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9dbc64e | 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 | import OAI.Computability.DepthThree.MomentBounds
import OAI.Computability.DepthThree.FiniteProbabilityBounds
import OAI.Computability.DepthThree.SparseTestCodes
import OAI.Computability.DepthThree.InterpolationSigns
namespace OAI
universe uDepth1 uDepth2 uDepth3 uDepth4 uDepth5 uDepth6 uDepth7 uDepth8 uDepth9 uDepth10
noncomputable section
open scoped BigOperators Classical
namespace DepthThreeLowerBound
theorem finiteProb_abs_avg_gt_le {Ω : Type uDepth1} {Z : Type uDepth2} [Fintype Ω] [Nonempty Ω]
[Fintype Z] [Nonempty Z] [DecidableEq Z]
(h : ℕ) (hh : 0 < h) (bits : Ω → Z → Bool)
(hdist : UniformBitsUpTo (2 * h) bits) (J : Z → Bool)
(δ : ℝ) (hδ : 0 < δ) :
finiteProb (fun ω => δ < |finiteAvg (fun z => sign (bits ω z) * indicator (J z))|) ≤
(((2 * h : ℕ) : ℝ) ^ (2 * h) * (Fintype.card Z : ℝ) ^ h) /
(δ * (Fintype.card Z : ℝ)) ^ (2 * h) := by
have hN : 0 < (Fintype.card Z : ℝ) := by exact_mod_cast Fintype.card_pos
have hnorm (ω : Ω) :
|finiteAvg (fun z => sign (bits ω z) * indicator (J z))| =
|∑ z, sign (bits ω z) * indicator (J z)| / (Fintype.card Z : ℝ) := by
rw [finiteAvg, abs_mul, abs_inv, abs_of_nonneg hN.le, div_eq_inv_mul]
have hevent :
finiteProb (fun ω => δ < |finiteAvg (fun z => sign (bits ω z) * indicator (J z))|) ≤
finiteProb (fun ω => δ * (Fintype.card Z : ℝ) <
|∑ z, sign (bits ω z) * indicator (J z)|) := by
apply finiteProb_mono
intro ω hω
rw [hnorm ω] at hω
exact (lt_div_iff₀ hN).mp hω
exact hevent.trans (finiteProb_abs_gt_le_moment
(fun ω => ∑ z, sign (bits ω z) * indicator (J z))
(2 * h) (Nat.mul_pos (by decide) hh)
(δ * (Fintype.card Z : ℝ))
(((2 * h : ℕ) : ℝ) ^ (2 * h) * (Fintype.card Z : ℝ) ^ h)
(mul_pos hδ hN) (even_moment_le_manuscript h bits hdist J))
def sparseMomentBound (m b L h : ℕ) (δ : ℝ) : ℝ :=
((((2 * m + 1) ^ b + 1) ^ L : ℕ) : ℝ) *
((((2 * h : ℕ) : ℝ) ^ (2 * h) * ((2 ^ m : ℕ) : ℝ) ^ h) /
(δ * ((2 ^ m : ℕ) : ℝ)) ^ (2 * h))
theorem finiteProb_sparse_codes_gt_le {Ω : Type uDepth3} {V : Type uDepth4} [Fintype Ω] [Nonempty Ω]
[Fintype V] (b L h : ℕ) (hh : 0 < h)
(bits : Ω → Cube V → Bool) (hdist : UniformBitsUpTo (2 * h) bits)
(δ : ℝ) (hδ : 0 < δ) :
finiteProb (fun ω => ∃ c : SparseTestCode V b L,
δ < |finiteAvg (fun x => sign (bits ω x) * indicator (c.eval x))|) ≤
sparseMomentBound (Fintype.card V) b L h δ := by
let P : SparseTestCode V b L → Ω → Prop := fun c ω =>
δ < |finiteAvg (fun x => sign (bits ω x) * indicator (c.eval x))|
