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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Brownian.IntervalMesh
import OAI.Probability.SKGap.Localization.ChainingCore
namespace OAI
section
noncomputable section
namespace SKGap
open MeasureTheory ProbabilityTheory Real Set Filter
open scoped BigOperators ENNReal Topology
variable {α : Type*} [Fintype α] [DecidableEq α]
def boxMeshEdges (k : ℕ) : Finset ((α→Fin (4^k+1))×(α→Fin (4^(k+1)+1))) :=
Finset.univ.filter (fun e=>dist (boxMeshPoint (k+1) e.2) (boxMeshPoint k e.1)≤2*(1/4:ℝ)^k)
lemma boxMesh_base_exp : (Fintype.card (α→Fin (4^0+1)):ℝ)≤exp (log 16*(Fintype.card α:ℝ)) := by
have hh : (Fintype.card (α→Fin (4^0+1)):ℝ)≤(16:ℝ)^(Fintype.card α) := by
simp only [Fintype.card_fun,Fintype.card_fin,pow_zero];norm_num
exact pow_le_pow_left₀ (by norm_num) (by norm_num) _
apply hh.trans_eq
rw [←exp_log (by norm_num : (0:ℝ)<16),←exp_nat_mul]
simp only [log_exp]
congr 1;ring
theorem box_chaining_probability {Ω E : Type*} [MeasurableSpace Ω]
[AddCommGroup E] [TopologicalSpace E] (μ : Measure Ω)
(p : E→ℝ) (hc : Continuous p) (hp : ∀ x y,p (x+y)≤p x+p y)
(F : (α→Icc (0:ℝ) 1)→Ω→E) (hF : ∀ ω,Continuous (fun t=>F t ω))
{n : ℕ} (hn : 1≤n) {A B : ℝ} (hB : 0≤B) (hA : B+1≤A)
(hdim : log 16*(Fintype.card α:ℝ)≤B*(n:ℝ))
(b : ℝ) (r : ℕ→ℝ) (hr : ∀ k,0≤r k) (hrs : Summable r)
(hb : ∀ θ,μ {ω | b<p (F θ ω)}≤ENNReal.ofReal (exp (-A*(n:ℝ))))
(hi : ∀ k θ η,dist θ η≤2*(1/4:ℝ)^k→
μ {ω | r k<p (F θ ω-F η ω)}≤ENNReal.ofReal (exp (-A*((k:ℝ)+1)^2*(n:ℝ)))) :
μ {ω | ∃ θ,b+(∑' k,r k)<p (F θ ω)}≤ENNReal.ofReal (3*exp (-(n:ℝ))) := by
classical
have hq := continuous_chaining_probability μ p hc hp F hF univ
(fun k=>α→Fin (4^k+1)) boxMeshPoint boxMeshEdges (by
intro θ _
refine ⟨boxMeshIndex θ,?_,boxMeshIndex_tendsto θ⟩
intro k
simpa only [boxMeshEdges,Finset.mem_filter,Finset.mem_univ,true_and] using boxMeshIndex_increment θ k)
r hr hrs b (ENNReal.ofReal (exp (-A*(n:ℝ))))
(fun k=>ENNReal.ofReal (exp (-A*((k:ℝ)+1)^2*(n:ℝ))))
(fun i=>hb (boxMeshPoint 0 i)) (by
intro k e he
exact hi k _ _ (Finset.mem_filter.mp he).2)
simp only [mem_univ,true_and] at hq
apply hq.trans
have hc1 : log 2≤(1:ℝ) := by
have hh := log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)
norm_num at hh ⊢;exact hh
have hbcard : (Fintype.card (α→Fin (4^0+1)):ℝ≥0∞)≤ENNReal.ofReal (exp (B*(n:ℝ))) := by
rw [←ENNReal.ofReal_natCast]
exact ENNReal.ofReal_le_ofReal (boxMesh_base_exp.trans (exp_le_exp.mpr hdim))
have hbcost := chaining_entropy_tail_product hn hB (by norm_num : (0:ℝ)<1) hA 0
(m := (Fintype.card (α→Fin (4^0+1)):ℝ≥0∞))
(q := ENNReal.ofReal (exp (-A*(n:ℝ))))
(by simpa only [Nat.cast_zero,zero_add,mul_one] using hbcard) (by simp)
have hecard (k : ℕ) : ((boxMeshEdges (α:=α) k).card:ℝ≥0∞)≤ENNReal.ofReal (exp (B*((k:ℝ)+1)*(n:ℝ))) := by
rw [←ENNReal.ofReal_natCast]
apply ENNReal.ofReal_le_ofReal
apply (boxMesh_edges_exp k _).trans
apply exp_le_exp.mpr
have hh := mul_le_mul_of_nonneg_right hdim (by positivity : 0≤(k:ℝ)+1)
nlinarith only [hh]
have hicost := chaining_entropy_series hn hB hc1 hA
(fun k=>((boxMeshEdges (α:=α) k).card:ℝ≥0∞))
(fun k=>ENNReal.ofReal (exp (-A*((k:ℝ)+1)^2*(n:ℝ)))) hecard (fun _=>le_rfl)
apply (add_le_add hbcost hicost).trans_eq
simp only [Nat.cast_zero,zero_add,mul_one,one_mul,neg_mul]
rw [←ENNReal.ofReal_add (exp_pos _).le (by positivity)]
congr 1;ring
end SKGap
end
end
end OAI
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