File size: 4,681 Bytes
135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.FiniteProduct
import OAI.Probability.SKBarriers.SpinGlass.FiniteIndicators
import OAI.Probability.SKBarriers.Dynamics.GreedyRecursion
import OAI.Probability.SKBarriers.Dynamics.GridThreshold
namespace OAI
section
noncomputable section
open scoped BigOperators
open Classical MeasureTheory Set
namespace SK.Analytic
def gridThresholds (n p : ℕ) (t ρ : ℝ) (z : Fin p → Fin n) (j : ℕ) : ℝ :=
if h : j<p then thresholdGrid n (t+4*j*ρ) ρ (z ⟨j,h⟩) else t+4*j*ρ
theorem gridThresholds_prefix (n p j : ℕ) (t ρ : ℝ) (z : Fin p → Fin n) (hj : j<p) :
∀ i<j, gridThresholds n p t ρ z i =
gridThresholds n j t ρ (fun a => z ⟨a.val,a.isLt.trans hj⟩) i := by
intro i hij
simp only [gridThresholds,dite_eq_left hij,dite_eq_left (hij.trans hj)]
theorem gridThresholds_good_mass {n : ℕ} (hn : 0<n) (μ : ProbabilityMeasure ℝ)
(p : ℕ) (t ρ : ℝ) (hρ : 0<ρ) :
((FiniteLaw.uniformFin n hn).iid (Fin p)).prob (fun z => ∃ i : Fin p,
ρ^4 < (μ:Measure ℝ).real (Icc (thresholdGrid n (t+4*i.val*ρ) ρ (z i)-2*ρ^8)
(thresholdGrid n (t+4*i.val*ρ) ρ (z i)+2*ρ^8))) ≤
p*((4*ρ^8/ρ+1/(n:ℝ))/ρ^4) := by
classical
let P := FiniteLaw.uniformFin n hn
refine (FiniteLaw.prob_exists _ _).trans ?_
calc
_ ≤ ∑ i : Fin p, (4*ρ^8/ρ+1/(n:ℝ))/ρ^4 := by
apply Finset.sum_le_sum
intro i _
rw [P.iid_prob_single i (fun u => ρ^4 < (μ:Measure ℝ).real
(Icc (thresholdGrid n (t+4*i.val*ρ) ρ u-2*ρ^8) (thresholdGrid n (t+4*i.val*ρ) ρ u+2*ρ^8)))]
have HM := P.markov (fun u => (μ:Measure ℝ).real
(Icc (thresholdGrid n (t+4*i.val*ρ) ρ u-2*ρ^8) (thresholdGrid n (t+4*i.val*ρ) ρ u+2*ρ^8)))
(fun _ => measureReal_nonneg) (ρ^4) (pow_pos hρ _)
have HE := thresholdGrid_mass_expect hn μ hρ (by positivity : 0≤2*ρ^8) (t+4*i.val*ρ)
apply (le_div_iff₀ (pow_pos hρ 4)).mpr
have HR : 2*(2*ρ^8)/ρ=4*ρ^8/ρ := by ring
rw [HR] at HE
simpa only [mul_comm] using HM.trans HE
_ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
theorem prod_block_marker (B s : ℕ) (hB : 0<B) (γ : ℝ) :
(∏ i : Fin (B*s), if i.val%B=0 then γ else 1)=γ^s := by
classical
let : NeZero B := ⟨hB.ne'⟩
have hfirst : (∏ i : Fin B, if i.val%B=0 then γ else 1)=γ := by
have he (i : Fin B) : i.val%B=0 ↔ i=0 := by rw [Nat.mod_eq_of_lt i.isLt]; exact (Fin.ext_iff (a := i) (b := (0:Fin B))).symm
simp only [he]
simp
induction s with
| zero => simp
| succ s ih =>
rw [Nat.mul_succ,Fin.prod_univ_add]
simp only [Fin.val_castAdd,Fin.val_natAdd]
rw [show (∏ i : Fin B, if ((B*s)+i.val)%B=0 then γ else 1)=γ from by
simpa only [Nat.add_mod,Nat.mul_mod_right,zero_add,Nat.mod_mod] using hfirst]
rw [ih,pow_succ]
theorem greedy_grid_windows_bound {n : ℕ} (hn : 0<n) (base : ℕ → Config n)
(pick : ℕ → (ℕ → Config n) → ℕ → Config n) (x : ℕ → Config n)
(B s m : ℕ) (hB : 0<B) (t ρ b w : ℝ) (hρ : 0<ρ) (hw : 0≤w) (k : ℕ) :
((FiniteLaw.uniformFin n hn).iid (Fin (B*s))).prob (fun z => ∀ i : Fin (B*s), i.val%B=0 →
|overlap (greedyReference base pick x (gridThresholds n (B*s) t ρ z) b m i.val) (x k)-
thresholdGrid n (t+4*i.val*ρ) ρ (z i)|≤w) ≤ (2*w/ρ+1/(n:ℝ))^s := by
classical
let P := FiniteLaw.uniformFin n hn
let γ := 2*w/ρ+1/(n:ℝ)
let E (j : ℕ) (z : Fin j → Fin n) (u : Fin n) := j%B=0 →
|overlap (greedyReference base pick x (gridThresholds n j t ρ z) b m j) (x k)-
thresholdGrid n (t+4*j*ρ) ρ u|≤w
have hE (j : ℕ) (z : Fin j → Fin n) : P.prob (E j z) ≤ if j%B=0 then γ else 1 := by
by_cases hj : j%B=0
· rw [ite_eq_left hj]
have H := thresholdGrid_window hn hρ hw (t+4*j*ρ)
(overlap (greedyReference base pick x (gridThresholds n j t ρ z) b m j) (x k))
simpa only [E,hj,true_implies,abs_sub_comm] using H
· exact (P.prob_le_one _).trans_eq (ite_eq_right hj).symm
have H := P.iid_predictable_bound E (fun j => if j%B=0 then γ else 1)
(fun j => by dsimp only [γ]; split_ifs <;> positivity) hE (B*s)
rw [prod_block_marker B s hB] at H
refine (FiniteLaw.prob_mono _ (fun z hz i hi => ?_)).trans H
have hr := gridThresholds_prefix n (B*s) i.val t ρ z i.isLt
have he := greedyReference_predictable base pick x (gridThresholds n (B*s) t ρ z)
(gridThresholds n i.val t ρ (fun a => z ⟨a.val,a.isLt.trans i.isLt⟩)) b m i.val hr
rw [← he]
exact hz i hi
end SK.Analytic
end
end
end OAI
|