| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.KernelCaps |
| import OAI.Probability.MemoryPrecision.KernelIntegral |
| import OAI.Probability.MemoryPrecision.KernelVersion |
| import OAI.Probability.MemoryPrecision.SeedAverage |
| import OAI.Probability.MemoryPrecision.StochasticMass |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
|
|
| noncomputable section |
| universe u v |
|
|
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision.Kernel |
| open MemoryPrecision NoiselessRegression |
|
|
| def CubePrior {d : ℕ} (hd : 2 ≤ d) : Measure (MemoryPrecision.Kernel.Sphere d) := |
| (CubeLaw d).map (CubeSphere hd) |
| instance {d : ℕ} (hd : 2 ≤ d) : IsProbabilityMeasure (CubePrior hd) := by |
| unfold CubePrior |
| infer_instance |
| |
| lemma cubeSphere_preserving {d : ℕ} (hd : 2 ≤ d) : |
| MeasurePreserving (CubeSphere hd) (CubeLaw d) (CubePrior hd) := |
| ⟨cubeSphere_measurable hd, rfl⟩ |
| |
| namespace Policy |
| variable {d M T : ℕ} (K : Policy d M T) |
| |
| lemma eval_comp {X Y : Type*} (w : ℕ → MemoryPrecision.Kernel.State M → X → ℝ≥0∞) (ι : Y → X) |
| (n k : ℕ) (j : MemoryPrecision.Kernel.State M) (s : Y) (obs : Observations d k) : |
| K.eval w n k j (ι s) obs = K.eval (fun n j s => w n j (ι s)) n k j s obs := by |
| induction k generalizing n j with |
| | zero => rfl |
| | succ k ih => |
| simp only [eval] |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [ih] |
| lemma pathValue_comp {X Y : Type*} (w : ℕ → MemoryPrecision.Kernel.State M → X → ℝ≥0∞) (ι : Y → X) |
| (s : Y) (obs : Observations d T) : |
| K.pathValue w (ι s) obs = K.pathValue (fun n j s => w n j (ι s)) s obs := by |
| simp only [pathValue, K.eval_comp w ι] |
| |
| lemma borel_cube_bound (hK : K.Borel) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) : |
| (∫⁻ z : Cube d × Rows d T, |
| K.pathValue (K.outputTest (chordTest ε)) (CubeSphere (d := d) (by omega) z.1) |
| (records (CubeSphere (d := d) (by omega) z.1) z.2) |
| ∂(CubeLaw d).prod (RowLaw d T)) ≤ |
| (T+1:ℝ≥0∞)*ENNReal.ofReal ((12:ℝ)^(d/2)*ε^((d/2:ℕ)/4:ℝ) * |
| CubeStep d (2^M)^MemoryPrecision.BlockCount T ((d/2)/4)) := by |
| have hm₁ : Measurable (fun z : Cube d × Rows d T => |
| (CubeSphere (d := d) (by omega) z.1,z.2)) := |
| ((cubeSphere_measurable (d := d) (by omega)).comp measurable_fst).prodMk measurable_snd |
| have hm₀ := K.experiment_fixed_measurable (f := chordTest (d := d) ε) hK (measurable_chordTest ε) |
| have hm := hm₀.comp hm₁ |
| have hprod := lintegral_prod (μ := CubeLaw d) (ν := RowLaw d T) _ hm.aemeasurable |
| simp only [Function.comp_apply] at hprod |
| rw [hprod] |
| simp_rw [K.pathValue_comp _ (CubeSphere (d := d) (by omega))] |
| apply K.integral_pathValue_bound (fun n j u => K.outputTest (chordTest ε) n j (CubeSphere (d := d) (by omega) u)) |
| (CubeEmbedding d) hK |
| (fun n j => (K.outputTest_measurable (measurable_chordTest ε) n j).comp (cubeSphere_measurable (d := d) (by omega))) |
