| import OAI.Probability.PosteriorReplicas.Learners.CompletedKernelLaw | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory Set Filter Function | |
| open scoped ENNReal NNReal RealInnerProductSpace unitInterval | |
| noncomputable section | |
| namespace PosteriorRegression | |
| namespace CompletedKernelLearner | |
| open Learner | |
| def uniformSuccess {d M : ℕ} (K : CompletedKernelLearner d M) (T : ℕ) (ε : ℝ) : ℝ≥0∞ := | |
| ∫⁻ z : UnitSphere d × Samples d T, | |
| K.estimateLaw z.1 z.2 {ŝ | angularError z.1 ŝ ≤ ε} | |
| ∂((uniformSphere d).prod (sampleLaw d T)) | |
| lemma angularBall_measurable {d : ℕ} (s : UnitSphere d) (ε : ℝ) : | |
| MeasurableSet {ŝ : UnitSphere d | angularError s ŝ ≤ ε} := by | |
| exact measurableSet_le (Real.continuous_arccos.measurable.comp | |
| (measurable_subtype_coe.inner measurable_const)) measurable_const | |
| lemma Realization.uniformSuccess_eq {d M T : ℕ} (hd : 3≤d) {K : CompletedKernelLearner d M} | |
| (R : Realization K) (ε : ℝ) : | |
| (R.toLearner T).uniformSuccess (coinsLaw T) ε = K.uniformSuccess T ε := by | |
| have : NeZero d := ⟨by omega⟩ | |
| let S : Set (UnitSphere d × (Coins T × Samples d T)) := | |
| {z | angularError z.1 ((R.toLearner T).estimate z.1 z.2.1 z.2.2) ≤ ε} | |
| have hS : NullMeasurableSet S ((uniformSphere d).prod ((coinsLaw T).prod (sampleLaw d T))) := | |
| (R.uniformAdmissible hd).angularSuccess_nullMeasurable ε | |
| let e : ((UnitSphere d × Samples d T) × Coins T) ≃ᵐ | |
| UnitSphere d × (Coins T × Samples d T) := | |
| MeasurableEquiv.prodAssoc.trans ((MeasurableEquiv.refl _).prodCongr MeasurableEquiv.prodComm) | |
| have he : MeasurePreserving e | |
| (((uniformSphere d).prod (sampleLaw d T)).prod (coinsLaw T)) | |
| ((uniformSphere d).prod ((coinsLaw T).prod (sampleLaw d T))) := | |
| ((MeasurePreserving.id (uniformSphere d)).prod | |
| (Measure.measurePreserving_swap (μ := sampleLaw d T) (ν := coinsLaw T))).comp | |
| (measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d T) (coinsLaw T)) | |
| have hQ := hS.preimage he.quasiMeasurePreserving | |
| have hI := (aemeasurable_const : AEMeasurable (fun _ : (UnitSphere d × Samples d T) × Coins T => (1 : ℝ≥0∞)) | |
| (((uniformSphere d).prod (sampleLaw d T)).prod (coinsLaw T))).indicator₀ hQ | |
| change ((uniformSphere d).prod ((coinsLaw T).prod (sampleLaw d T))) S = _ | |
| rw [← he.measure_preimage hS] | |
| rw [← one_mul ((((uniformSphere d).prod (sampleLaw d T)).prod (coinsLaw T)) (e ⁻¹' S)), | |
| ← lintegral_indicator_const₀ hQ 1, lintegral_prod _ hI] | |
| unfold uniformSuccess | |
| apply lintegral_congr | |
| intro z | |
| have hball := angularBall_measurable z.1 ε | |
| rw [← R.estimate_law z.1 z.2, Measure.map_apply (R.measurable_estimate z.1 z.2) hball] | |
| have hm : MeasurableSet {ω : Coins T | angularError z.1 ((R.toLearner T).estimate z.1 ω z.2) ≤ ε} := | |
| (R.measurable_estimate z.1 z.2) hball | |
| change (∫⁻ ω, ({ω : Coins T | angularError z.1 ((R.toLearner T).estimate z.1 ω z.2) ≤ ε}).indicator | |
| (fun _ => (1 : ℝ≥0∞)) ω ∂coinsLaw T) = _ | |
| rw [lintegral_indicator_const hm] | |
| simp only [one_mul] | |
| rfl | |
| lemma uniformSuccess_le_one {d M : ℕ} [NeZero d] (K : CompletedKernelLearner d M) (T : ℕ) (ε : ℝ) : | |
| K.uniformSuccess T ε ≤ 1 := by | |
| apply (lintegral_mono (fun z => prob_le_one)).trans | |
| simp | |
| lemma fixed_quadratic_memory {d M T : ℕ} {A : ℝ} (hA : 0<A) (hd : 32≤d) | |
| (hM : (M : ℝ)≤A*(d : ℝ)^2) {ε : ℝ} (hε : 0<ε) (hε10 : ε≤1/10) | |
| (K : CompletedKernelLearner d M) (hs : (2/3 : ℝ≥0∞) ≤ K.uniformSuccess T ε) : | |
| (1/(4*(400000+100*A))) * (d : ℝ) * Real.log (1/ε) ≤ (T : ℝ) := by | |
| have he := K.realization.uniformSuccess_eq (T := T) (by omega) ε | |
| exact source_fixed_quadratic_memory hA hd hM hε hε10 (coinsLaw T) | |
| (K.realization.toLearner T) (K.realization.uniformAdmissible (by omega)) (by simpa only [he] using hs) | |
| end CompletedKernelLearner | |
| end PosteriorRegression | |
| end | |
| end OAI | |