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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 | import OAI.Combinatorics.EditApproximation.Numerics.VectorOptimizerWork
import OAI.Combinatorics.EditApproximation.Geometry.BandDimension
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Reindexing the exact finite optimizer onto an explicit, duplicate-free
coordinate array. The array order is retained throughout the evaluation. -/
namespace EditApproximation
open Finset
def restrictOptimizerVector {ι : Type*} {d : ℕ} (coordinates : Vector ι d)
(values : ι → ℚ) : Vector ℚ d :=
Vector.ofFn fun i => values (coordinates.get i)
theorem restrictOptimizerVector_get {ι : Type*} {d : ℕ} (coordinates : Vector ι d)
(values : ι → ℚ) (i : Fin d) :
(restrictOptimizerVector coordinates values).get i = values (coordinates.get i) :=
vector_get_ofFn _ _
theorem coordinateVector_sum {ι : Type*} [DecidableEq ι] {d : ℕ}
(coordinates : Vector ι d) (hnodup : coordinates.toList.Nodup) (f : ι → ℚ) :
(∑ i : Fin d, f (coordinates.get i)) = ∑ c ∈ coordinates.toList.toFinset, f c := by
rw [List.sum_toFinset f hnodup]
have hcoordinates : coordinates.toList = List.ofFn coordinates.get := by
rw [← Vector.toList_ofFn]
congr 1
exact (Vector.ofFn_getElem).symm
rw [hcoordinates, List.map_ofFn, List.sum_ofFn]
rfl
theorem rationalLinearizationMaximizer_restrict {ι : Type*} [DecidableEq ι] {d : ℕ}
(coordinates : Vector ι d) (hnodup : coordinates.toList.Nodup)
(vertices : List (ι → ℚ)) (gradient : ι → ℚ) :
rationalLinearizationMaximizer univ
(vertices.map fun v => fun i => v (coordinates.get i))
(fun i => gradient (coordinates.get i)) =
fun i => rationalLinearizationMaximizer coordinates.toList.toFinset vertices gradient
(coordinates.get i) := by
have hsum (v : ι → ℚ) :
(∑ i : Fin d, gradient (coordinates.get i) * v (coordinates.get i)) =
∑ c ∈ coordinates.toList.toFinset, gradient c * v c :=
coordinateVector_sum coordinates hnodup (fun c => gradient c * v c)
unfold rationalLinearizationMaximizer
rw [argmax_map_commutes]
simp only [hsum]
cases h : vertices.argmax (fun v => ∑ c ∈ coordinates.toList.toFinset, gradient c * v c) with
| none => rfl
| some v => rfl
theorem finiteRationalOptimizer_restrict {ι : Type*} [DecidableEq ι] {d : ℕ}
(coordinates : Vector ι d) (hnodup : coordinates.toList.Nodup)
(vertices : List (ι → ℚ)) (linear : ι → ℚ) (G j : ℕ) :
finiteRationalOptimizer univ (vertices.map fun v => fun i => v (coordinates.get i))
(fun i => linear (coordinates.get i)) G j =
fun i => finiteRationalOptimizer coordinates.toList.toFinset vertices linear G j
(coordinates.get i) := by
induction j with
| zero => cases vertices <;> rfl
| succ j ih =>
have hgradient :
rationalSmoothedGradient G (fun i => linear (coordinates.get i))
(finiteRationalOptimizer univ (vertices.map fun v => fun i => v (coordinates.get i))
(fun i => linear (coordinates.get i)) G j) =
fun i => rationalSmoothedGradient G linear
(finiteRationalOptimizer coordinates.toList.toFinset vertices linear G j)
(coordinates.get i) := by
rw [ih]
rfl
funext i
rw [finiteRationalOptimizer, finiteRationalOptimizer, hgradient,
rationalLinearizationMaximizer_restrict coordinates hnodup]
simp only [ih]
theorem materializedOptimizerWithWork_restrict_value {ι : Type*} [DecidableEq ι] {d : ℕ}
(coordinates : Vector ι d) (hnodup : coordinates.toList.Nodup)
(vertices : List (ι → ℚ)) (linear : ι → ℚ) (G j : ℕ) (i : Fin d) :
(materializedOptimizerWithWork (vertices.map (restrictOptimizerVector coordinates))
(restrictOptimizerVector coordinates linear) G j).1.get i =
finiteRationalOptimizer coordinates.toList.toFinset vertices linear G j (coordinates.get i) := by
rw [materializedOptimizerWithWork_value]
have hvertices : (vertices.map (restrictOptimizerVector coordinates)).map Vector.get =
vertices.map (fun v => fun k => v (coordinates.get k)) := by
simp only [List.map_map]
congr 1
funext v k
exact restrictOptimizerVector_get coordinates v k
have hlinear : (restrictOptimizerVector coordinates linear).get =
fun k => linear (coordinates.get k) := by
funext k
exact restrictOptimizerVector_get coordinates linear k
rw [hvertices, hlinear, finiteRationalOptimizer_restrict coordinates hnodup]
/-- Concrete ordered coordinates; no enumeration of the ambient state
type and no choice of an equivalence is used by the evaluator. -/
theorem listedBandCoordinates_nodup {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) : (listedBandCoordinates band).Nodup :=
List.nodup_dedup _
theorem listedBandCoordinates_toFinset {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) :
(listedBandCoordinates band).toFinset = bandCoordinates band.toFinset := by
ext coordinate
constructor
· intro hc
obtain ⟨action, ha, hc⟩ := List.mem_flatMap.mp
(List.mem_dedup.mp (List.mem_toFinset.mp hc))
obtain ⟨child, hchild⟩ := List.mem_ofFn.mp hc
rw [← hchild]
exact action_coordinate_mem band.toFinset (List.mem_toFinset.mpr ha) child
· intro hc
obtain ⟨action, ha, hc⟩ := mem_biUnion.mp hc
obtain ⟨child, _, heq⟩ := mem_image.mp hc
apply List.mem_toFinset.mpr
apply List.mem_dedup.mpr
apply List.mem_flatMap.mpr
exact ⟨action, List.mem_toFinset.mp ha, List.mem_ofFn.mpr ⟨child, heq⟩⟩
