| 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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|