let B : ℝ :=
(((2 * h : ℕ) : ℝ) ^ (2 * h) * (Fintype.card (Cube V) : ℝ) ^ h) /
(δ * (Fintype.card (Cube V) : ℝ)) ^ (2 * h)
have hsingle (c : SparseTestCode V b L) : finiteProb (P c) ≤ B :=
finiteProb_abs_avg_gt_le h hh bits hdist c.eval δ hδ
calc
_ ≤ ∑ c : SparseTestCode V b L, finiteProb (P c) := by
simpa only [P, Finset.mem_univ, true_and] using
(finiteProb_exists_le_sum (Finset.univ : Finset (SparseTestCode V b L)) P)
_ ≤ ∑ _c : SparseTestCode V b L, B :=
Finset.sum_le_sum (fun c _ => hsingle c)
_ = (Fintype.card (SparseTestCode V b L) : ℝ) * B := by
simp [nsmul_eq_mul]
_ = sparseMomentBound (Fintype.card V) b L h δ := by
simp [B, sparseMomentBound, Cube,
Fintype.card_bool, mul_comm]
theorem finiteProb_sparse_cnfs_gt_le {Ω : Type uDepth5} {V : Type uDepth6} [Fintype Ω] [Nonempty Ω]
[Fintype V] (b L h : ℕ) (hh : 0 < h)
(bits : Ω → Cube V → Bool) (hdist : UniformBitsUpTo (2 * h) bits)
(δ : ℝ) (hδ : 0 < δ) :
finiteProb (fun ω => ∃ H : CNF V, H.WidthAtMost b ∧ H.length ≤ L ∧
δ < |finiteAvg (fun x => sign (bits ω x) * indicator (H.eval x))|) ≤
sparseMomentBound (Fintype.card V) b L h δ := by
apply le_trans _ (finiteProb_sparse_codes_gt_le b L h hh bits hdist δ hδ)
apply finiteProb_mono
rintro ω ⟨H, hw, hlen, hbad⟩
obtain ⟨c, hc⟩ := SparseTestCode.exists_eval_eq H hw hlen
refine ⟨c, ?_⟩
simpa only [hc] using hbad
theorem acceptEval_uniformBitsUpTo {F : Type uDepth7} {Z : Type uDepth8} [Field F]
[Module BinaryAlgebra.F2 F] [Fintype F] [DecidableEq Z]
(ell : F →ₗ[BinaryAlgebra.F2] BinaryAlgebra.F2) (hOne : ell 1 = 1)
(t : ℕ) (a : Z → F) (ha : Function.Injective a) :
UniformBitsUpTo t (Interpolation.acceptEval ell t a) := by
intro s hs Φ
exact Interpolation.finiteAvg_acceptEval_finset ell hOne a s hs
(fun _ _ _ _ hxy => ha hxy) Φ
theorem coefficient_sparse_cnfs_gt_le {F : Type uDepth9} {V : Type uDepth10} [Field F]
[Module BinaryAlgebra.F2 F] [Fintype F] [Fintype V]
(ell : F →ₗ[BinaryAlgebra.F2] BinaryAlgebra.F2) (hOne : ell 1 = 1)
(a : Cube V → F) (ha : Function.Injective a)
(b L h : ℕ) (hh : 0 < h) (δ : ℝ) (hδ : 0 < δ) :
finiteProb (fun β : Fin (2 * h) → F =>
∃ H : CNF V, H.WidthAtMost b ∧ H.length ≤ L ∧
δ < |finiteAvg (fun x =>
sign (Interpolation.acceptEval ell (2 * h) a β x) * indicator (H.eval x))|) ≤
sparseMomentBound (Fintype.card V) b L h δ := by
exact finiteProb_sparse_cnfs_gt_le b L h hh _
(acceptEval_uniformBitsUpTo ell hOne (2 * h) a ha) δ hδ
end DepthThreeLowerBound
end
end OAI
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