def listedBandCoordinateVector {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) : Vector (Fin M × σ) (listedBandCoordinates band).length :=
Vector.ofFn (listedBandCoordinates band).get
theorem listedBandCoordinateVector_toList {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) :
(listedBandCoordinateVector band).toList = listedBandCoordinates band := by
simp only [listedBandCoordinateVector, Vector.toList_ofFn, List.ofFn_get]
theorem listedBandCoordinates_length {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) :
(listedBandCoordinates band).length = (bandCoordinates band.toFinset).card := by
rw [← listedBandCoordinates_toFinset band,
List.toFinset_card_of_nodup (listedBandCoordinates_nodup band)]
def materializedBandActionWithWork {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (mass : Fin M → ℚ)
(linear : Fin M × σ → ℚ) (G j : ℕ) : Option (Fin M → σ) × ℕ :=
let coordinates := listedBandCoordinateVector band
let vertices := band.map fun action =>
(action, restrictOptimizerVector coordinates (actionVector mass action))
materializedSelectionWithWork vertices (restrictOptimizerVector coordinates linear) G j
/-- Materializing coordinates and vertices preserves the actual action
label selected by the complete finite optimizer, including all ties. -/
theorem materializedBandActionWithWork_value {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (hband : band ≠ []) (mass : Fin M → ℚ)
(linear : Fin M × σ → ℚ) (G j : ℕ) :
(materializedBandActionWithWork band mass linear G j).1.getD (band.head hband) =
finiteOptimizerAction band hband mass linear G j := by
let coordinates := listedBandCoordinateVector band
have hnodup : coordinates.toList.Nodup := by
rw [show coordinates.toList = listedBandCoordinates band from
listedBandCoordinateVector_toList band]
exact listedBandCoordinates_nodup band
have hcoordinates : coordinates.toList.toFinset = bandCoordinates band.toFinset := by
rw [show coordinates.toList = listedBandCoordinates band from
listedBandCoordinateVector_toList band, listedBandCoordinates_toFinset]
have hvertexMaps :
(((band.map fun action => (action, restrictOptimizerVector coordinates (actionVector mass action))).map
Prod.snd).map Vector.get) =
(band.map (actionVector mass)).map (fun v => fun i => v (coordinates.get i)) := by
simp only [List.map_map]
congr 1
funext action i
exact restrictOptimizerVector_get coordinates _ i
have hlinear : (restrictOptimizerVector coordinates linear).get =
fun i => linear (coordinates.get i) := by
funext i
exact restrictOptimizerVector_get coordinates linear i
have hscore (action : Fin M → σ) :
(∑ i : Fin (listedBandCoordinates band).length,
rationalSmoothedGradient G (restrictOptimizerVector coordinates linear).get
(finiteRationalOptimizer univ
(((band.map fun action => (action, restrictOptimizerVector coordinates (actionVector mass action))).map
Prod.snd).map Vector.get) (restrictOptimizerVector coordinates linear).get G j) i *
(restrictOptimizerVector coordinates (actionVector mass action)).get i) =
∑ c ∈ bandCoordinates band.toFinset,
rationalSmoothedGradient G linear
(finiteRationalOptimizer (bandCoordinates band.toFinset) (band.map (actionVector mass))
linear G j) c * actionVector mass action c := by
rw [hvertexMaps, hlinear, finiteRationalOptimizer_restrict coordinates hnodup]
simp only [restrictOptimizerVector_get]
change (∑ i, rationalSmoothedGradient G linear
(finiteRationalOptimizer coordinates.toList.toFinset (band.map (actionVector mass)) linear G j)
(coordinates.get i) * actionVector mass action (coordinates.get i)) = _
have hs := coordinateVector_sum coordinates hnodup (fun c =>
rationalSmoothedGradient G linear
(finiteRationalOptimizer coordinates.toList.toFinset (band.map (actionVector mass)) linear G j) c *
actionVector mass action c)
simpa only [hcoordinates] using hs
unfold materializedBandActionWithWork
rw [materializedSelectionWithWork_value, argmax_map_commutes]
simp only [Function.comp_def, Option.map_map]
rw [show (fun x : Fin M → σ => x) = id from rfl, Option.map_id]
simp only [finiteOptimizerAction]
apply congrArg (fun result : Option (Fin M → σ) => result.getD (band.head hband))
apply congrArg (fun score : (Fin M → σ) → ℚ => band.argmax score)
funext action
exact hscore action
/-- Work of the optimizer after preparing its coordinate/vertex arrays.
The finite preparation loops are charged separately. -/
theorem materializedBandActionWithWork_optimizer_cost {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (mass : Fin M → ℚ)
(linear : Fin M × σ → ℚ) (G j : ℕ) (hj : j ≤ G ^ 80) :
(materializedBandActionWithWork band mass linear G j).2 ≤
(band.length + 1) * ((bandCoordinates band.toFinset).card + 1) * (G + 2) ^ 210 := by
unfold materializedBandActionWithWork
simpa only [List.length_map, listedBandCoordinates_length] using
materializedSelectionWithWork_polynomial
(band.map fun action => (action, restrictOptimizerVector
(listedBandCoordinateVector band) (actionVector mass action)))
(restrictOptimizerVector (listedBandCoordinateVector band) linear) G j hj
end EditApproximation
end OAI
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