| import OAI.Probability.GaussianRegression.Sphere |
|
|
| namespace OAI |
|
|
| /-! |
| Fubini bounds and rotational lower bounds for sequential estimation. |
| -/ |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| open InnerProductSpace |
| open scoped Pointwise |
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|
| noncomputable section |
|
|
| universe u |
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|
| namespace NoiselessRegression |
|
|
| section FubiniAux |
| variable {X Y Z : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace Z] |
| {μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν] |
| lemma ae_fiber_aemeasurable {f : X × Y → Z} (hf : AEMeasurable f (μ.prod ν)) : |
| ∀ᵐ y ∂ν, AEMeasurable (fun x => f (x,y)) μ := by |
| have he := Measure.ae_ae_of_ae_prod hf.prod_swap.ae_eq_mk |
| filter_upwards [he] with y hy |
| have hm : Measurable (fun x => hf.prod_swap.mk _ (y,x)) := |
| hf.prod_swap.measurable_mk.comp (measurable_const.prodMk measurable_id) |
| exact hm.aemeasurable.congr (Filter.EventuallyEq.symm hy) |
|
|
| lemma prod_null_set_bound {S : Set (X × Y)} (hS : NullMeasurableSet S (μ.prod ν)) |
| {c : ℝ≥0∞} (h : ∀ᵐ y ∂ν, μ {x | (x,y) ∈ S} ≤ c) : |
| (μ.prod ν) S ≤ c * ν univ := by |
| have hm : AEMeasurable (S.indicator (fun _ => (1 : ℝ≥0∞))) (μ.prod ν) := |
| aemeasurable_const.indicator₀ hS |
| rw [← one_mul ((μ.prod ν) S), ← lintegral_indicator_const₀ hS 1, lintegral_prod_symm _ hm] |
| apply (lintegral_mono_ae ?_).trans_eq (lintegral_const c) |
| filter_upwards [h, ae_fiber_aemeasurable hm] with y hy hmy |
| have hs : NullMeasurableSet {x | (x,y) ∈ S} μ := by |
| have heq : {x | (x,y) ∈ S} = {x | S.indicator (fun _ => (1 : ℝ≥0∞)) (x,y) = 1} := by |
| ext x |
| by_cases hx : (x,y) ∈ S <;> simp [hx] |
| rw [heq] |
| exact nullMeasurableSet_eq_fun hmy aemeasurable_const |
| change (∫⁻ x, ({x | (x,y) ∈ S}).indicator (fun _ => (1 : ℝ≥0∞)) x ∂μ) ≤ c |
| simpa [lintegral_indicator_const₀ hs] using hy |
| end FubiniAux |
|
|
| instance sampleLaw_probability (d T : ℕ) : IsProbabilityMeasure (sampleLaw d T) := by |
| unfold sampleLaw |
| infer_instance |
| instance uniformSphere_probability (d : ℕ) [NeZero d] : IsProbabilityMeasure (uniformSphere d) := by |
| change IsProbabilityMeasure (sphereLaw (E := Vector d)) |
| infer_instance |
|
|
| namespace Learner |
| lemma run_congr_labels {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (s t : UnitSphere d) (ω : Ω) (x : Samples d T) |
| (h : ∀ i, ⟪x i,s.val⟫ = ⟪x i,t.val⟫) (n : ℕ) : |
| L.run s ω x n = L.run t ω x n := by |
| induction n with |
| | zero => rfl |
| | succ n ih => |
| simp only [run, ih] |
| split |
| · rfl |
| · split |
| · rw [h] |
| · rfl |
|
|
| lemma estimate_congr_labels {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (s t : UnitSphere d) (ω : Ω) (x : Samples d T) |
| (h : ∀ i, ⟪x i,s.val⟫ = ⟪x i,t.val⟫) : L.estimate s ω x = L.estimate t ω x := by |
| simp only [estimate, terminal, run_congr_labels L s t ω x h] |
|
|
