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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 | import OAI.Computability.DepthThree.CircuitNormalization
import OAI.Computability.DepthThree.CircuitDeletion
import Mathlib.Tactic.Linarith
namespace OAI
universe uDepth1
noncomputable section
open scoped BigOperators
open scoped Classical
namespace DepthThreeLowerBound
variable {V : Type uDepth1} [instFintypeV : Fintype V]
namespace CNF
theorem mass_le_of_width_correlation (H : CNF V) (hH : H.Normalized)
(f : Cube V → Bool) (hsupport : ∀ x, H.eval x = true → f x = true)
(k : ℕ) (ε : ℝ)
(hcorr : ∀ J : CNF V, J.WidthAtMost k →
|finiteAvg (fun x : Cube V => sign (f x) * indicator (J.eval x))| ≤ ε) :
finiteAvg (fun x : Cube V => indicator (H.eval x)) ≤
ε + (H.length : ℝ) * ((2 : ℝ) ^ k)⁻¹ := by
let J := H.deleteWide k
have hpoint (x : Cube V) : indicator (H.eval x) ≤
sign (f x) * indicator (J.eval x) +
(indicator (J.eval x) - indicator (H.eval x)) := by
have hm := H.indicator_eval_le_deleteWide k x
have hs := hsupport x
change indicator (H.eval x) ≤ indicator (J.eval x) at hm
cases hh : H.eval x <;> cases hj : J.eval x <;> cases hf : f x <;>
simp_all [indicator, sign]
calc
_ ≤ finiteAvg (fun x : Cube V => sign (f x) * indicator (J.eval x) +
(indicator (J.eval x) - indicator (H.eval x))) := finiteAvg_mono hpoint
_ = finiteAvg (fun x : Cube V => sign (f x) * indicator (J.eval x)) +
finiteAvg (fun x : Cube V => indicator (J.eval x) - indicator (H.eval x)) :=
finiteAvg_add _ _
_ ≤ |finiteAvg (fun x : Cube V => sign (f x) * indicator (J.eval x))| +
finiteAvg (fun x : Cube V => indicator (J.eval x) - indicator (H.eval x)) :=
add_le_add_left (le_abs_self _) _
_ ≤ ε + (H.length : ℝ) * ((2 : ℝ) ^ k)⁻¹ :=
add_le_add (hcorr J (H.deleteWide_widthAtMost k)) (H.deletion_error_bound hH k)
end CNF
namespace Circuit3
theorem indicator_output_le_sum_middle
{V : Type uDepth1}
[Fintype V]
(C : Circuit3 V) (x : Cube V) :
indicator (C.eval x) ≤ ∑ j ∈ C.top, indicator ((C.middleCNF j).eval x) := by
classical
cases h : C.eval x with
| false =>
simpa using Finset.sum_nonneg (fun j (_ : j ∈ C.top) =>
indicator_nonneg ((C.middleCNF j).eval x))
| true =>
obtain ⟨j, hj, hx⟩ := (C.eval_eq_true_iff_middleCNF x).mp h
have hb := Finset.single_le_sum
(fun i (_ : i ∈ C.top) => indicator_nonneg ((C.middleCNF i).eval x)) hj
simpa [hx] using hb
theorem acceptance_density_le (C : Circuit3 V) (f : Cube V → Bool)
(hC : C.Computes f) (k : ℕ) (ε : ℝ) (hε : 0 ≤ ε)
(hcorr : ∀ H : CNF V, H.WidthAtMost k →
|finiteAvg (fun x : Cube V => sign (f x) * indicator (H.eval x))| ≤ ε) :
finiteAvg (fun x : Cube V => indicator (f x)) ≤
(C.gateCount : ℝ) * ε + (C.gateCount : ℝ) ^ 2 * ((2 : ℝ) ^ k)⁻¹ := by
classical
let q : ℝ := ((2 : ℝ) ^ k)⁻¹
have hq : 0 ≤ q := inv_nonneg.mpr (pow_nonneg (by norm_num) _)
have hm (j : Fin C.middleCount) (hj : j ∈ C.top) :
finiteAvg (fun x : Cube V => indicator ((C.middleCNF j).eval x)) ≤
ε + (C.gateCount : ℝ) * q := by
have hs (x : Cube V) (hx : (C.middleCNF j).eval x = true) : f x = true := by
rw [← hC x]
exact C.middleCNF_implies_output hj hx
have hfirst := (C.middleCNF j).mass_le_of_width_correlation
(C.middleCNF_normalized j) f hs k ε hcorr
refine hfirst.trans (add_le_add_right (mul_le_mul_of_nonneg_right ?_ hq) ε)
exact_mod_cast C.middleCNF_length_le_gateCount j
have hB : 0 ≤ ε + (C.gateCount : ℝ) * q :=
add_nonneg hε (mul_nonneg (Nat.cast_nonneg _) hq)
calc
_ = finiteAvg (fun x : Cube V => indicator (C.eval x)) :=
finiteAvg_congr (fun x => congrArg indicator (hC x).symm)
_ ≤ finiteAvg (fun x : Cube V =>
∑ j ∈ C.top, indicator ((C.middleCNF j).eval x)) :=
finiteAvg_mono (C.indicator_output_le_sum_middle)
_ = ∑ j ∈ C.top, finiteAvg (fun x : Cube V =>
indicator ((C.middleCNF j).eval x)) := finiteAvg_sum _ _
_ ≤ ∑ j ∈ C.top, (ε + (C.gateCount : ℝ) * q) :=
Finset.sum_le_sum hm
_ = (C.top.card : ℝ) * (ε + (C.gateCount : ℝ) * q) := by
simp only [Finset.sum_const, nsmul_eq_mul]
_ ≤ (C.gateCount : ℝ) * (ε + (C.gateCount : ℝ) * q) := by
apply mul_le_mul_of_nonneg_right _ hB
exact_mod_cast C.top_card_le_gateCount
_ = _ := by dsimp [q]; ring
end Circuit3
end DepthThreeLowerBound
end
end OAI
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