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a2ee19a | 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 | 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
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