File size: 1,849 Bytes
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 | import OAI.Computability.DepthThree.HardSliceEventual
import OAI.Computability.DepthThree.WideCorrelation
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace DepthThreeLowerBound
theorem exists_language_slice_small_correlations (s : ℝ) (hs : 0 ≤ s) :
∃ N : ℕ, 20480 ≤ N ∧ ∀ n : ℕ, N ≤ n →
∃ α : Fin n → Sum (Fin (dataDimension n)) Bool,
∀ H : CNF (Fin (dataDimension n)),
H.WidthAtMost (degreeCutoff s (dataDimension n)) →
|finiteAvg (fun x =>
sign (language (List.ofFn (GateInput.assignment α x))) * indicator (H.eval x))| ≤
6 * (2 : ℝ) ^ (-(64 * (s + 1) * Real.sqrt (dataDimension n : ℝ)) / 16) := by
let B : ℝ := 64 * (s + 1)
have hB : 0 < B := by dsimp [B]; linarith
obtain ⟨b, hb, D, hD, hparams⟩ := exists_final_parameter_width B s hB hs
obtain ⟨Ng, hNg, hgood⟩ := exists_good_language_slice_eventually b hb B hB
refine ⟨max Ng (5 * D), hNg.trans (le_max_left _ _), ?_⟩
intro n hn
have hng : Ng ≤ n := (le_max_left _ _).trans hn
have hnd : 5 * D ≤ n := (le_max_right _ _).trans hn
have hDd : D ≤ dataDimension n := by unfold dataDimension; omega
obtain ⟨hp0, hphalf, hθ⟩ := hparams (dataDimension n) hDd
have hp1 : liveProbability B (dataDimension n) < 1 := by linarith
have hθ' : restrictionTheta (degreeCutoff s (dataDimension n)) b
(liveProbability B (dataDimension n)) ≤ 1 / 2 := by
simpa only [restrictionTheta_eq_reverseTheta] using hθ
obtain ⟨α, hα⟩ := hgood n hng
refine ⟨α, ?_⟩
exact wide_correlation_of_good_restrictions hp0 hp1
(Real.rpow_nonneg (by norm_num) _) b (degreeCutoff s (dataDimension n)) hb hθ'
(fun x => language (List.ofFn (GateInput.assignment α x))) hα
end DepthThreeLowerBound
end
end OAI
|