| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.KernelSphere |
|
|
| 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 |
| open NoiselessRegression |
|
|
| def uniformChordSuccess {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) : ℝ≥0∞ := |
| ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) |
| {z | dist (L.estimate z.1 z.2.1 z.2.2).val z.1.val ≤ ε} |
| end MemoryPrecision |
| namespace MemoryPrecision |
| open NoiselessRegression |
| lemma real_route_cap_bound {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N) |
| {δ : ℕ → Fin N → (Space d × ℝ) → Fin N} (hδ : ∀ n j, Measurable (δ n j)) |
| (y : Fin N → Space d) {ε α : ℝ} (hε : 0 < ε) |
| (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) (k : ℕ) (j : Fin N) : |
| ((SpherePrior d).prod (Measure.pi (fun _ : Fin k => stdGaussian (Space d)))) |
| {z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} ≤ |
| ENNReal.ofReal ((3*4^d:ℝ)*ε^((d:ℝ)-1-α) * RealStep d N ^ BlockCount k (d/8)) := by |
| let : NeZero d := ⟨by omega⟩ |
| let : NeZero N := ⟨by omega⟩ |
| let : IsProbabilityMeasure (SpherePrior d) := |
| (show IsProbabilityMeasure (sphereLaw (E := Space d)) from inferInstance) |
| let w : ℕ → Fin N → Space d → ℝ≥0∞ := fun _ j => capWeight (y j) ε |
| have he (s : Space d) : continuation δ w 0 k j s = |
| ∫⁻ A : Fin k → Space d, capWeight (y (streamRoute δ 0 k j (A,fun i => ⟪s,A i⟫))) ε s |
| ∂Measure.pi (fun _ => stdGaussian (Space d)) := by |
| simpa [w, blockValue, continuation] using continuation_unroll (w := w) hδ 0 k 0 j s |
| let S : Set (Sphere d × (Fin k → Space d)) := |
| {z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} |
| have hS : MeasurableSet S := by |
| apply measurableSet_le _ measurable_const |
| exact measurable_subtype_coe.comp measurable_fst |>.dist |
| ((measurable_of_finite y).comp ((measurable_streamRoute hδ 0 k j).comp (by fun_prop))) |
| change ((SpherePrior d).prod _) S ≤ _ |
| rw [← one_mul (((SpherePrior d).prod _) S), ← lintegral_indicator_const hS 1, |
| lintegral_prod _ (measurable_const.indicator hS).aemeasurable] |
| have hf : ∀ s : Sphere d, |
| (∫⁻ A, S.indicator (fun _ => (1 : ℝ≥0∞)) (s,A) ∂Measure.pi (fun _ => stdGaussian (Space d))) = |
| continuation δ w 0 k j s.val := by |
| intro s |
| rw [he] |
| apply lintegral_congr |
| intro A |
| classical |
| change (if dist s.val (y (streamRoute δ 0 k j (A,fun i => ⟪s.val,A i⟫))) ≤ ε then 1 else 0) = _ |
| unfold capWeight |
| rw [Set.indicator_apply] |
| rfl |
| simp_rw [hf] |
| exact real_continuation_cap_integral hd hN hδ (fun _ => y) hε hαlo hαhi 0 k j |
|
|
| lemma real_prefix_route_cap_bound {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N) |
| {δ : ℕ → Fin N → (Space d × ℝ) → Fin N} (hδ : ∀ n j, Measurable (δ n j)) |
| (y : Fin N → Space d) {ε α : ℝ} (hε : 0 < ε) |
| (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) {k T : ℕ} (hk : k ≤ T) (j : Fin N) : |
