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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.GradientStorageBits
import OAI.Combinatorics.EditApproximation.Geometry.BandOptimizerWork
import OAI.Combinatorics.EditApproximation.Numerics.OptimizerWorkspace
namespace OAI
/-! The probability vectors actually stored for band actions have polynomial
encodings in the input child masses. No accuracy event is needed. -/
namespace EditApproximation
open Finset
def bandVertexEncodingBudget (M V B : ℕ) : ℕ :=
B + (Nat.size (M * V) + 2 * (M * B + 1) + 1) + 3
theorem actionVector_encoding_le {M : ℕ} {σ : Type*} [DecidableEq σ]
(mass : Fin M → ℚ) (action : Fin M → σ) (coordinate : Fin M × σ)
(V B : ℕ) (habs : ∀ i, |mass i| ≤ V)
(hbits : ∀ i, rationalEncodingBits (mass i) ≤ B) :
rationalEncodingBits (actionVector mass action coordinate) ≤ bandVertexEncodingBudget M V B := by
have hsum := rational_finite_sum_encoding_le (univ : Finset (Fin M)) mass V B
(fun i _ => habs i) (fun i _ => (by
unfold rationalEncodingBits at hbits
have h := hbits i
omega))
simp only [Finset.card_univ, Fintype.card_fin] at hsum
unfold actionVector
split_ifs
· have hdiv := rationalEncodingBits_div (mass coordinate.1) (∑ i, mass i)
have hmass := hbits coordinate.1
unfold bandVertexEncodingBudget
omega
· norm_num [rationalEncodingBits, bandVertexEncodingBudget]
theorem actionVector_zero_one {M : ℕ} {σ : Type*} [DecidableEq σ]
(mass : Fin M → ℚ) (action : Fin M → σ) (coordinate : Fin M × σ)
(hpos : ∀ i, 0 < mass i) :
0 ≤ actionVector mass action coordinate ∧ actionVector mass action coordinate ≤ 1 := by
have hsum : 0 < ∑ i, mass i :=
(hpos coordinate.1).trans_le (Finset.single_le_sum (fun i _ => (hpos i).le) (mem_univ _))
unfold actionVector
split_ifs
· exact ⟨(div_pos (hpos _) hsum).le,
(div_le_one₀ hsum).2 (single_le_sum (fun i _ => (hpos i).le) (mem_univ _))⟩
· norm_num
def storedBandVertices {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (mass : Fin M → ℚ) :
List (Vector ℚ (listedBandCoordinates band).length) :=
band.map fun action => restrictOptimizerVector (listedBandCoordinateVector band) (actionVector mass action)
theorem storedBandVertices_nonempty {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (mass : Fin M → ℚ) (hband : band ≠ []) :
storedBandVertices band mass ≠ [] := by
intro hempty
exact hband (List.map_eq_nil_iff.mp hempty)
theorem storedBandVertices_input_bounds {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (mass : Fin M → ℚ) (V B : ℕ)
(hpos : ∀ i, 0 < mass i) (habs : ∀ i, |mass i| ≤ V)
(hbits : ∀ i, rationalEncodingBits (mass i) ≤ B)
(vertex : Vector ℚ (listedBandCoordinates band).length)
(hvertex : vertex ∈ storedBandVertices band mass)
(i : Fin (listedBandCoordinates band).length) :
(0 ≤ vertex.get i ∧ vertex.get i ≤ 1) ∧
Nat.size (vertex.get i).den ≤ bandVertexEncodingBudget M V B := by
obtain ⟨action, _, rfl⟩ := List.mem_map.mp hvertex
rw [restrictOptimizerVector_get]
constructor
· exact actionVector_zero_one mass action _ hpos
· have h := actionVector_encoding_le mass action
((listedBandCoordinateVector band).get i) V B habs hbits
unfold rationalEncodingBits at h
omega
theorem storedBandOptimizer_history_bits {M : ℕ} {σ : Type*} [DecidableEq σ]
(band : List (Fin M → σ)) (hband : band ≠ []) (mass : Fin M → ℚ)
(linear : Fin M × σ → ℚ) (G K V B : ℕ)
(hpos : ∀ i, 0 < mass i) (habs : ∀ i, |mass i| ≤ V)
(hbits : ∀ i, rationalEncodingBits (mass i) ≤ B) :
rationalVectorListStorageBits (optimizerStoredHistory (storedBandVertices band mass)
(restrictOptimizerVector (listedBandCoordinateVector band) linear) G K) ≤
(K + 1) * ((listedBandCoordinates band).length *
(2 * ((K + 1) * (2 * Nat.size (K + 1) + 2 + bandVertexEncodingBudget M V B) + 1) + 1)) := by
exact optimizerStoredHistory_storage_le _ _ G K _ (storedBandVertices_nonempty band mass hband)
(fun vertex hvertex i => storedBandVertices_input_bounds band mass V B hpos habs hbits vertex hvertex i)
end EditApproximation
end OAI
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