| 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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