| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.KernelCaps |
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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 NoiselessRegression |
| namespace Policy |
| variable {d M T : ℕ} (K : Policy d M T) |
|
|
| lemma eval_pair_le_one {X : Type*} (w v : ℕ → MemoryPrecision.Kernel.State M → X → ℝ≥0∞) |
| (s t : X) (hpair : ∀ n j, w n j s + v n j t ≤ 1) |
| (n k : ℕ) (j : MemoryPrecision.Kernel.State M) (obs : Observations d k) : |
| K.eval w n k j s obs + K.eval v n k j t obs ≤ 1 := by |
| induction k generalizing n j with |
| | zero => exact hpair n j |
| | succ k ih => |
| simp only [eval, ← Finset.sum_add_distrib, ← mul_add] |
| calc |
| _ ≤ ∑ a : Action M, K.transition (n+1) j (obs 0) a * 1 := by |
| apply Finset.sum_le_sum |
| intro a _ |
| apply mul_le_mul' le_rfl |
| split_ifs |
| · exact hpair _ _ |
| · exact ih _ _ _ |
| _ = 1 := by simpa only [mul_one] using K.transition_sum (n+1) j (obs 0) |
|
|
| lemma pathValue_pair_le_one {X : Type*} (w v : ℕ → MemoryPrecision.Kernel.State M → X → ℝ≥0∞) |
| (s t : X) (hpair : ∀ n j, w n j s + v n j t ≤ 1) (obs : Observations d T) : |
| K.pathValue w s obs + K.pathValue v t obs ≤ 1 := by |
| simp only [pathValue, ← Finset.sum_add_distrib, ← mul_add] |
| calc |
| _ ≤ ∑ a : Action M, K.initial a * 1 := by |
| apply Finset.sum_le_sum |
| intro a _ |
| apply mul_le_mul' le_rfl |
| split_ifs |
| · exact hpair _ _ |
| · exact K.eval_pair_le_one w v s t hpair _ _ _ _ |
| _ = 1 := by simpa only [mul_one] using K.initial_sum |
|
|
| lemma chord_pair_le_one {ε : ℝ} (hε : ε ≤ 1/10) |
| (s t v : MemoryPrecision.Kernel.Sphere d) (hst : 1/5 < dist s.val t.val) : |
| chordTest ε (s,v) + chordTest ε (t,v) ≤ 1 := by |
| classical |
| have hn : ¬(dist v.val s.val ≤ ε ∧ dist v.val t.val ≤ ε) := by |
| rintro ⟨hs,ht⟩ |
| have htri := dist_triangle s.val v.val t.val |
| rw [dist_comm s.val v.val] at htri |
| linarith |
| simp only [chordTest, Set.indicator_apply, Set.mem_ofPred_eq] |
| split_ifs <;> simp_all |
|
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| lemma output_chord_pair_le_one {ε : ℝ} (hε : ε ≤ 1/10) |
| (s t : MemoryPrecision.Kernel.Sphere d) (hst : 1/5 < dist s.val t.val) |
| (n : ℕ) (j : MemoryPrecision.Kernel.State M) : |
| K.outputTest (chordTest ε) n j s + K.outputTest (chordTest ε) n j t ≤ 1 := by |
| have hm : Measurable (fun v : MemoryPrecision.Kernel.Sphere d => chordTest ε (s,v)) := |
| (measurable_chordTest ε).comp (measurable_const.prodMk measurable_id) |
| simp only [outputTest] |
| rw [← lintegral_add_left hm] |
| exact (lintegral_mono (fun v => chord_pair_le_one hε s t v hst)).trans_eq (by simp) |
|
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| lemma path_reflection_pair {ε : ℝ} (hε : ε ≤ 1/10) |
| (A : Rows d T) (s : MemoryPrecision.Kernel.Sphere d) |
| (U : Submodule ℝ (MemoryPrecision.Kernel.Vector d)) (hA : ∀ i, A i ∈ U) |
| (hst : 1/5 < dist s.val (U.reflection s.val)) : |
| K.pathValue (K.outputTest (chordTest ε)) s (records s A) + |
| K.pathValue (K.outputTest (chordTest ε)) (sphereIsometry U.reflection s) |
| (records (sphereIsometry U.reflection s) A) ≤ 1 := by |
| have he : records (sphereIsometry U.reflection s) A = records s A := by |
| funext i |
| change (A i, ⟪U.reflection s.val, A i⟫) = (A i, ⟪s.val, A i⟫) |
| apply Prod.ext |
| · rfl |
| · have h := U.reflection.inner_map_map s.val (A i) |
| rw [(Submodule.reflection_eq_self_iff (A i)).2 (hA i)] at h |
| exact h |
| rw [he] |
| have hpair : ∀ n j, K.outputTest (chordTest ε) n j s + |
| K.outputTest (chordTest ε) n j (sphereIsometry U.reflection s) ≤ 1 := |
| fun n j => K.output_chord_pair_le_one hε s (sphereIsometry U.reflection s) hst n j |
| exact K.pathValue_pair_le_one (K.outputTest (chordTest ε)) |
| (K.outputTest (chordTest ε)) s (sphereIsometry U.reflection s) hpair (records s A) |
| end Policy |
|
|
| lemma sphere_reflection_function_bound {d : ℕ} [NeZero d] |
| (U : Submodule ℝ (MemoryPrecision.Kernel.Vector d)) (hU : 4*Module.finrank ℝ U ≤ d) |
| {f : MemoryPrecision.Kernel.Sphere d → ℝ≥0∞} (hf : AEMeasurable f (UniformSphere d)) |
| (hf1 : ∀ s, f s ≤ 1) |
| (hpair : ∀ s, 1/5 < dist s.val (U.reflection s.val) → |
