| 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 |
|
|
