| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.OwnListEnergy |
| import OAI.Computability.VertexCover.Analysis.SelectedIntegralAbs |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
| open MeasureTheory |
|
|
| theorem canonicalGradient_update_integrable (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (j : J) (s : Fin (Φ.WeightDimension d) → ℝ) (k : Fin (Φ.WeightDimension d)) : |
| Integrable (fun other : Φ.BatchWeights J => |
| Φ.canonicalGradient seed J c0 A hA (Function.update other j s) j k) (Φ.batchLaw J) := by |
| apply Integrable.of_bound |
| (Φ.canonicalGradient_update_measurable seed J c0 A hA j s k).aestronglyMeasurable 1 |
| apply Filter.Eventually.of_forall |
| intro other |
| simpa only [Real.norm_eq_abs] using VertexCover.Threshold.coordinate_bound |
| (Φ.canonicalGradient_block_bound seed J c0 A hA (Function.update other j s) j) k |
|
|
| theorem ownList_jensen (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (j : J) (i : Φ.Query d) (s : Fin (Φ.WeightDimension d) → ℝ) (β : ℝ) : |
| (∑ a ∈ (Φ.ownList J frozen c0 A hA j i s β).toFinset, |
| |Φ.ownGradient J frozen c0 A hA j i s (i.slot a)|) ≤ |
| VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j i => |
| ∫ other, ∑ a ∈ (Φ.ownList J frozen c0 A hA j i s β).toFinset, |
| |Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA |
| (Function.update other j s) j (i.slot a)| ∂Φ.batchLaw J) := by |
| classical |
| exact VertexCover.Average.selected_integral_abs_le (Φ.batchLaw J) |
| (Φ.ownList J frozen c0 A hA j i s β).toFinset |
| (fun (hidden : Φ.ownFiber J frozen j i) other a => |
| Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA |
| (Function.update other j s) j (i.slot a)) |
| (fun hidden a => Φ.canonicalGradient_update_integrable _ J c0 A hA j s (i.slot a)) |
|
|
| theorem ownGradient_energy_le_averaged_mass (Φ : LabelCover) {d : ℕ} |
| (J : Finset (Fin d)) (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (j : J) (i : Φ.Query d) (s : Fin (Φ.WeightDimension d) → ℝ) |
| {β : ℝ} (hβ : 0 < β) : |
| (∑ k, (Φ.ownGradient J frozen c0 A hA j i s k)^2) ≤ β/2 + |
| VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j i => |
| ∫ other, ∑ a ∈ (Φ.ownList J frozen c0 A hA j i s β).toFinset, |
| |Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA |
| (Function.update other j s) j (i.slot a)| ∂Φ.batchLaw J) := by |
| exact (Φ.ownList_energy J frozen c0 A hA j i s hβ).trans |
| (add_le_add le_rfl (Φ.ownList_jensen J frozen c0 A hA j i s β)) |
|
|
| theorem ownFiber_tower (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (j : J) (F : Φ.Query d → Φ.HiddenSeeds J → ℝ) : |
| VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J => |
| VertexCover.finiteMean (fun hidden' : Φ.ownFiber J frozen j |
| (Φ.query (Φ.spliceSeeds J frozen hidden) j) => |
| F (Φ.query (Φ.spliceSeeds J frozen hidden) j) hidden')) = |
| VertexCover.finiteMean (fun hidden : Φ.HiddenSeeds J => |
| F (Φ.query (Φ.spliceSeeds J frozen hidden) j) hidden) := by |
| classical |
| exact VertexCover.Average.finiteMean_fiber_tower_of_mem |
| (fun hidden : Φ.HiddenSeeds J => Φ.query (Φ.spliceSeeds J frozen hidden) j) F |
| (Φ.ownFiber J frozen j) (fun i hidden => Φ.mem_ownFiber J frozen j i hidden) |
|
|
| end VertexCover.LabelCover |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