| (cubeEmbedding_measurable d) (CubeLaw d) |
| intro r hr j |
| apply cube_mass_bound |
| exact stoch_cube_mass hd (show 1 ≤ 2^M from one_le_pow₀ (by norm_num)) |
| (K.marginal_measurable hK) (fun n j obs => (K.marginal_sum n j obs).le) |
| (fun n j => (K.outputTest_measurable (measurable_chordTest ε) n j).comp (cubeSphere_measurable (d := d) (by omega))) |
| (fun n j u => K.outputTest_le_one (chordTest_le_one ε) n j (CubeSphere (d := d) (by omega) u)) |
| (by positivity) (K.output_chord_cube_local (by omega) hε) |
| (MemoryPrecision.BlockCount T ((d/2)/4)) r |
| (hr.trans (blockCount_mul_ge T (by omega))) 0 j |
| |
| lemma completed_cube_bound (hK : K.Completed) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) : |
| (∫⁻ z : Cube d × Rows d T, |
| K.pathValue (K.outputTest (chordTest ε)) (CubeSphere (d := d) (by omega) z.1) |
| (records (CubeSphere (d := d) (by omega) z.1) z.2) |
| ∂(CubeLaw d).prod (RowLaw d T)) ≤ |
| (T+1:ℝ≥0∞)*ENNReal.ofReal ((12:ℝ)^(d/2)*ε^((d/2:ℕ)/4:ℝ) * |
| CubeStep d (2^M)^MemoryPrecision.BlockCount T ((d/2)/4)) := by |
| have he := K.version_pathValue_ae ((CubeLaw d).prod (RowLaw d T)) hK |
| (fun z : Cube d × Rows d T => CubeSphere (d := d) (by omega) z.1) |
| (fun z : Cube d × Rows d T => records (CubeSphere (d := d) (by omega) z.1) z.2) |
| (fun i => show Measure.QuasiMeasurePreserving _ _ _ from cube_sample_observation_quasi hd i) |
| (chordTest ε) |
| rw [← lintegral_congr_ae he] |
| exact (K.version hK).borel_cube_bound (K.version_borel hK) hd hε |
| |
| lemma source_cube_chord_bound {Ω : Type u} [MeasurableSpace Ω] (L : Ω → Policy d M T) |
| (ρ : Measure Ω) [IsProbabilityMeasure ρ] (hd : 16 ≤ d) |
| (hL : Admissible L ρ (CubePrior (by omega))) {ε : ℝ} (hε : 0 < ε) : |
| ChordSuccess L ρ (CubePrior (by omega)) ε ≤ |
| (T+1:ℝ≥0∞)*ENNReal.ofReal ((12:ℝ)^(d/2)*ε^((d/2:ℕ)/4:ℝ) * |
| CubeStep d (2^M)^MemoryPrecision.BlockCount T ((d/2)/4)) := by |
| have hp := (cubeSphere_preserving (by omega : 2 ≤ d)).prod |
| (MeasurePreserving.id (ρ.prod (RowLaw d T))) |
| have hf := hL.experiment (chordTest ε) (measurable_chordTest ε) (chordTest_le_one ε) |
| have he : (∫⁻ z : Cube d × (Ω × Rows d T), |
| experiment L (chordTest ε) (CubeSphere (d := d) (by omega) z.1,z.2) |
| ∂(CubeLaw d).prod (ρ.prod (RowLaw d T))) = ChordSuccess L ρ (CubePrior (by omega)) ε := by |
| calc |
| _ = ∫⁻ z, experiment L (chordTest ε) z ∂((CubeLaw d).prod (ρ.prod (RowLaw d T))).map |
| (Prod.map (CubeSphere (d := d) (by omega)) id) := |
| (lintegral_map' (by simpa only [hp.map_eq] using hf) hp.measurable.aemeasurable).symm |
| _ = _ := by rw [hp.map_eq]; rfl |
| rw [← he] |
| apply seed_average_bound (CubeLaw d) (RowLaw d T) ρ |
| (hf.comp_quasiMeasurePreserving hp.quasiMeasurePreserving) |
| filter_upwards [hL.completed] with ω hω |
| exact (L ω).completed_cube_bound hω hd hε |
| end Policy |
| end MemoryPrecision.Kernel |
| |
| |
| end |
| |
| end OAI |
| |