| ((SpherePrior d).prod (Measure.pi (fun _ : Fin T => stdGaussian (Space d)))) |
| {z | dist z.1.val (y (streamRoute δ 0 k j |
| ((fun i => z.2 (Fin.castLE hk i)),fun i => ⟪z.1.val,z.2 (Fin.castLE hk i)⟫))) ≤ ε} ≤ |
| ENNReal.ofReal ((3*4^d:ℝ)*ε^((d:ℝ)-1-α) * RealStep d N ^ BlockCount T (d/8)) := by |
| let : NeZero d := ⟨by omega⟩ |
| let : NeZero N := ⟨by omega⟩ |
| let : IsProbabilityMeasure (SpherePrior d) := |
| (show IsProbabilityMeasure (sphereLaw (E := Space d)) from inferInstance) |
| have hp := (MeasurePreserving.id (SpherePrior d)).prod |
| (measurePreserving_pi_prefix (stdGaussian (Space d)) hk) |
| have hs : MeasurableSet |
| {z : Sphere d × (Fin k → Space d) | |
| dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} := by |
| apply measurableSet_le _ measurable_const |
| exact (measurable_subtype_coe.comp measurable_fst).dist |
| ((measurable_of_finite y).comp ((measurable_streamRoute hδ 0 k j).comp (by fun_prop))) |
| apply (le_of_eq (hp.measure_preimage hs.nullMeasurableSet)).trans |
| apply (real_route_cap_bound hd hN hδ y hε hαlo hαhi k j).trans |
| exact ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left |
| (pow_le_pow_right₀ (realStep_one d hN) (blockCount_mono hk)) (by positivity)) |
|
|
| open NoiselessRegression.Learner |
|
|
| lemma frozen_chord_real_bound {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) |
| (ω : Ω) (hd : 16 ≤ d) |
| (htrans : ∀ t j, AEMeasurable (fun obs => (L.transition t ω j obs).2) (observationReference d)) |
| {ε : ℝ} (hε : 0 < ε) {α : ℝ} (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) : |
| ((uniformSphere d).prod (sampleLaw d T)) |
| {z | dist z.1.val (L.estimate z.1 ω z.2).val ≤ 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 |
| classical |
| let : NeZero d := ⟨by omega⟩ |
| let δ : ℕ → State M → (Vector d × ℝ) → State M := fun t j => (htrans t j).mk _ |
| have hδ (t : ℕ) (j : State M) : Measurable (δ t j) := (htrans t j).measurable_mk |
| let μ := (uniformSphere d).prod (sampleLaw d T) |
| have hgood : ∀ᵐ z ∂μ, ∀ t j (i : Fin T), |
| (L.transition t ω j (z.2 i,⟪z.1.val,z.2 i⟫)).2 = δ t j (z.2 i,⟪z.1.val,z.2 i⟫) := by |
| simp only [ae_all_iff] |
| intro t j i |
| exact (sphere_sample_observation_quasi (by simpa using (show 3 ≤ d by omega)) i).ae (htrans t j).ae_eq_mk |
| let S : Fin (T+1) → Set (UnitSphere d × Samples d T) := fun t => |
| {z | dist z.1.val (L.output t ω (streamRoute δ 0 t (L.initialChoice ω).2 |
| ((fun i => z.2 (Fin.castLE (by omega) i)),fun i => ⟪z.1.val,z.2 (Fin.castLE (by omega) i)⟫))).val ≤ ε} |
| have hmono : {z : UnitSphere d × Samples d T | dist z.1.val (L.estimate z.1 ω z.2).val ≤ ε} |
| ≤ᵐ[μ] ⋃ t, S t := by |
| filter_upwards [hgood] with z hz |
| intro hs |
| rcases L.terminal_forced_witness z.1 ω z.2 with ⟨t,ht,he⟩ |
| have hf : L.forcedRun z.1 ω z.2 t = streamRoute δ 0 t (L.initialChoice ω).2 |
| ((fun i => z.2 (Fin.castLE ht i)),fun i => ⟪z.1.val,z.2 (Fin.castLE ht i)⟫) := by |