| f s + f (sphereIsometry U.reflection s) ≤ 1) : |
| (∫⁻ s, f s ∂UniformSphere d) ≤ ENNReal.ofReal (21/32:ℝ) := by |
| let R := sphereIsometry U.reflection |
| let B : Set (MemoryPrecision.Kernel.Sphere d) := {s | (4/5:ℝ) ≤ ‖U.orthogonalProjectionOnto s.val‖^2} |
| have hB : MeasurableSet B := by dsimp [B]; exact measurableSet_le measurable_const (by fun_prop) |
| have hp : MeasurePreserving R (UniformSphere d) (UniformSphere d) := sphereIsometry_preserving U.reflection |
| have he : (∫⁻ s, f (R s) ∂UniformSphere d) = ∫⁻ s, f s ∂UniformSphere d := by |
| calc |
| _ = ∫⁻ s, f s ∂(UniformSphere d).map R := |
| (lintegral_map' (by simpa only [hp.map_eq] using hf) hp.measurable.aemeasurable).symm |
| _ = _ := by rw [hp.map_eq] |
| have hb : UniformSphere d B ≤ ENNReal.ofReal (5/16:ℝ) := |
| sphere_projection_large U (by simpa using hU) |
| have hsum : ∀ s, f s + f (R s) ≤ 1 + B.indicator (fun _ => 1) s := by |
| intro s |
| classical |
| by_cases hs : s ∈ B |
| · simpa [hs] using add_le_add (hf1 s) (hf1 (R s)) |
| · have hst : 1/5 < dist s.val (U.reflection s.val) := by |
| by_contra hn |
| have hd : dist s.val (U.reflection s.val) ≤ 1/5 := le_of_not_gt hn |
| rw [reflection_dist] at hd |
| have hn := U.norm_sq_eq_add_norm_sq_starProjection s.val |
| rw [mem_sphere_zero_iff_norm.mp s.property] at hn |
| have he : ‖U.starProjection s.val‖ = ‖U.orthogonalProjectionOnto s.val‖ := rfl |
| rw [he] at hn |
| apply hs |
| change (4/5:ℝ) ≤ ‖U.orthogonalProjectionOnto s.val‖^2 |
| nlinarith [norm_nonneg (Uᗮ.starProjection s.val)] |
| simpa [hs] using hpair s hst |
| have hi : (∫⁻ s, f s ∂UniformSphere d) + (∫⁻ s, f s ∂UniformSphere d) ≤ |
| 1 + ENNReal.ofReal (5/16:ℝ) := by |
| calc |
| _ = (∫⁻ s, f s ∂UniformSphere d) + (∫⁻ s, f (R s) ∂UniformSphere d) := by rw [he] |
| _ = ∫⁻ s, f s + f (R s) ∂UniformSphere d := (lintegral_add_left' hf _).symm |
| _ ≤ ∫⁻ s, 1 + B.indicator (fun _ => (1:ℝ≥0∞)) s ∂UniformSphere d := lintegral_mono hsum |
| _ = 1 + UniformSphere d B := by rw [lintegral_add_left measurable_const]; simp [lintegral_indicator_const hB] |
| _ ≤ _ := add_le_add le_rfl hb |
| calc |
| _ = ENNReal.ofReal (1/2:ℝ) * ((∫⁻ s, f s ∂UniformSphere d) + |
| (∫⁻ s, f s ∂UniformSphere d)) := by |
| rw [← two_mul, ← mul_assoc] |
| have hc : ENNReal.ofReal (1/2:ℝ) * 2 = 1 := by |
| rw [← ENNReal.ofReal_ofNat, ← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 1/2)] |
| norm_num |
| rw [hc, one_mul] |
| _ ≤ ENNReal.ofReal (1/2:ℝ) * (1 + ENNReal.ofReal (5/16:ℝ)) := mul_le_mul' le_rfl hi |
| _ = ENNReal.ofReal (21/32:ℝ) := by |
| rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_add (by norm_num : (0:ℝ) ≤ 1) |
| (by norm_num : (0:ℝ) ≤ 5/16), |
| ← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 1/2)] |
| norm_num |
|
|
| namespace Policy |
| lemma source_linear_success_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Ω → Policy d M T) (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (hd : 0 < d) (hT : 4*T ≤ d) {ε : ℝ} (hε : ε ≤ 1/10) |
| (hL : Admissible L ρ (UniformSphere d)) : |
| Success L ρ (UniformSphere d) ε ≤ ENNReal.ofReal (21/32:ℝ) := by |
| let : NeZero d := ⟨hd.ne'⟩ |
| apply (success_le_chord L ρ (UniformSphere d) ε).trans |
| have hm := hL.experiment (chordTest ε) (measurable_chordTest ε) (chordTest_le_one ε) |
| unfold ChordSuccess |
| rw [lintegral_prod_symm _ hm] |
| apply le_trans (b := ∫⁻ _z : Ω × Rows d T, ENNReal.ofReal (21/32:ℝ) |
| ∂ρ.prod (RowLaw d T)) ?_ (by simp) |
| apply lintegral_mono_ae |
| filter_upwards [ae_fiber_aemeasurable hm] with z hz |
| let U : Submodule ℝ (MemoryPrecision.Kernel.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) |
| apply sphere_reflection_function_bound U ((Nat.mul_le_mul_left 4 hdim).trans hT) hz |
| · intro s |
| exact (L z.1).pathValue_le_one _ ((L z.1).outputTest_le_one (chordTest_le_one ε)) _ _ |
| · intro s hs |
| exact (L z.1).path_reflection_pair hε z.2 s U |
| (fun i => Submodule.subset_span (Set.mem_range_self i)) hs |
| end Policy |
| end MemoryPrecision.Kernel |
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| end |
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| end OAI |
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