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e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Numerics.VectorOptimizerWork
namespace OAI
/-! Representation bounds for the actual stored optimizer iterates. -/
namespace EditApproximation
open Finset
theorem finiteRationalOptimizer_coordinate_bounds {ι : Type*}
(coordinates : Finset ι) (vertices : List (ι → ℚ)) (linear : ι → ℚ)
(G j : ℕ) (hvertices : vertices ≠ []) (i : ι)
(hinput : ∀ vertex ∈ vertices, 0 ≤ vertex i ∧ vertex i ≤ 1) :
0 ≤ finiteRationalOptimizer coordinates vertices linear G j i ∧
finiteRationalOptimizer coordinates vertices linear G j i ≤ 1 := by
induction j with
| zero =>
cases vertices with
| nil => exact False.elim (hvertices rfl)
| cons vertex rest => exact hinput vertex List.mem_cons_self
| succ j ih =>
have hselected := (rationalLinearizationMaximizer_spec coordinates vertices
(rationalSmoothedGradient G linear (finiteRationalOptimizer coordinates vertices linear G j))
hvertices).1
have hbounds := hinput _ hselected
have hden : (0 : ℚ) < (j : ℚ) + 2 := by positivity
have hweight : (0 : ℚ) ≤ 2 / ((j : ℚ) + 2) := by positivity
have hweightOne : (2 : ℚ) / ((j : ℚ) + 2) ≤ 1 :=
(div_le_one₀ hden).2 (by have hj : (0 : ℚ) ≤ j := Nat.cast_nonneg _; linarith)
change 0 ≤ (1 - 2 / ((j : ℚ) + 2)) * _ + 2 / ((j : ℚ) + 2) * _ ∧
(1 - 2 / ((j : ℚ) + 2)) * _ + 2 / ((j : ℚ) + 2) * _ ≤ 1
have hcomplement : 0 ≤ 1 - (2 : ℚ) / ((j : ℚ) + 2) := sub_nonneg.mpr hweightOne
constructor
· exact add_nonneg (mul_nonneg hcomplement ih.1) (mul_nonneg hweight hbounds.1)
· calc
_ ≤ (1 - 2 / ((j : ℚ) + 2)) * 1 + (2 : ℚ) / ((j : ℚ) + 2) * 1 :=
add_le_add (mul_le_mul_of_nonneg_left ih.2 hcomplement)
(mul_le_mul_of_nonneg_left hbounds.2 hweight)
_ = 1 := by ring
theorem materializedOptimizerWithWork_denominator {d : ℕ}
(vertices : List (Vector ℚ d)) (linear : Vector ℚ d) (G K B : ℕ)
(hvertices : vertices ≠ []) (hK : 0 < K) (i : Fin d)
(hinput : ∀ vertex ∈ vertices, Nat.size (vertex.get i).den ≤ B) :
Nat.size ((materializedOptimizerWithWork vertices linear G K).1.get i).den ≤
K * (2 * Nat.size K + 2 + B) + 1 := by
rw [materializedOptimizerWithWork_value, finiteRationalOptimizer_eq_iterate]
apply rationalOptimizerIterate_denominator_bit_length _ _ K B hK i
intro j _
have hlist : vertices.map Vector.get ≠ [] := by
intro hempty
exact hvertices (List.map_eq_nil_iff.mp hempty)
have hmember := (rationalLinearizationMaximizer_spec univ (vertices.map Vector.get)
(rationalSmoothedGradient G linear.get
(finiteRationalOptimizer univ (vertices.map Vector.get) linear.get G j)) hlist).1
obtain ⟨vertex, hv, heq⟩ := List.mem_map.mp hmember
change Nat.size (finiteOptimizerSelection univ (vertices.map Vector.get) linear.get G j i).den ≤ B
unfold finiteOptimizerSelection
rw [← heq]
exact hinput vertex hv
theorem materializedOptimizerWithWork_encoding {d : ℕ}
(vertices : List (Vector ℚ d)) (linear : Vector ℚ d) (G K B : ℕ)
(hvertices : vertices ≠ []) (hK : 0 < K) (i : Fin d)
(hinput : ∀ vertex ∈ vertices,
(0 ≤ vertex.get i ∧ vertex.get i ≤ 1) ∧ Nat.size (vertex.get i).den ≤ B) :
rationalEncodingBits ((materializedOptimizerWithWork vertices linear G K).1.get i) ≤
2 * (K * (2 * Nat.size K + 2 + B) + 1) + 1 := by
have hlist : vertices.map Vector.get ≠ [] := by
intro hempty
exact hvertices (List.map_eq_nil_iff.mp hempty)
have hpoint := finiteRationalOptimizer_coordinate_bounds univ (vertices.map Vector.get)
linear.get G K hlist i (by
intro vertex hv
obtain ⟨stored, hs, rfl⟩ := List.mem_map.mp hv
exact (hinput stored hs).1)
rw [← materializedOptimizerWithWork_value vertices linear G K] at hpoint
have habs : |(materializedOptimizerWithWork vertices linear G K).1.get i| ≤ 1 := by
rw [abs_of_nonneg hpoint.1]
exact hpoint.2
have hnum := Nat.size_le_size (rational_numerator_bound _ 1 habs)
simp only [Nat.one_mul] at hnum
have hden := materializedOptimizerWithWork_denominator vertices linear G K B hvertices hK i
(fun vertex hv => (hinput vertex hv).2)
unfold rationalEncodingBits
omega
end EditApproximation
end OAI
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