| rw [L.forcedRun_eq_route z.1 ω z.2 ht] |
| exact streamRoute_congr _ _ 0 t _ _ _ (fun n j i => hz n j (Fin.castLE ht i)) |
| refine Set.mem_iUnion.mpr ⟨⟨t,by omega⟩, ?_⟩ |
| change dist z.1.val (L.output t ω _) ≤ ε |
| change dist z.1.val (L.estimate z.1 ω z.2).val ≤ ε at hs |
| simpa only [estimate, he, hf] using hs |
| let B : ℝ≥0∞ := ENNReal.ofReal ((3*4^d:ℝ)*ε^((d:ℝ)-1-α) * RealStep d (2^M) ^ BlockCount T (d/8)) |
| have hb (t : Fin (T+1)) : μ (S t) ≤ B := by |
| simpa only [B, μ, S, uniformSphere, SpherePrior, MemoryPrecision.Kernel.UniformSphere, |
| sampleLaw] using |
| real_prefix_route_cap_bound (N := 2^M) hd (one_le_pow₀ (by norm_num)) hδ (fun j => (L.output t ω j).val) hε |
| hαlo hαhi (by omega : (t : ℕ) ≤ T) (L.initialChoice ω).2 |
| calc |
| _ ≤ μ (⋃ t, S t) := measure_mono_ae hmono |
| _ ≤ ∑' t, μ (S t) := measure_iUnion_le S |
| _ ≤ ∑' _ : Fin (T+1), B := ENNReal.tsum_le_tsum hb |
| _ = _ := by simp [B, tsum_fintype, nsmul_eq_mul, mul_assoc] |
|
|
| lemma chord_real_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ] |
| (hL : L.Admissible ρ) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) |
| {α : ℝ} (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) : |
| uniformChordSuccess L ρ 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 |
| let : NeZero d := ⟨by omega⟩ |
| let : SigmaFinite (observationReference d) := by unfold observationReference; infer_instance |
| have hω : ∀ᵐ ω ∂ρ, ∀ t j, |
| AEMeasurable (fun obs => (L.transition t ω j obs).2) (observationReference d) := by |
| simp only [ae_all_iff] |
| intro t j |
| filter_upwards [ae_fiber_aemeasurable (hL.transition t j).prod_swap] with ω hω |
| exact measurable_snd.comp_aemeasurable hω |
| have hp := ((MeasurePreserving.id (uniformSphere d)).prod |
| (Measure.measurePreserving_swap (μ := sampleLaw d T) (ν := ρ))).comp |
| (measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d T) ρ) |
| let S := {z : UnitSphere d × (Ω × Samples d T) | |
| dist (L.estimate z.1 z.2.1 z.2.2).val z.1.val ≤ ε} |
| have hS : NullMeasurableSet S ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := |
| nullMeasurableSet_le |
| ((measurable_subtype_coe.comp_aemeasurable hL.uniform_experiment).dist |
| (measurable_subtype_coe.comp measurable_fst).aemeasurable) aemeasurable_const |
| have he := hp.measure_preimage hS |
| have hb := prod_null_set_bound (hS.preimage hp.quasiMeasurePreserving) |
| (c := (T+1 : ℝ≥0∞) * ENNReal.ofReal ((3*4^d:ℝ)*ε^((d:ℝ)-1-α) * RealStep d (2^M) ^ BlockCount T (d/8))) |
| (by |
| filter_upwards [hω] with ω hω |
| change ((uniformSphere d).prod (sampleLaw d T)) |
| {z | dist (L.estimate z.1 ω z.2).val z.1.val ≤ ε} ≤ _ |
| simpa only [dist_comm] using frozen_chord_real_bound L ω hd hω hε hαlo hαhi) |
| simpa only [he, measure_univ, mul_one, S, uniformChordSuccess] using hb |
|
|
| end MemoryPrecision |
|
|
|
|
| end |
|
|
| end OAI |
|
|
