| 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 |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision.Kernel |
| open MemoryPrecision NoiselessRegression |
| namespace Policy |
| variable {d M T : ℕ} (K : Policy d M T) |
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| lemma borel_sphere_bound (hK : K.Borel) (hd : 16 ≤ d) {ε α : ℝ} (hε : 0 < ε) |
| (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) : |
| (∫⁻ z : MemoryPrecision.Kernel.Sphere d × Rows d T, |
| K.pathValue (K.outputTest (chordTest ε)) z.1 (records z.1 z.2) |
| ∂(UniformSphere d).prod (RowLaw d T)) ≤ |
| (T+1:ℝ≥0∞)*ENNReal.ofReal |
| ((3*4^d:ℝ)*ε^((d:ℝ)-1-α)*RealStep d (2^M)^BlockCount T (d/8)) := by |
| have : NeZero d := ⟨by omega⟩ |
| rw [lintegral_prod _ (K.experiment_fixed_measurable hK (measurable_chordTest ε)).aemeasurable] |
| apply K.integral_pathValue_bound (K.outputTest (chordTest ε)) Subtype.val hK |
| (K.outputTest_measurable (measurable_chordTest ε)) measurable_subtype_coe (UniformSphere d) |
| intro r hr j |
| have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd |
| have hb := stoch_sphere_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) |
| (K.outputTest_measurable (measurable_chordTest ε)) |
| (K.outputTest_le_one (chordTest_le_one ε)) |
| (H := (3*4^d)*ε^((d:ℝ)-1-α)) (by positivity) hαlo hαhi |
| (K.output_chord_sphere_local (by omega) hε (by linarith) (by linarith)) |
| (BlockCount T (d/8)) r (hr.trans (blockCount_mul_ge T (by omega))) 0 j |
| exact sphere_mass_bound hb |
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| lemma completed_sphere_bound (hK : K.Completed) (hd : 16 ≤ d) {ε α : ℝ} (hε : 0 < ε) |
| (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) : |
| (∫⁻ z : MemoryPrecision.Kernel.Sphere d × Rows d T, |
| K.pathValue (K.outputTest (chordTest ε)) z.1 (records z.1 z.2) |
| ∂(UniformSphere d).prod (RowLaw d T)) ≤ |
| (T+1:ℝ≥0∞)*ENNReal.ofReal |
| ((3*4^d:ℝ)*ε^((d:ℝ)-1-α)*RealStep d (2^M)^BlockCount T (d/8)) := by |
| have : NeZero d := ⟨by omega⟩ |
| have he := K.version_pathValue_ae ((UniformSphere d).prod (RowLaw d T)) hK Prod.fst |
| (fun z : MemoryPrecision.Kernel.Sphere d × Rows d T => records z.1 z.2) |
| (fun i => show Measure.QuasiMeasurePreserving _ _ _ from |
| sphere_sample_observation_quasi (by simpa using (show 3 ≤ d by omega)) i) (chordTest ε) |
| rw [← lintegral_congr_ae he] |
| exact (K.version hK).borel_sphere_bound (K.version_borel hK) hd hε hαlo hαhi |
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| lemma chord_real_bound {Ω : Type u} [MeasurableSpace Ω] (L : Ω → Policy d M T) |
| (ρ : Measure Ω) [IsProbabilityMeasure ρ] (hL : Admissible L ρ (UniformSphere d)) |
| (hd : 16 ≤ d) {ε α : ℝ} (hε : 0 < ε) |
| (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) : |
| ChordSuccess L ρ (UniformSphere d) ε ≤ |
| (T+1:ℝ≥0∞)*ENNReal.ofReal |
| ((3*4^d:ℝ)*ε^((d:ℝ)-1-α)*RealStep d (2^M)^BlockCount T (d/8)) := by |
| have : NeZero d := ⟨by omega⟩ |
| apply seed_average_bound (UniformSphere d) (RowLaw d T) ρ |
| (hL.experiment (chordTest ε) (measurable_chordTest ε) (chordTest_le_one ε)) |
| filter_upwards [hL.completed] with ω hω |
| exact (L ω).completed_sphere_bound hω hd hε hαlo hαhi |
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| lemma zero_fixed_sphere_bound {d M : ℕ} (K : Policy d M 0) (hd : 1 ≤ d) |
| {ε : ℝ} (hε : 0 < ε) : |
| (∫⁻ z : MemoryPrecision.Kernel.Sphere d × Rows d 0, |
| K.pathValue (K.outputTest (chordTest ε)) z.1 (records z.1 z.2) |
| ∂(UniformSphere d).prod (RowLaw d 0)) ≤ ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by |
| have : NeZero d := ⟨by omega⟩ |
| have he (s : MemoryPrecision.Kernel.Sphere d) (obs : Observations d 0) : |
| K.pathValue (K.outputTest (chordTest ε)) s obs = |
| ∑ a : Action M, K.initial a * K.outputTest (chordTest ε) 0 a.2 s := by |
| unfold pathValue |
| simp only [eval, ite_self] |
| simp_rw [he] |
| have hm : Measurable (fun s : MemoryPrecision.Kernel.Sphere d => |
| ∑ a : Action M, K.initial a * K.outputTest (chordTest ε) 0 a.2 s) := |
| Finset.measurable_sum _ (fun a _ => (K.outputTest_measurable (measurable_chordTest ε) 0 a.2).const_mul _) |
| have hm' : Measurable (fun z : MemoryPrecision.Kernel.Sphere d × Rows d 0 => |
| ∑ a : Action M, K.initial a * K.outputTest (chordTest ε) 0 a.2 z.1) := |
| hm.comp measurable_fst |
| rw [lintegral_prod _ hm'.aemeasurable] |
| simp only [lintegral_const, measure_univ, mul_one] |
| rw [lintegral_finsetSum _ (fun a _ => |
| (K.outputTest_measurable (measurable_chordTest ε) 0 a.2).const_mul _)] |
| calc |
| _ = ∑ a : Action M, K.initial a * |
| (∫⁻ s, K.outputTest (chordTest ε) 0 a.2 s ∂UniformSphere d) := by |
| apply Finset.sum_congr rfl |
| intro a _ |
| exact lintegral_const_mul _ (K.outputTest_measurable (measurable_chordTest ε) 0 a.2) |
| _ ≤ ∑ a : Action M, K.initial a * ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := |
| Finset.sum_le_sum (fun a _ => mul_le_mul' le_rfl (K.output_chord_sphere_total hd hε 0 a.2)) |
| _ = _ := by rw [← Finset.sum_mul, K.initial_sum, one_mul] |
|
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| lemma chord_zero_bound {d M : ℕ} {Ω : Type u} [MeasurableSpace Ω] (L : Ω → Policy d M 0) |
| (ρ : Measure Ω) [IsProbabilityMeasure ρ] (hL : Admissible L ρ (UniformSphere d)) |
| (hd : 1 ≤ d) {ε : ℝ} (hε : 0 < ε) : |
| ChordSuccess L ρ (UniformSphere d) ε ≤ ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by |
| have : NeZero d := ⟨by omega⟩ |
| apply seed_average_bound (UniformSphere d) (RowLaw d 0) ρ |
| (hL.experiment (chordTest ε) (measurable_chordTest ε) (chordTest_le_one ε)) |
| exact Filter.Eventually.of_forall (fun ω => (L ω).zero_fixed_sphere_bound hd hε) |
| end Policy |
| end MemoryPrecision.Kernel |
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| end |
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| end OAI |
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