_div,Function.comp_def] using tendsto_inv_atTop_zero.comp hh
variable {X : Type*} [MeasurableSpace X] (μ : Measure X)
(d : ℕ→ℕ→X→ℝ)
def diagonalDeviationMass (J : ℕ→ℕ) (n : ℕ) : ℝ :=
∑ p∈Finset.range (J n), ∫ v, d n p v ∂μ
def diagonalDeviationThreshold (J : ℕ→ℕ) (n : ℕ) : ℝ :=
Real.sqrt (diagonalDeviationMass μ d J n)+1/(n+1:ℝ)
def diagonalGoodSet (J : ℕ→ℕ) (n : ℕ) : Set X :=
{v | ∀ p∈Finset.range (J n), d n p v≤diagonalDeviationThreshold μ d J n}
lemma diagonalDeviationThreshold_pos (J : ℕ→ℕ) (n : ℕ) :
0<diagonalDeviationThreshold μ d J n := by
unfold diagonalDeviationThreshold
exact add_pos_of_nonneg_of_pos (Real.sqrt_nonneg _) (by positivity)
lemma diagonalGoodSet_measurable (hm : ∀ n p, Measurable (d n p)) (J : ℕ→ℕ) (n : ℕ) :
MeasurableSet (diagonalGoodSet μ d J n) := by
unfold diagonalGoodSet
simp only [ofPred_forall]
exact MeasurableSet.iInter fun p => MeasurableSet.iInter fun _ => measurableSet_le (hm n p) measurable_const
lemma diagonalGoodSet_compl_bound [IsProbabilityMeasure μ] (hn : ∀ n p v, 0≤d n p v)
(hi : ∀ n p, Integrable (d n p) μ) (J : ℕ→ℕ) (n : ℕ) :
μ.real (diagonalGoodSet μ d J n)ᶜ ≤ Real.sqrt (diagonalDeviationMass μ d J n) := by
let S := fun v => ∑ p∈Finset.range (J n), d n p v
have hSn (v) : 0≤S v := Finset.sum_nonneg fun p _ => hn n p v
have hSi : Integrable S μ := integrable_finsetSum _ (fun p _ => hi n p)
have hSm : ∫ v, S v ∂μ = diagonalDeviationMass μ d J n := integral_finsetSum _ (fun p _ => hi n p)
have ha : 0≤diagonalDeviationMass μ d J n := hSm ▸ integral_nonneg hSn
have hs : (diagonalGoodSet μ d J n)ᶜ ⊆ {v | diagonalDeviationThreshold μ d J n≤S v} := by
intro v hv
simp only [diagonalGoodSet,mem_compl_iff,mem_ofPred_eq,not_forall,not_le] at hv
obtain ⟨p,hp,hpv⟩ := hv
exact hpv.le.trans (Finset.single_le_sum (fun q _ => hn n q v) hp)
have hb := mul_meas_ge_le_integral_of_nonneg (ae_of_all μ hSn) hSi
(diagonalDeviationThreshold μ d J n)
rw [hSm] at hb
have he := diagonalDeviationThreshold_pos μ d J n
calc
_ ≤ μ.real {v | diagonalDeviationThreshold μ d J n≤S v} := measureReal_mono hs
_ ≤ diagonalDeviationMass μ d J n/diagonalDeviationThreshold μ d J n :=
(le_div_iff₀ he).2 (by simpa [mul_comm] using hb)
_ ≤ Real.sqrt (diagonalDeviationMass μ d J n) := by
apply (div_le_iff₀ he).2
dsimp [diagonalDeviationThreshold]
nlinarith [Real.sq_sqrt ha, mul_nonneg (Real.sqrt_nonneg (diagonalDeviationMass μ d J n))
(by positivity : (0:ℝ)≤1/(n+1:ℝ))]
lemma diagonalGoodSet_compl_tendsto [IsProbabilityMeasure μ] (hn : ∀ n p v, 0≤d n p v)
(hi : ∀ n p, Integrable (d n p) μ) (J : ℕ→ℕ)
(hJ : Tendsto (diagonalDeviationMass μ d J) atTop (𝓝 0)) :
Tendsto (fun n => μ.real (diagonalGoodSet μ d J n)ᶜ) atTop (𝓝 0) := by
apply squeeze_zero (fun n => measureReal_nonneg)
(diagonalGoodSet_compl_bound μ d hn hi J)
simpa only [Function.comp_def,Real.sqrt_zero] using Real.continuous_sqrt.continuousAt.tendsto.comp hJ
lemma diagonalDeviationThreshold_tendsto (J : ℕ→ℕ)
(hJ : Tendsto (diagonalDeviationMass μ d J) atTop (𝓝 0)) :
Tendsto (diagonalDeviationThreshold μ d J) atTop (𝓝 0) := by
have h1 : Tendsto (fun n : ℕ => 1/(n+1:ℝ)) atTop (𝓝 0) := by
have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop :=
(tendsto_natCast_atTop_atTop (R:=ℝ)).atTop_add (tendsto_const_nhds (x:=(1:ℝ)))
simpa only [one_div,Function.comp_def] using tendsto_inv_atTop_zero.comp hh
change Tendsto (fun n => Real.sqrt (diagonalDeviationMass μ d J n)+1/(n+1:ℝ)) atTop (𝓝 0)
simpa only [Function.comp_def,Real.sqrt_zero,zero_add] using
(Real.continuous_sqrt.continuousAt.tendsto.comp hJ).add h1
lemma diagonalGoodSet_deviation_tendsto (hn : ∀ n p v, 0≤d n p v) (J : ℕ→ℕ)
(hJ : Tendsto J atTop atTop) (hmass : Tendsto (diagonalDeviationMass μ d J) atTop (𝓝 0))
(v : ℕ→X) (hv : ∀ᶠ n in atTop, v n∈diagonalGoodSet μ d J n) (p : ℕ) :
Tendsto (fun n => d n p (v n)) atTop (𝓝 0) := by
apply squeeze_zero' (Eventually.of_forall fun n => hn n p (v n)) _
(diagonalDeviationThreshold_tendsto μ d J hmass)
filter_upwards [hv, hJ.eventually (eventually_gt_atTop p)] with n hn hnp
exact hn p (Finset.mem_range.2 hnp)
end SphericalPerceptronFreeEnergy
end
end OAI
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