| lemma angularSuccess_nullMeasurable {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) (hL : L.Admissible ρ) : |
| NullMeasurableSet {z : UnitSphere d × (Ω × Samples d T) | |
| angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} |
| ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := by |
| apply nullMeasurableSet_le _ aemeasurable_const |
| exact Real.continuous_arccos.measurable.comp_aemeasurable |
| ((measurable_subtype_coe.comp_aemeasurable hL.uniform_experiment).inner |
| (measurable_subtype_coe.comp measurable_fst).aemeasurable) |
|
|
| lemma linear_success_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (hd : 0 < d) (hT : 4*T ≤ d) {ε : ℝ} (hε : ε ≤ 1/10) (hL : L.Admissible ρ) : |
| L.uniformSuccess ρ ε ≤ ENNReal.ofReal (21/32 : ℝ) := by |
| classical |
| let : NeZero d := ⟨hd.ne'⟩ |
| have hmeas := ae_fiber_aemeasurable hL.uniform_experiment |
| unfold uniformSuccess |
| refine (prod_null_set_bound (L.angularSuccess_nullMeasurable ρ ε hL) |
| (c := ENNReal.ofReal (21/32 : ℝ)) ?_).trans_eq (by simp) |
| filter_upwards [hmeas] with z hz |
| let U : Submodule ℝ (Vector d) := Submodule.span ℝ (Set.range z.2) |
| have hdim : Module.finrank ℝ U ≤ T := by |
| have h := finrank_span_le_card (R := ℝ) (Set.range z.2) |
| apply h.trans |
| rw [Set.toFinset_range] |
| exact (Finset.card_image_le).trans_eq (by simp) |
| have hU : 4 * Module.finrank ℝ U ≤ Module.finrank ℝ (Vector d) := by |
| simpa only [Vector, finrank_euclideanSpace_fin] using (Nat.mul_le_mul_left 4 hdim).trans hT |
| change sphereLaw {s : UnitSphere d | angularError s (L.estimate s z.1 z.2) ≤ ε} ≤ _ |
| apply sphere_reflection_success_bound U hU |
| · exact nullMeasurableSet_le |
| (Real.continuous_arccos.measurable.comp_aemeasurable |
| ((measurable_subtype_coe.comp_aemeasurable hz).inner measurable_subtype_coe.aemeasurable)) |
| aemeasurable_const |
| · intro s hs ht |
| let t := sphereIsometry U.reflection s |
| have he : L.estimate s z.1 z.2 = L.estimate t z.1 z.2 := by |
| apply L.estimate_congr_labels |
| intro i |
| have hi : z.2 i ∈ U := Submodule.subset_span (Set.mem_range_self i) |
| have he := U.reflection.inner_map_map (z.2 i) s.val |
| rw [(Submodule.reflection_eq_self_iff (z.2 i)).2 hi] at he |
| exact he.symm |
| have hs' : dist s.val (L.estimate s z.1 z.2).val ≤ ε := by |
| apply (chord_le_arccos s (L.estimate s z.1 z.2)).trans |
| simpa only [Set.mem_ofPred_eq, angularError, real_inner_comm] using hs |
| have ht' : dist t.val (L.estimate t z.1 z.2).val ≤ ε := by |
| apply (chord_le_arccos t (L.estimate t z.1 z.2)).trans |
| simpa only [Set.mem_ofPred_eq, angularError, real_inner_comm] using ht |
| calc |
| _ = dist s.val t.val := rfl |
| _ ≤ dist s.val (L.estimate s z.1 z.2).val + dist (L.estimate s z.1 z.2).val t.val := dist_triangle _ _ _ |
| _ ≤ ε + ε := by rw [he, dist_comm (L.estimate t z.1 z.2).val t.val] at *; exact add_le_add hs' ht' |
| _ ≤ 1/5 := by linarith |
| end Learner |
| |
| end NoiselessRegression |
| |
| end |
| |
| end OAI |
| |
