File size: 3,329 Bytes
8769036 | 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 | import OAI.NumberTheory.TwoPoint.Circuits.CircuitDyadicBudgets
namespace OAI
/-! Specialize the proved approximation construction to dyadic size
budgets and circuit depth twenty-two. -/
namespace TwoPointCorrelations
open scoped Classical
lemma braverman_approx_degree_bound {m j d : ℕ} (hm : m ≤ 2 ^ j) (hd : d ≤ 22) :
(bravermanSamples j * (Nat.log 2 m + 3)) ^ d ≤ bravermanBase j ^ 22 := by
have hbase : bravermanSamples j * (Nat.log 2 m + 3) ≤ bravermanBase j :=
Nat.mul_le_mul_left _ (Nat.add_le_add_right (nat_log_le_of_dyadic hm) 3)
exact (Nat.pow_le_pow_left hbase _).trans
(Nat.pow_le_pow_right (bravermanBase_pos j) hd)
lemma braverman_norm_power_bound {m j d : ℕ} (hm : m ≤ 2 ^ j) (hd : d ≤ 22) :
(2 * (bravermanSamples j * (Nat.log 2 m + 3)) + 2) ^ d ≤
(2 * bravermanBase j + 2) ^ 22 := by
have hbase : bravermanSamples j * (Nat.log 2 m + 3) ≤ bravermanBase j :=
Nat.mul_le_mul_left _ (Nat.add_le_add_right (nat_log_le_of_dyadic hm) 3)
exact (Nat.pow_le_pow_left (Nat.add_le_add_right (Nat.mul_le_mul_left 2 hbase) 2) _).trans
(Nat.pow_le_pow_right (by omega) hd)
lemma braverman_norm_base_bound {m j : ℕ} (hm : m ≤ 2 ^ j) :
2 * ((m : ℝ) + 1) ^ 2 ≤ (2 : ℝ) ^ (2 * j + 3) := by
have hmreal : (m : ℝ) ≤ (2 : ℝ) ^ j := by exact_mod_cast hm
have htwo : (1 : ℝ) ≤ 2 ^ j := one_le_pow₀ (by norm_num)
have hsum : (m : ℝ) + 1 ≤ 2 * (2 : ℝ) ^ j := by linarith only [hmreal, htwo]
calc
_ ≤ 2 * (2 * (2 : ℝ) ^ j) ^ 2 :=
mul_le_mul_of_nonneg_left
(pow_le_pow_left₀ (add_nonneg (Nat.cast_nonneg m) zero_le_one) hsum 2) (by norm_num)
_ = _ := by
rw [pow_add, show 2 * j = j * 2 by omega, pow_mul]
ring
theorem AC0Circuit.dyadic_approximation {n j : ℕ}
(ν : FiniteLaw (BooleanCube n)) (c : AC0Circuit n)
(hc : c.depth ≤ 22) (hm : c.size ≤ 2 ^ j) :
∃ (P : BooleanCube n → ℝ) (E : AC0Circuit n),
WalshDegreeLE P (bravermanBase j ^ 22) ∧
E.depth ≤ 89 ∧ E.size ≤ 2 ^ bravermanErrorExponent j ∧
ν.probability (fun x => E.eval x = true) ≤
(2 : ℝ) ^ j * (7 / 8 : ℝ) ^ bravermanSamples j ∧
(∀ x, E.eval x ≠ true → P x = c.indicator x) ∧
∀ x, |P x| ≤ (2 : ℝ) ^ bravermanNormExponent j := by
obtain ⟨P, E, hP, hd, hsize, hprob, he, hnorm⟩ :=
c.polynomial_error_approximation ν (bravermanSamples j) (bravermanSamples_pos j)
have hmreal : (c.size : ℝ) ≤ (2 : ℝ) ^ j := by exact_mod_cast hm
have h89 : E.depth ≤ 89 := hd.trans (Nat.add_le_add_right (Nat.mul_le_mul_left 4 hc) 1)
refine ⟨P, E, hP.mono (braverman_approx_degree_bound hm hc), h89,
hsize.trans (braverman_error_size_budget hm), ?_, he, ?_⟩
· exact hprob.trans (mul_le_mul_of_nonneg_right hmreal (by positivity))
· intro x
calc
|P x| ≤ _ := hnorm x
_ ≤ ((2 : ℝ) ^ (2 * j + 3)) ^
((2 * (bravermanSamples j * (Nat.log 2 c.size + 3)) + 2) ^ c.depth) :=
pow_le_pow_left₀ (by positivity) (braverman_norm_base_bound hm) _
_ ≤ ((2 : ℝ) ^ (2 * j + 3)) ^ ((2 * bravermanBase j + 2) ^ 22) :=
pow_le_pow_right₀ (one_le_pow₀ (by norm_num)) (braverman_norm_power_bound hm hc)
_ = _ := by rw [← pow_mul]; rfl
end TwoPointCorrelations
end OAI
|